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1. The image is 10 cm behind the mirror and you are 30 cm in front of the mirror. You

must focus your eyes for a distance of 10 cm + 30 cm = 40 cm.

2. The bird is a distance *d*2 in front of the mirror; the plane of its image is that same

distance *d*2 behind the mirror. The lateral distance between you and the bird is *d*3 = 5.00

m. We denote the distance from the camera to the mirror as *d*1, and we construct a right

triangle out of *d*3 and the distance between the camera and the image plane (*d*1 + *d*2).

Thus, the focus distance is

( ) ( ) ( )2 2 221 2 3 4.30m+3.30m 5.00m 9.10m.*d d d d*= + + = + =

3. The intensity of light from a point source varies as the inverse of the square of the

distance from the source. Before the mirror is in place, the intensity at the center of the

screen is given by *IP* = *A*/*d
*2
, where *A* is a constant of proportionality. After the mirror is

in place, the light that goes directly to the screen contributes intensity *IP*, as before.

Reflected light also reaches the screen. This light appears to come from the image of the

source, a distance *d* behind the mirror and a distance 3*d* from the screen. Its contribution

to the intensity at the center of the screen is

( )2 2 .

9 93

*P
*

*r
*

*IA A
I
*

*dd
*

= = =

The total intensity at the center of the screen is

10 .

9 9

*P
*

*P r P P
*

*I
I I I I I*= + = + =

The ratio of the new intensity to the original intensity is *I*/*IP* = 10/9 = 1.11.

4. When *S* is barely able to see *B* the light rays from *B* must reflect to *S* off the edge of the

mirror. The angle of reflection in this case is 45°, since a line drawn from *S* to the

mirror’s edge makes a 45° angle relative to the wall. By the law of reflection, we find

*x
*

*d
x
*

*d
*

/ tan

. .

2 45

2

3 0 15= ° = = =m

2 m.

5. We apply the law of refraction, assuming all angles are in radians:

sin

sin ,

θ θ ′

= *n
n
*

*w
*

air

which in our case reduces to θ*'* ≈ θ/*nw* (since both θ and θ *'* are small, and *n*air ≈ 1). We
refer to our figure below.

The object *O* is a vertical distance *d*1 above the water, and the water surface is a vertical

distance *d*2 above the mirror. We are looking for a distance *d* (treated as a positive

number) below the mirror where the image *I* of the object is formed. In the triangle *O AB*

1 1| | tan ,*AB d d*θ θ= ≈

and in the triangle *CBD*

2 2 2

2 | | 2 tan 2 .

*w
*

*d
BC d d
*

*n
*

θθ θ′ ′= ≈ ≈

Finally, in the triangle *ACI*, we have |*AI*| = *d + d*2. Therefore,

( )

1 2 2 2 2 2 2 1 2

2 2| | | | | | 1 | |

tan

2 200cm 250cm 200cm 351cm.

1.33

*w w
*

*d d dAC AB BC
d AI d d d d d d
*

*n n
*

θ θ θ θ θ

+= − = − ≈ − = + − = + −

= + − =

6. We note from Fig. 34-32 that *m* =
1

2
when *p *= 5 cm. Thus Eq. 34-7 (the magnification

equation) gives us *i* = −10 cm in that case. Then, by Eq. 34-9 (which applies to mirrors
and thin-lenses) we find the focal length of the mirror is *f* = 10 cm. Next, the problem

asks us to consider *p *= 14 cm. With the focal length value already determined, then Eq.

34-9 yields *i* = 35 cm for this new value of object distance. Then, using Eq. 34-7 again,

we find *m = i/p *= −2.5.

**7**. We use Eqs. 34-3 and 34-4, and note that *m = –i*/*p*. Thus,

1 1 1 2

*p pm f r
*− = = .

We solve for *p*:

*p
r
*

*m
*= −FHG

I KJ = −

F HG

I KJ =2 1

1 35 0 1

1

2 50 10 5

.

. .

cm

2 cm.

8. The graph in Fig. 34-33 implies that *f* = 20 cm, which we can plug into Eq. 34-9 (with

*p *= 70 cm) to obtain *i* = +28 cm.

9. We recall that a concave mirror has a positive value of focal length.

(a) Then (with *f* = +12 cm and *p *= +18 cm), the radius of curvature is 2 24 cm*r f*= = + .

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* )= +36 cm.

(c)Then, by Eq. 34-7, the lateral magnification is *m* = −*i*/*p* = −2.0.

(d) Since the image distance computation produced a positive value, the image is real (R).

(e) The magnification computation produced a negative value, so it is inverted (I).

(f) For a mirror, the side where a real image forms is the same as the side where the

object is.

10. A concave mirror has a positive value of focal length.

(a) Then (with *f* = +10 cm and *p *= +15 cm), the radius of curvature is 2 20 cm*r f*= = + .

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* ) = +30 cm.

(c)Then, by Eq. 34-7, *m* = −*i*/*p* = –2.0.

(d) Since the image distance computation produced a positive value, the image is real (R).

(e) The magnification computation produced a negative value, so it is inverted (I).

(f) For a mirror, the side where a real image forms is the same as the side where the

object is.

11. A concave mirror has a positive value of focal length.

(a) Then (with *f* = +18 cm and *p *= +12 cm) , the radius of curvature is *r* = 2*f* = + 36 cm.

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* ) = –36 cm.

(c) Then, by Eq. 34-7, *m* = −*i*/*p* = +3.0.

(d) Since the image distance is negative, the image is virtual (V).

(e) The magnification computation produced a positive value, so it is upright [not

inverted] (NI).

(f) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

12. A concave mirror has a positive value of focal length.

(a) Then (with *f* = +36 cm and *p *= +24 cm), the radius of curvature is *r* = 2*f* = + 72 cm.

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* ) = –72 cm.

(c) Then, by Eq. 34-7, *m* = −*i*/*p* = +3.0.

(d) Since the image distance is negative, the image is virtual (V).

(e) The magnification computation produced a positive value, so it is upright [not

inverted] (NI).

(f) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

13. A convex mirror has a negative value of focal length.

(a) Then (with *f* = –10 cm and *p *= +8 cm), the radius of curvature is *r* = 2*f* = –20 cm.

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* ) = – 4.4 cm.

(c) Then, by Eq. 34-7, *m* = −*i*/*p* = +0.56.

(d) Since the image distance is negative, the image is virtual (V).

(e) The magnification computation produced a positive value, so it is upright [not

inverted] (NI).

(f) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

14. A convex mirror has a negative value of focal length.

(a) Then (with *f* = –35 cm and *p *= +22 cm), the radius of curvature is *r* = 2*f* = –70 cm.

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* ) = –14 cm.

(c) Then, by Eq. 34-7, *m* = −*i*/*p* = +0.61.

(d) Since the image distance is negative, the image is virtual (V).

(e) The magnification computation produced a positive value, so it is upright [not

inverted] (NI).

(f) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

15. A convex mirror has a negative value of focal length.

(a) Then (with *f* = –8 cm and *p *= +10 cm), the radius of curvature is *r* = 2*f* = –16 cm.

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* ) = –4.4 cm.

(c) Then, by Eq. 34-7, *m* = −*i*/*p* = +0.44.

(d) Since the image distance is negative, the image is virtual (V).

(e) The magnification computation produced a positive value, so it is upright [not

inverted] (NI).

(f) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

16. A convex mirror has a negative value of focal length.

(a) Then (with *f* = –14 cm and *p *= +17 cm), the radius of curvature is *r* = 2*f* = –28 cm.

(b) Eq. 34-9 yields *i* = *pf* /( *p*−*f* ) = –7.7 cm.

(c) Then, by Eq. 34-7, *m* = −*i*/*p* = +0.45.

(d) Since the image distance is negative, the image is virtual (V).

(e) The magnification computation produced a positive value, so it is upright [not

inverted] (NI).

(f) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

17. (a) From Eqs. 34-3 and 34-4, we obtain

*i = pf*/(*p – f* ) = *pr*/(2*p – r*).

Differentiating both sides with respect to time and using *vO* = –*dp*/*dt*, we find

*v
di
*

*dt
*

*d
*

*dt
*

*pr
*

*p r
*

*rv p r v pr
*

*p r
*

*r
*

*p r
v
*

*I
*

*O O
*

*O
*= =

− F HG

I KJ =

− − + −

= −

F HG

I KJ2

2 2

2 2 2

2b g b g

.

(b) If *p* = 30 cm, we obtain

*v
I
*=

− L NM

O QP

=15 15

50 056

2

cm

2 30 cm cm cm / s cm / s.b g b g. .

(c) If *p* = 8.0 cm, we obtain

*v
I
*=

− L NM

O QP

= ×15 15

50 11 10

2

3cm

2 8.0 cm cm cm / s cm / s.b g b g. .

(d) If *p* = 1.0 cm, we obtain

( ) ( ) 2

15cm 5.0cm/s 6.7cm/s.

2 1.0cm 15cm
*I
*

*v *= =
−

18. We note that there is “singularity” in this graph (Fig. 34-34) like there was in Fig. 34-

33), which tells us that there is no point where *p *= *f* (which causes Eq. 34-9 to “blow

up”). Since *p *> 0, as usual, then this means that the focal length is not positive. We

know it is not a flat mirror since the curve shown does decrease with *p*, so we conclude it

is a convex mirror. We examine the point where *m* = 0.50 and *p *= 10 cm. Combining

Eq. 34-7 and Eq. 34-9 we obtain *m* = *– i /p = –f/(p – f)*. This yields *f* = *–*10 cm (verifying

our expectation that the mirror is convex). Now, for *p *= 21 cm, we find *m* = *–f/(p – f)*

= +0.32.

19. (a) The mirror is concave.

(b) *f* = +20 cm (positive, because the mirror is concave).

(c) *r* = 2*f* = 2(+20 cm) = +40 cm.

(d) The object distance *p* = +10 cm, as given in the Table.

(e) The image distance is *i* = (1/*f* – 1/*p*)
–1

= (1/20 cm – 1/10 cm) –1

= –20 cm.

(f) *m = –i*/*p* = –(–20 cm/10 cm) = +2.0.

(g) The image is virtual (V).

(h) The image is upright or not inverted (NI).

(i) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

20. (a) The fact that the magnification is 1 means that the mirror is flat (plane).

(b) Flat mirrors (and flat “lenses” such as a window pane) have *f* = ∞ (or *f* = –∞ since the
sign does not matter in this extreme case).

(c) The radius of curvature is *r* = 2*f* = ∞ (or *r* = –∞) by Eq. 34-3.

(d) *p* = + 10 cm, as given in the Table.

(e) Eq. 34-4 readily yields *i* = *pf* /( *p*−*f* ) = –10 cm.

(f) The magnification is *m**= –i*/*p* = +1.0.

(g) The image is virtual since *i *< 0.

(h) The image is upright, or not inverted (NI).

(i) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

21. (a) Since *f* > 0, the mirror is concave.

(b) *f* = + 20 cm, as given in the Table.

(c) Using Eq. 34-3, we obtain *r* = 2*f* = +40 cm.

(d) *p* = + 10 cm, as given in the Table.

(e) Eq. 34-4 readily yields *i* = *pf* /( *p*−*f* ) = +60 cm.

(f) Eq. 34-6 gives *m**= –i*/*p* = –2.0.

(g) Since *i* > 0, the image is real (R).

(h) Since *m* < 0, the image is inverted (I).

(i) For a mirror, the side where a real image forms is the same as the side where the

object is.

22. (a) Since *m* = − 1/2 < 0, the image is inverted. With that in mind, we examine the
various possibilities in Figs. 34-7, 34-9 and 34-10, and note that an inverted image (for

reflections from a single mirror) can only occur if the mirror is concave (and if *p > f *).

(b) Next, we find *i* from Eq. 34-6 (which yields *i* = *mp* = 30 cm) and then use this value

(and Eq. 34-4) to compute the focal length; we obtain *f* = +20 cm.

(c) Then, Eq. 34-3 gives *r* = 2*f* = +40 cm.

(d) *p* = 60 cm, as given in the Table.

(e) As already noted, *i* = +30 cm.

(f) *m* = − 1/2, as given.

(g) Since *i* > 0, the image is real (R).

(h) As already noted, the image is inverted (I).

(i) For a mirror, the side where a real image forms is the same as the side where the

object is.

23. (a) Since *r* < 0 then (by Eq. 34-3) *f* < 0, which means the mirror is convex.

(b) The focal length is *f = r*/2 = –20 cm.

(c) *r* = – 40 cm, as given in the Table.

(d) Eq. 34-4 leads to *p* = +20 cm.

(e) *i* = –10 cm, as given in the Table.

(f) Eq. 34-6 gives *m* = +0.50.

(g) The image is virtual (V).

(h) The image is upright, or not inverted (NI).

(i) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

24. (a) Since 0 < *m* < 1, the image is upright but smaller than the object. With that in

mind, we examine the various possibilities in Figs. 34-7, 34-9 and 34-10, and note that

such an image (for reflections from a single mirror) can only occur if the mirror is convex.

(b) Thus, we must put a minus sign in front of the “20” value given for *f*, i.e., *f* = – 20 cm.

(c) Eq. 34-3 then gives *r* = 2*f *= –40 cm.

(d) To solve for *i* and *p* we must set up Eq. 34-4 and Eq. 34-6 as a simultaneous set and

solve for the two unknowns. The results are *p* = +180 cm = +1.8 m, and

(e) *i* = –18 cm.

(f) *m* = 0.10, as given in the Table.

(g) The image is virtual (V) since *i* < 0.

(h) The image is upright, or not inverted (NI), as already noted.

(i) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

25. (a) The mirror is convex, as given.

(b) Knowing the mirror is convex means we must put a minus sign in front of the “40”

value given for *r*. Then, Eq. 34-3 yields *f = r*/2 = –20 cm.

(c) *r* = – 40 cm.

(d) The fact that the mirror is convex also means that we need to insert a minus sign in

front of the “4.0” value given for *i*, since the image in this case must be virtual (see Figs.

34-7, 34-9 and 34-10). Eq. 34-4 leads to *p* = +5.0 cm.

(e) As noted above, *i* = – 4.0.

(f) Eq. 34-6 gives *m* = +0.8.

(g) The image is virtual (V) since *i* < 0.

(h) The image is upright, or not inverted (NI).

(i) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

26. (a) Since the image is inverted, we can scan Figs. 34-7, 34-9 and 34-10 in the

textbook and find that the mirror must be concave.

(b) This also implies that we must put a minus sign in front of the “0.50” value given for

*m*. To solve for *f*, we first find *i* = *–pm* = +12 cm from Eq. 34-6 and plug into Eq. 34-4;

the result is *f* = +8 cm.

(c) Thus, *r* = 2*f* = +16 cm.

(d) *p* = +24 cm, as given in the Table.

(e) As shown above, *i* = *–pm* = +12 cm.

(f) *m* = *–*0.50, with a minus sign.

(g) The image is real (R) since *i* > 0.

(h) The image is inverted (I), as noted above.

(i) For a mirror, the side where a real image forms is the same as the side where the

object is.

27. (a) The fact that the focal length is given as a negative value means the mirror is

convex.

(b) *f* = *–*30 cm, as given in the Table.

(c) The radius of curvature is *r* = 2*f* = *–*60 cm.

(d) Eq. 34-9 gives *p *= *if /*(*i – f*) = +30 cm.

(e)* i *= *–*15, as given in the Table.

(f) From Eq. 34-7, we get *m* = +1/2 = 0.50.

(g) The image distance is given as a negative value (as it would have to be, since the

mirror is convex), which means the image is virtual (V).

(h) Since *m* > 0, the image is upright (not inverted: NI).

(i) The image is on the side of the mirror opposite to the object.

28. (a) We are told that the image is on the same side as the object; this means the image

is real (R) and further implies that the mirror is concave.

(b) The focal distance is *f* = +20 cm.

(c) The radius of curvature is *r* = 2*f* = +40 cm.

(d) *p* = +60 cm, as given in the Table.

(e) Eq. 34-9 gives *i *= *pf/*(*p – f*) = +30 cm.

(f) Eq. 34-7 gives *m* = −*i/p *= −0.50.

(g) As noted above, the image is real (R).

(h) The image is inverted (I) since *m* < 0.

(i) For a mirror, the side where a real image forms is the same as the side where the

object is.

29. (a) As stated in the problem, the image is inverted (I) which implies that it is real (R).

It also (more directly) tells us that the magnification is equal to a negative value: *m* =

−0.40. By Eq. 34-7, the image distance is consequently found to be *i* = +12 cm. Real
images don’t arise (under normal circumstances) from convex mirrors, so we conclude

that this mirror is concave.

(b) The focal length is* f* = +8.6 cm, using Eq. 34-9 *f* = +8.6 cm.

(c) The radius of curvature is *r* = 2*f* = +17.2 cm ≈ 17 cm.

(d) *p* = +30 cm, as given in the Table.

(e) As noted above, *i* = +12 cm.

(f) Similarly, *m* = −0.40, with a minus sign.

(g) The image is real (R).

(h) The image is inverted (I).

(i) For a mirror, the side where a real image forms is the same as the side where the

object is.

30. (a) From Eq. 34-7, we get *i* = −*mp* = +28 cm, which implies the image is real (R) and
on the same side as the object. Since *m* < 0, we know it was inverted (I). From Eq. 34-9,

we obtain *f* = *ip/*(*i + p*) = +16 cm, which tells us (among other things) that the mirror is

concave.

(b) *f* = *ip/*(*i + p*) = +16 cm.

(c) *r* = 2*f* = +32 cm.

(d) *p* = +40 cm, as given in the Table.

(e) *i* = −*mp* = +28 cm.

(f) *m* = −0.70, as given in the Table.

(g) The image is real (R).

(h) The image is inverted (I).

(i) For a mirror, the side where a real image forms is the same as the side where the

object is.

31. (a) The fact that the magnification is equal to a positive value means that the image is

upright (not inverted: NI), and further implies (by Eq. 34-7) that the image distance (*i*) is

equal to a negative value the image is virtual (V). Looking at the discussion of

mirrors in sections 34-3 and 34-4, we see that a positive magnification of magnitude less

than unity is only possible for convex mirrors.

(b) For 0< *m* < 1 this will only give a positive value for *p *= *f /*(1 – 1/*m*) if *f* < 0. Thus,

with a minus sign, we have *f* = −30 cm.

(c) *r* = 2*f* = –60 cm.

(d) *p *= *f /*(1 – 1/*m*) = + 120 cm = 1.2 m.

(e) *i* = –*mp* = –24 cm.

(f) *m* = +0.20, as given in the Table.

(g) The image is virtual (V).

(h) The image is upright, or not inverted (NI).

(i) For a mirror, the side where a virtual image forms is opposite from the side where the

object is.

32. (a) We use Eq. 34-8 and note that *n*1 = *n*air = 1.00, *n*2 = *n, p* = ∞, and *i* = 2*r*:

100

2

1. .

∞
+ = −*n
*

*r
*

*n
*

*r
*

We solve for the unknown index: *n* = 2.00.

(b) Now *i = r* so Eq. 34-8 becomes

*n
*

*r
*

*n
*

*r
*= −1 ,

which is not valid unless *n *→∞ or .*r *→∞ It is impossible to focus at the center of the
sphere.

33. We use Eq. 34-8 (and Fig. 34-10(d) is useful), with *n*1 = 1.6 and *n*2 = 1 (using the

rounded-off value for air).

16 1 1 16. .

*p i r
*+ = −

Using the sign convention for *r* stated in the paragraph following Eq. 34-8 (so that

5.0 cm*r *= − ), we obtain *i* = –2.4 cm for objects at *p* = 3.0 cm. Returning to Fig. 34-36
(and noting the location of the observer), we conclude that the tabletop seems 7.4 cm

away.

34. In addition to *n*1 =1.0, we are given (a) *n*2=1.5, (b) *p* = +10 and (c) *r* = +30.

(d) Eq. 34-8 yields

*i n
n n
*

*r
*

*n
*

*p
*= − −
F
HG

I KJ =

− − F HG

I KJ = −

−

2 2 1 1

1

15 15 10

30

10

10 18.

. . .

cm cm cm.

(e) The image is virtual (V) and upright since 0*i *< .

(f) The object and its image are in the same side. The ray diagram would be similar to Fig.

34-11(c) in the textbook.

35. In addition to *n*1 =1.0, we are given (a) *n*2=1.5, (b) *p* = +10 and (d) 13*i *= − .

(c) Eq. 34-8 yields

( ) ( ) 1 1

1 2 2 1

1.0 1.5 1.5 1.0 32.5cm 33 cm

10 13

*n n
r n n
*

*p i
*

− −

= − + = − + = − ≈ − −

.

(e) The image is virtual (V) and upright.

(f) The object and its image are in the same side. The ray diagram would be similar to Fig.

34-11(e).

36. In addition to *n*1 =1.0, we are given (a) *n*2=1.5, (c) *r* = +30 and (d) 600*i *= + .

(b) Eq. 34-8 gives

1

2 1 2

1.0 71cm.

1.5 1.0 1.5 30 600

*n
p
*

*n n n
*

*r i
*

= = =− − −−

(d) With 0*i *> , the image is real (R) and inverted.

(e) The object and its image are in the opposite side. The ray diagram would be similar to

Fig. 34-11(a) in the textbook.

37. In addition to *n*1 =1.5, we are given (a) *n*2=1.0, (b) *p* = +10 and (d) 6.0*i *= − .

(c) We manipulate Eq. 34-8 to find *r*:

*r n n
n
*

*p
*

*n
*

*i
*= − +

F HG

I KJ = − + −

F HG

I KJ =

− −

2 1 1 2

1 1

10 15 15

10

10

6 0 30b g b g. . . .

. cm.

(e) The image is virtual (V) and upright.

(f) The object and its image are in the same side. The ray diagram would be similar to Fig.

34-11(f) in the textbook, but with the object and the image located closer to the surface.

38. In addition to *n*1 =1.5, we are given (a) *n*2=1.0, (c) *r* = −30 and (d) 7.5*i *= − .

(b)We manipulate Eq. 34-8 to find *p*:

1

2 1 2

1.5 10cm.

1.0 1.5 1.0 30 7.5

*n
p
*

*n n n
*

*r i
*

= = =− − −− − −

(e) The image is virtual (V) and upright.

(f) The object and its image are in the same side. The ray diagram would be similar to Fig.

34-11(d) in the textbook.

39. In addition to *n*1 =1.5, we are given (a) *n*2=1.0, (b) *p* = +70 and (c) *r* = +30.

(d) We manipulate Eq. 34-8 to find the image distance:

*i n
n n
*

*r
*

*n
*

*p
*= − −
F
HG

I KJ =

− − F HG

I KJ = −

− −

2 2 1 1

1 1

10 10 15

30

15

70 26.

. . .

cm cm cm.

(e) The image is virtual (V) and upright.

(f) The object and its image are in the same side. The ray diagram would be similar to Fig.

34-11(f) in the textbook.

40. In addition to *n*1 =1.5, we are given (b) *p* = +100, (c) *r* = −30 and (d) 600*i *= + .

(a) We manipulate Eq. 34-8 to separate the indices:

( )1 12 2 2 1 1 1 1 1.5 1.5

0.035 0.035 30 600 100 30

*n n
n n n
*

*r i p r
*− = + − = + − = −

− −

which implies *n*2 = 1.0.

(e) The image is real (R) and inverted.

(f) The object and its image are in the opposite side. The ray diagram would be similar to

Fig. 34-11(b) in the textbook.

41. Let the diameter of the Sun be *ds* and that of the image be *di*. Then, Eq. 34-5 leads to

( )( )( )2 8 3

11

20.0 10 m 2 6.96 10 m | | 1.86 10 m

1.50 10 m

1.86mm.

*i s s s
*

*i f
d m d d d
*

*p p
*

− −

× × = = ≈ = = ×

× =

42. The singularity the graph (where the curve goes to ±∞) is at *p *= 30 cm, which implies
(by Eq. 34-9) that *f* = 30 cm > 0 (converging type lens). For *p *= 100 cm, Eq. 34-9 leads

to *i* = +43 cm.

43. We use the lens maker’s equation, Eq. 34-10:

1 1

1 1

1 2*f
n
*

*r r
*= − −

F HG

I KJb g

where *f* is the focal length, *n* is the index of refraction, *r*1 is the radius of curvature of the

first surface encountered by the light and *r*2 is the radius of curvature of the second

surface. Since one surface has twice the radius of the other and since one surface is

convex to the incoming light while the other is concave, set *r*2 = –2*r*1 to obtain

1 1

1 1

2

3 1

21 1 1*f
n
*

*r r
*

*n
*

*r
*= − +

F HG

I KJ =

− ( )

( ) .

(a) We solve for the smaller radius *r*1:

*r
n f
*

1

3 1

2

3 15 1 60

2 45= − = − =( ) ( . )( mm) mm.

(b) The magnitude of the larger radius is 2 1| | 2 90 mm*r r*= = .

44. Since the focal length is a constant for the whole graph, then 1/*p* + 1/*i* = constant.

Consider the value of the graph at *p *= 20 cm; we estimate its value there to be –10 cm.

Therefore, 1/20 + 1/(–10) = 1/70 + 1/*i*new . Thus, *i*new = –16 cm.

45. (a) We use Eq. 34-10:

*f n
r r
*

= − − F HG

I KJ

L NM

O QP

= − ∞ − −

F HG

I KJ

L NM

O QP

= + − −

( ) ( . )1 1 1

15 1 1 1

20 40

1 2

1 1

cm cm.

(b) From Eq. 34-9,

*i
f p
*

= − F HG

I KJ = −

F HG

I KJ = ∞

− − 1 1 1

40

1

40

1 1

cm cm .

46. Combining Eq. 34-7 and Eq. 34-9, we have *m*( *p* – *f *) = –*f*. The graph in Fig. 34-39

indicates that *m* = 2 where *p *= 5 cm, so our expression yields *f* = 10 cm. Plugging this

back into our expression and evaluating at *p *= 14 cm yields *m* = –2.5.

47. We solve Eq. 34-9 for the image distance:

1

1 1 .

*fp
i
*

*f p p f
*

−

= − = −

The height of the image is thus

*h mh
i
*

*p
h
*

*fh
*

*p f
i p p
*

*p*= =
F
HG
I
KJ = − = − =

(

. .

75

27 0 075 5 0

mm)(1.80 m)

m m mm.

48. Combining Eq. 34-7 and Eq. 34-9, we have *m*( *p* – *f *) = –*f*. The graph in Fig. 34-40

indicates that *m* = 0.5 where *p *= 15 cm, so our expression yields *f* = –15 cm. Plugging

this back into our expression and evaluating at *p *= 35 cm yields *m* = +0.30.

49. Using Eq. 34-9 and noting that *p + i = d* = 44 cm, we obtain *p
*2
– *dp + df* = 0.

Therefore,

*p d d df*= ± − = ± − =1
2

4 22 44 4 44 222( ) ( (cm 1

2 cm) cm)(11 cm) cm.2

50. We recall that for a converging (C) lens, the focal length value should be positive ( *f* =

+4 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = +5.3 cm.

(b) Eq. 34-7 give *m* = /*i p*− = −0.33.

(c) The fact that the image distance *i *is a positive value means the image is real (R).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image is on the side opposite from the object (see Fig. 34-14).

51. We recall that for a converging (C) lens, the focal length value should be positive ( *f* =

+16 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = – 48 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +4.0.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image is on the same side as the object (see Fig. 34-14).

52. We recall that for a diverging (D) lens, the focal length value should be negative ( *f*

= –6 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = –3.8 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +0.38.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image is on the same side as the object (see Fig. 34-14).

53. We recall that for a diverging (D) lens, the focal length value should be negative ( *f*

= –12 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = –4.8 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +0.60.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image is on the same side as the object (see Fig. 34-14).

54. We recall that for a converging (C) lens, the focal length value should be positive ( *f* =

+35 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = –88 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +3.5.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image is on the same side as the object (see Fig. 34-14).

55. We recall that for a diverging (D) lens, the focal length value should be negative ( *f*

= –14 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = –8.6 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +0.39.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image is on the same side as the object (see Fig. 34-14).

56. We recall that for a diverging (D) lens, the focal length value should be negative ( *f*

= –31 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = –8.7 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +0.72.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image is on the same side as the object (see Fig. 34-14).

57. We recall that for a converging (C) lens, the focal length value should be positive ( *f* =

+20 cm).

(a) Eq. 34-9 gives *i* = *pf/*(*p*–*f*) = +36 cm.

(b) Eq. 34-7 give *m* = /*i p*− = −0.80.

(c) The fact that the image distance is a positive value means the image is real (R).

(d) A negative value of magnification means the image is inverted (I).

(e) The image is on the opposite side of the object (see Fig. 34-14).

58. (a) A convex (converging) lens, since a real image is formed.

(b) Since *i = d – p* and *i*/*p* = 1/2,

*p
d*= = =2
3

2 40 0

3 26 7

. .

cm cm.

b g

(c) The focal length is

*f
i p d d
*

*d*= +
F
HG

I KJ = +

F HG

I KJ = = =

− − 1 1 1

3

1

2 3

2

9

2 40 0

9 889

1 1

/ /

. .

cm cm.

b g

59. (a) Combining Eq. 34-9 and Eq. 34-10 gives *i* = +84 cm.

(b) Eq. 34-7 give *m* = /*i p*− = −1.4.

(c) The fact that the image distance is a positive value means the image is real (R).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image is on the side opposite from the object (see Fig. 34-14).

60. (a) Combining Eq. 34-9 and Eq. 34-10 gives *i* = –26 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +4.3.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image it is on the same side as the object (see Fig. 34-14).

61. (a) Combining Eq. 34-9 and Eq. 34-10 gives *i* = –18 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +0.76.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image it is on the same side as the object (see Fig. 34-14).

62. (a) Combining Eq. 34-9 and Eq. 34-10 gives *i* = –9.7 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +0.54.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image it is on the same side as the object (see Fig. 34-14).

63. (a) Combining Eq. 34-9 and Eq. 34-10 gives *i* = –30 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +0.86.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image it is on the same side as the object (see Fig. 34-14).

64. (a) Combining Eq. 34-9 and Eq. 34-10 gives *i* = –63 cm.

(b) Eq. 34-7 give *m* = /*i p*− = +2.2.

(c) The fact that the image distance is a negative value means the image is virtual (V).

(d) A positive value of magnification means the image is not inverted (NI).

(e) The image it is on the same side as the object (see Fig. 34-14).

65. (a) Combining Eq. 34-9 and Eq. 34-10 gives *i* = +55 cm.

(b) Eq. 34-7 give *m* = /*i p*− = −0.74.

(c) The fact that the image distance is a positive value means the image is real (R).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image is on the side opposite from the object (see Fig. 34-14).

66. (a) Eq. 34-10 yields *f r r
n
*

= − = +− −1

1 1 2

11 1 30( / / ) cm. Since *f* > 0, this must be a

converging (“C”) lens. From Eq. 34-9, we obtain

1 1 15cm.

1 1 1 1 30 10

*i
*

*f p
*

= = = − − −

(b) Eq. 34-6 yields *m* = /*i p*− = –(–15)/10 = +1.5.

(c) Since *i* < 0, the image is virtual (V).

(d) Since *m* > 0, the image is upright, or not inverted (NI).

(e) The image it is on the same side as the object. The ray diagram would be similar to

Fig. 34-15(b) in the textbook.

67. (a) Eq. 34-10 yields *f r r
n
*

= − = −− −1

1 1 2

11 1 30( / / ) cm. Since *f* < 0, this must be a

diverging (“D”) lens. From Eq. 34-9, we obtain

1 1 7.5cm.

1 1 1 1 30 10

*i
*

*f p
*

= = = − − −−

(b) Eq. 34-6 yields *m* = /*i p*− = –(–7.5)/10 = +0.75.

(c) Since *i* < 0, the image is virtual (V).

(d) Since *m* > 0, the image is upright, or not inverted (NI).

(e) The image it is on the same side as the object. The ray diagram would be similar to

Fig. 34-15(c) in the textbook.

68. (a) Eq. 34-10 yields *f r r
n
*

= − = −− −1

1 1 2

11 1 120( / / ) cm. Since *f* < 0, this must be a

diverging (“D”) lens. From Eq. 34-9, we obtain

1 1 9.2cm.

1 1 1 1 120 10

*i
*

*f p
*

= = = − − −−

(b) Eq. 34-6 yields *m* = /*i p*− = –(–9.2)/10 = +0.92.

(c) Since *i* < 0, the image is virtual (V).

(d) Since *m* > 0, the image is upright, or not inverted (NI).

(e) The image it is on the same side as the object. The ray diagram would be similar to

Fig. 34-15(c) in the textbook.

69. (a) The fact that *m* > 1 means the lens is of the converging type (C) (it may help to

look at Fig. 34-14 to illustrate this).

(b) A converging lens implies *f* = +20 cm, with a plus sign.

(d) Eq. 34-9 then gives *i* = –13 cm.

(e) Eq. 34-7 gives *m* = /*i p*− = +1.7.

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(g) A positive value of magnification means the image is not inverted (NI).

(h) The image it is on the same side as the object (see Fig. 34-14).

**7**0. (a) The fact that *m* < 1 and that the image is upright (not inverted: NI) means the lens

is of the diverging type (D) (it may help to look at Fig. 34-14 to illustrate this).

(b) A diverging lens implies that *f* = –20 cm, with a minus sign.

(d) Eq. 34-9 gives *i* = –5.7 cm.

(e) Eq. 34-7 gives *m* = /*i p*− = +0.71.

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(h) The image it is on the same side as the object (see Fig. 34-14).

71. (a) Eq. 34-7 yields *i *= –*mp* = –(0.25)(16) = –4.0 cm. Eq. 34-9 gives *f* = –5.3 cm,

which implies the lens is of the diverging type (D).

(b) From (a), we have *f* = –5.3 cm.

(d) Similarly, *i *= –4.0 cm.

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(g) A positive value of magnification means the image is not inverted (NI).

(h) The image it is on the same side as the object (see Fig. 34-14).

72. (a) Eq. 34-7 readily yields *i *= +4.0 cm. Then Eq. 34-9 gives *f* = +3.2 cm, which

implies the lens is of the converging type (C).

(b) From (a), we have *f* = +3.2 cm.

(d) Similarly, *i *= +4.0 cm.

(f) The fact that the image distance is a positive value means the image is real (R).

(g) The fact that the magnification is a negative value means the image is inverted (I).

(h) The image is on the side opposite from the object.

73. (a) Eq. 34-7 readily yields *i *= –20 cm. Then Eq. 34-9 gives *f* = +80 cm, which

implies the lens is of the converging type (C).

(b) From (a), we have *f* = +80 cm.

(d) Similarly, *i *= –20 cm.

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(g) A positive value of magnification means the image is not inverted (NI).

(h) The image it is on the same side as the object (see Fig. 34-14).

**7**4. (b) Since this is a converging lens (“C”) then *f* > 0, so we should put a plus sign in

front of the “10” value given for the focal length.

(d) Eq. 34-9 gives

1 1 20cm.

1 1 1 1 10 20

*i
*

*f p
*

= = = + − −

(e) From Eq. 34-6, *m* = –20/20 = –1.0.

(f) The fact that the image distance is a positive value means the image is real (R).

(g) The fact that the magnification is a negative value means the image is inverted (I).

(h) The image is on the side opposite from the object.

75. (a) Since *f* > 0, this is a converging lens (“C”).

(d) Eq. 34-9 gives

1 1 10cm.

1 1 1 1 10 5

*i
*

*f p
*

= = = − − −

(e) From Eq. 34-6, *m* = –(–10)/5 = +2.0.

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(g) A positive value of magnification means the image is not inverted (NI).

(h) The image it is on the same side as the object (see Fig. 34-14).

76. (a) We are told the magnification is positive and greater than 1. Scanning the single-

lens-image figures in the textbook (Figs. 34-14, 34-15 and 34-17), we see that such a

magnification (which implies an upright image larger than the object) is only possible if

the lens is of the converging (“C”) type (and if *p < f *).

(b) We should put a plus sign in front of the “10” value given for the focal length.

(d) Eq. 34-9 gives

1 1 10cm.

1 1 1 1 10 5

*i
*

*f p
*

= = = − − −

(e) / 2.0*m i p*= − = + .

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(g) A positive value of magnification means the image is not inverted (NI).

(h) The image it is on the same side as the object (see Fig. 34-14).

77. (a) We are told the magnification is less than 1, and we note that *p < | f |*). Scanning

Figs. 34-14, 34-15 and 34-17, we see that such a magnification (which implies an image

smaller than the object) and object position (being fairly close to the lens) are

simultaneously possible only if the lens is of the diverging (“D”) type.

(b) Thus, we should put a minus sign in front of the “10” value given for the focal length.

(d) Eq. 34-9 gives

1 1 3.3cm.

1 1 1 1 10 5

*i
*

*f p
*

= = = − − −−

(e) / 0.67*m i p*= − = + .

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(g) A positive value of magnification means the image is not inverted (NI).

78. (a) We are told the absolute value of the magnification is 0.5 and that the image was

upright (NI). Thus, *m* = +0.5. Using Eq. 34-6 and the given value of *p*, we find *i* = –5.0

cm; it is a virtual image. Eq. 34-9 then yields the focal length: *f* = –10 cm. Therefore, the

lens is of the diverging (“D”) type.

(b) From (a), we have *f* = –10 cm.

(d) Similarly, *i* = –5.0 cm.

(e) *m* = +0.5, with a plus sign

(f) The fact that the image distance *i *is a negative value means the image is virtual (V).

(h) The image it is on the same side as the object (see Fig. 34-14).

79. (a) Using Eq. 34-6 (which implies the image is inverted) and the given value of *p*, we

find *i = –mp* = +5.0 cm; it is a real image. Eq. 34-9 then yields the focal length: *f* = +3.3

cm. Therefore, the lens is of the converging (“C”) type.

(b) From (a), we have *f* = +3.3 cm.

(d) Similarly, *i = –mp* = +5.0 cm.

(f) The fact that the image distance is a positive value means the image is real (R).

(g) The fact that the magnification is a negative value means the image is inverted (I).

(h) The image is on the side opposite from the object.The ray diagram would be similar

to Fig. 34-15(a) in the textbook.

80. (a) The image from lens 1 (which has *f*1 = +15 cm) is at *i*1 = –30 cm (by Eq. 34-9).

This serves as an “object” for lens 2 (which has *f*2 = +8 cm) with *p*2 = *d* – *i*1 = 40 cm.

Then Eq. 34-9 (applied to lens 2) yields *i*2 = +10 cm.

(b) Eq. 34-11 yields *M* = *m*1*m*2 1 1 2 2 1 2 1 2( / )( / ) /*i p i p i i p p*= − − = = –0.75.

(c) The fact that the (final) image distance is a positive value means the image is real (R).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image it is on the side opposite from the object (relative to lens 2).

81. (a) The image from lens 1 (which has *f*1 = +8 cm) is at *i*1 = 24 cm (by Eq. 34-9). This

serves as an “object” for lens 2 (which has *f*2 = +6 cm) with *p*2 = *d* – *i*1 = 8 cm. Then Eq.

34-9 (applied to lens 2) yields *i*2 = +24 cm.

(b) Eq. 34-11 yields *M* = *m*1*m*2 1 1 2 2 1 2 1 2( / )( / ) /*i p i p i i p p*= − − = = +6.0.

(c)The fact that the (final) image distance is a positive value means the image is real (R).

(d) The fact that the magnification is positive means the image is not inverted (NI).

(e) The image it is on the side opposite from the object (relative to lens 2).

82. (a) The image from lens 1 (which has *f*1 = +12 cm) is at *i*1 = +60 cm (by Eq. 34-9).

This serves as an “object” for lens 2 (which has *f*2 = +10 cm) with *p*2 = *d* – *i*1 = 7 cm.

Then Eq. 34-9 (applied to lens 2) yields *i*2 = –23 cm.

(b) Eq. 34-11 yields *M* = *m*1*m*2 1 1 2 2 1 2 1 2( / )( / ) /*i p i p i i p p*= − − = = –13.

(c) The fact that the (final) image distance is negative means the image is virtual (V).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image it is on the same side as the object (relative to lens 2).

83. (a) The image from lens 1 (which has *f*1 = +9 cm) is at *i*1 = 16.4 cm (by Eq. 34-9).

This serves as an “object” for lens 2 (which has *f*2 = +5 cm) with *p*2 = *d* – *i*1 = –8.4 cm.

Then Eq. 34-9 (applied to lens 2) yields *i*2 = +3.1 cm.

(b) Eq. 34-11 yields *M* = *m*1*m*2 1 1 2 2 1 2 1 2( / )( / ) /*i p i p i i p p*= − − = = –0.31.

(c) The fact that the (final) image distance is a positive value means the image is real (R).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image it is on the side opposite from the object (relative to lens 2). Since this

result involves a negative value for *p*2 (and perhaps other “non-intuitive” features), we

offer a few words of explanation: lens 1 is converging the rays towards an image (that

never gets a chance to form due to the intervening presence of lens 2) that would be real

and inverted (and 8.4 cm beyond lens 2’s location). Lens 2, in a sense, just causes these

rays to converge a little more rapidly, and causes the image to form a little closer (to the

lens system) than if lens 2 were not present.

84.(a) The image from lens 1 (which has *f*1 = –6 cm) is at *i*1 = –3.4 cm (by Eq. 34-9).

This serves as an “object” for lens 2 (which has *f*2 = +6 cm) with *p*2 = *d* – *i*1 = 15.4 cm.

Then Eq. 34-9 (applied to lens 2) yields *i*2 = +9.8 cm.

(b) Eq. 34-11 yields *M* = –0.27.

(c) The fact that the (final) image distance is a positive value means the image is real (R).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image it is on the side opposite from the object (relative to lens 2).

85. (a) The image from lens 1 (which has *f*1 = +6 cm) is at *i*1 = –12 cm (by Eq. 34-9). This

serves as an “object” for lens 2 (which has *f*2 = –6 cm) with *p*2 = *d* – *i*1 = 20 cm. Then Eq.

34-9 (applied to lens 2) yields *i*2 = –4.6 cm.

(b) Eq. 34-11 yields *M* = +0.69.

(c) The fact that the (final) image distance is negative means the image is virtual (V).

(d) The fact that the magnification is positive means the image is not inverted (NI).

(e) The image it is on the same side as the object (relative to lens 2).

86. (a) The image from lens 1 (which has *f*1 = +8 cm) is at *i*1 = +24 cm (by Eq. 34-9). This

serves as an “object” for lens 2 (which has *f*2 = –8 cm) with *p*2 = *d* – *i*1 = 6 cm. Then Eq.

34-9 (applied to lens 2) yields *i*2 = –3.4 cm.

(b) Eq. 34-11 yields *M* = –1.1.

(c) The fact that the (final) image distance is negative means the image is virtual (V).

(d) The fact that the magnification is a negative value means the image is inverted (I).

(e) The image it is on the same side as the object (relative to lens 2).

87. (a) The image from lens 1 (which has *f*1 = –12 cm) is at *i*1 = –7.5 cm (by Eq. 34-9).

This serves as an “object” for lens 2 (which has *f*2 = –8 cm) with *p*2 = *d* – *i*1 = 17.5 cm.

Then Eq. 34-9 (applied to lens 2) yields *i*2 = –5.5 cm.

(b) Eq. 34-11 yields *M* = +0.12.

(c) The fact that the (final) image distance is negative means the image is virtual (V).

(d) The fact that the magnification is positive means the image is not inverted (NI).

(e) The image it is on the same side as the object (relative to lens 2).

88. The minimum diameter of the eyepiece is given by

*d
d
*

*m
*ey

ob mm

36 mm.= = =

θ

75 2 1.

89. (a) If *L* is the distance between the lenses, then according to Fig. 34-18, the tube

length is

*s = L – f*ob – *f*ey = 25.0 cm – 4.00 cm – 8.00 cm = 13.0 cm.

(b) We solve (1/*p*) + (1/*i*) = (1/*f*ob) for *p*. The image distance is

*i = f*ob + *s* = 4.00 cm + 13.0 cm = 17.0 cm,

so

( )( )ob ob

17.0 cm 4.00 cm 5.23 cm.

17.0 cm 4.00 cm

*if
p
*

*i f
*= = =

− −

(c) The magnification of the objective is

*m
i
*

*p
*= − = − = −17 0

5 23 3 25

.

. . .

cm

cm

(d) The angular magnification of the eyepiece is

*m
f
*

θ = = = 25 25

313 cm cm

8.00 cmey . .

(e) The overall magnification of the microscope is

*M mm*= = − = −θ 325 313 10 2. . . .b gb g

90. (a) Without the magnifier, θ = *h*/*Pn* (see Fig. 34-17). With the magnifier, letting *i = –
|i| = – Pn*, we obtain

1 1 1 1 1 1 1

*p f i f i f P
n
*

= − = + = + .

Consequently,

*m
h p
*

*h P
*

*f P
*

*P
*

*P
*

*f f
n
*

*n
*

*n
*

*n
*

θ θ θ

= ′ = = + = + = +/ /

/ /

/ .

1 1

1 1 1

25 cm

With *f* = 10 cm,
25cm

1 3.5 10cm

*m*θ = + = .

(b) In the case where the image appears at infinity, let | |*i i*= − →−∞ , so that
1/ 1/ 1/ 1/*p i p f*+ = = , we have

/ 1/ 25 cm .

/ 1/

*n
*

*n n
*

*Ph p f
m
*

*h P P f f
*θ

θ θ ′

= = = = =

With *f* = 10 cm,

25cm 2.5.

10cm
*m*θ = =

91. (a) When the eye is relaxed, its lens focuses far-away objects on the retina, a distance

*i* behind the lens. We set *p* = ∞ in the thin lens equation to obtain 1/*i* = 1/*f*, where *f* is the
focal length of the relaxed effective lens. Thus, *i = f* = 2.50 cm. When the eye focuses on

closer objects, the image distance *i* remains the same but the object distance and focal

length change. If *p* is the new object distance and *f '* is the new focal length, then

1 1 1

*p i f
*+ =

′ .

We substitute *i = f* and solve for *f '*:

′ = +

= =*f pf
f p
*

40 0 2 50

40 0 2 35

. .

. .

cm cm

cm + 2.50 cm cm.

b gb g

(b) Consider the lens maker’s equation

1 1

1 1

1 2*f
n
*

*r r
*= − −

F HG

I KJb g

where *r*1 and *r*2 are the radii of curvature of the two surfaces of the lens and *n* is the index

of refraction of the lens material. For the lens pictured in Fig. 34-43, *r*1 and *r*2 have about

the same magnitude, *r*1 is positive, and *r*2 is negative. Since the focal length decreases, the

combination (1/*r*1) – (1/*r*2) must increase. This can be accomplished by decreasing the

magnitudes of both radii.

92. We refer to Fig. 34-18. For the intermediate image *p* = 10 mm and

*i* = (*f*ob + *s + f*ey) – *f*ey = 300 m – 50 mm = 250 mm,

so

1 1 1 1

250

1

10 9 62

*f i p
f
*

ob

ob mm mm

mm,= + = + = .

and

*s* = (*f*ob + *s + f*ey) – *f*ob – *f*ey = 300 mm – 9.62 mm – 50 mm = 240 mm.

Then from Eq. 34-14,

*M
s
*

*f f
*= − = −

F HG

I KJ F HG

I KJ = −ob ey

cm mm

9.62 mm

mm

50 mm

25 240 150 125.

93. (a) Now, the lens-film distance is

*i
f p
*

= − F HG

I KJ = −

F HG

I KJ =

− − 1 1 1

5 0

1

100 5 3

1 1

. .

cm cm cm.

(b) The change in the lens-film distance is 5.3 cm – 5.0 cm = 0.30 cm.

94. (a) In the closest mirror M1, the “first” image *I*1 is 10 cm behind M1 and therefore

20 cm from the object *O*. This is the smallest distance between the object and an image

of the object.

(b) There are images from both *O* and *I*1 in the more distant mirror, M2: an image *I*2

located at 30 cm behind M2. Since *O* is 30 cm in front of it, *I*2 is 60 cm from *O*. This is

the second smallest distance between the object and an image of the object.

(c) There is also an image *I*3 which is 50 cm behind M2 (since *I*1 is 50 cm in front of it).

Thus, *I*3 is 80 cm from *O*. In addition, we have another image *I*4 which is 70 cm behind

M1 (since *I*2 is 70 cm in front of it). The distance from *I*4 to *O* for is 80 cm.

(d) Returning to the closer mirror M1, there is an image *I*5 which is 90 cm behind the

mirror (since *I*3 is 90 cm in front of it). The distances (measured from *O*) for *I*5 is 100 cm

= 1.0 m.

95. (a) Parallel rays are bent by positive-*f* lenses to their focal points *F*1, and rays that

come from the focal point positions *F*2 in front of positive-*f* lenses are made to emerge

parallel. The key, then, to this type of beam expander is to have the rear focal point *F*1 of

the first lens coincide with the front focal point *F*2 of the second lens. Since the triangles

that meet at the coincident focal point are similar (they share the same angle; they are

vertex angles), then *Wf*/*f*2 = *Wi*/*f*1 follows immediately. Substituting the values given, we

have

2

1

30.0 cm (2.5 mm) 6.0 mm.

12.5 cm
*f i
*

*f
W W
*

*f
*= = =

(b) The area is proportional to *W
*2
. Since intensity is defined as power *P* divided by area,

we have

22 2 2

21 1

2 2 2

2 2

1.6 kW/m .
*f f i
*

*f i
*

*i i f
*

*P WI W f f
I I
*

*I P W W f f
*= = = = =

(c) The previous argument can be adapted to the first lens in the expanding pair being of

the diverging type, by ensuring that the front focal point of the first lens coincides with

the front focal point of the second lens. The distance between the lenses in this case is

*f*2 – |*f*1| = 30.0 cm – 26.0 cm = 4.0 cm.

96. By Eq. 34-9, 1/*i* + 1/*p* is equal to constant (1/*f* ). Thus,

1/(–10) + 1/(15) = 1/*i*new + 1/(70).

This leads to *i*new = –21 cm.

97. (a) The “object” for the mirror which results in that box-image is equally in front of

the mirror (4 cm). This object is actually the first image formed by the system (produced

by the first transmission through the lens); in those terms, it corresponds to *i*1 = 10 – 4 =

6 cm. Thus, with *f*1 = 2 cm, Eq. 34-9 leads to

1 1 1 3 00

1 1 1

1
*p i f
*

*p*+ = = . cm.

(b) The previously mentioned box-image (4 cm behind the mirror) serves as an “object”

(at *p*3 = 14 cm) for the return trip of light through the lens (*f*3 = *f*1 = 2 cm). This time, Eq.

34-9 leads to

1 1 1 2 33

3 3 3

3
*p i f
*

*i*+ = = . cm.

98. (a) First, the lens forms a real image of the object located at a distance

*i
f p f f
*

*f*1
1 1

1

1 1

1

1

1 1 1 1

2 2= −

F HG

I KJ = −

F HG

I KJ =

− −

to the right of the lens, or at

*p*2 = 2(*f*1 + *f*2) – 2*f*1 = 2*f*2

in front of the mirror. The subsequent image formed by the mirror is located at a distance

*i
f p f f
*

*f*2
2 2

1

2 2

1

2

1 1 1 1

2 2= −

F HG

I KJ = −

F HG

I KJ =

− −

to the left of the mirror, or at

*p'*1 = 2(*f*1 + *f*2) – 2*f*2 = 2*f*1

to the right of the lens. The final image formed by the lens is at a distance *i'*1 to the left of

the lens, where

′ = − ′

F HG

I KJ = −

F HG

I KJ =

− −

*i
f p f f
*

*f*1
1 1

1

1 1

1

1

1 1 1 1

2 2 .

This turns out to be the same as the location of the original object.

(b) The lateral magnification is

*m
i
*

*p
*

*i
*

*p
*

*i
*

*p
*

*f
*

*f
*

*f
*

*f
*

*f
*

*f
*= −
F
HG

I KJ − F HG

I KJ −

′ ′

F HG

I KJ = − F HG

I KJ − F HG

I KJ − F HG

I KJ = −

1

1

2

2

1

1

1

1

2

2

1

1

2

2

2

2

2

2 10. .

(c) The final image is real (R).

(d) It is at a distance *i'*1 to the left of the lens,

(e) and inverted (I), as shown in the figure below.

99. We refer to Fig. 34-2 in the textbook. Consider the two light rays, *r* and *r'*, which are

closest to and on either side of the normal ray (the ray that reverses when it reflects).

Each of these rays has an angle of incidence equal to θ when they reach the mirror. Consider that these two rays reach the top and bottom edges of the pupil after they have

reflected. If ray *r* strikes the mirror at point *A* and ray *r'* strikes the mirror at *B*, the

distance between *A* and *B* (call it *x*) is

*x d
*

*o
*= 2 tanθ

where *do* is the distance from the mirror to the object. We can construct a right triangle

starting with the image point of the object (a distance *do* behind the mirror; see *I* in Fig.

34-2). One side of the triangle follows the extended normal axis (which would reach from

*I* to the middle of the pupil), and the hypotenuse is along the extension of ray *r* (after

reflection). The distance from the pupil to *I* is *d*ey + *do*, and the small angle in this triangle

is again θ. Thus,

tanθ =
+
*R
*

*d d
o*ey

where *R* is the pupil radius (2.5 mm). Combining these relations, we find

*x d
R
*

*d d
o
*

*o
*

= +

= +

2 2 100 2 5

300 100ey mm

mm

mm mm b g .

which yields *x* = 1.67 mm. Now, *x* serves as the diameter of a circular area *A* on the

mirror, in which all rays that reflect will reach the eye. Therefore,

*A x*= = =1
4 4

167 2 22 2 2π π . . .mm mmb g

100. We use Eq. 34-10, with the conventions for signs discussed in §34-6 and §34-7.

(a) For lens 1, the bi-convex (or double convex) case, we have

*f n
r r
*

= − − F HG

I KJ

L NM

O QP

= − − −

F HG

I KJ

L NM

O QP

= − −

1 1 1

15 1 1

40

1

40 40

1 2

1 1

b g b g. cm cm

cm.

(b) Since *f* > 0 the lens forms a real image of the Sun.

(c) For lens 2, of the planar convex type, we find

( ) 1

1 1 1.5 1 80cm.

40cm
*f
*

−

= − − = ∞ −

(d) The image formed is real (since *f* > 0).

(e) Now for lens 3, of the meniscus convex type, we have

( ) 1

1 1 1.5 1 240cm 2.4 m.

40cm 60cm
*f
*

−

= − − = =

(f) The image formed is real (since *f* > 0).

(g) For lens 4, of the bi-concave type, the focal length is

( ) 1

1 1 1.5 1 40cm.

40cm 40cm
*f
*

−

= − − = − −

(h) The image formed is virtual (since *f* < 0).

(i) For lens 5 (plane-concave), we have

( ) 1

1 1 1.5 1 80cm.

40cm
*f
*

−

= − − = − ∞

(j) The image formed is virtual (since *f* < 0).

(k) For lens 6 (meniscus concave),

( ) 1

1 1 1.5 1 240cm 2.4 m.

60cm 40cm
*f
*

−

= − − = − = −

(l) The image formed is virtual (since *f* < 0).

101. (a) The first image is figured using Eq. 34-8, with *n*1 = 1 (using the rounded-off

value for air) and *n*2 = 8/5.

1 8

5

16 1

*p i r
*+ = −.

For a “flat lens” *r* = ∞, so we obtain

*i* = – 8*p*/5 = – 64/5

(with the unit cm understood) for that object at *p* = 10 cm. Relative to the second surface,

this image is at a distance of 3 + 64/5 = 79/5. This serves as an object in order to find the

final image, using Eq. 34-8 again (and *r* = ∞) but with *n*1 = 8/5 and *n*2 = 4/3.

8

5

4

3 0

′ +

′ =

*p i
*

which produces (for *p'* = 79/5)

*i'* = – 5*p*/6 = – 79/6 ≈ – 13.2.

This means the observer appears 13.2 + 6.8 = 20 cm from the fish.

(b) It is straightforward to “reverse” the above reasoning, the result being that the final

fish-image is 7.0 cm to the right of the air-wall interface, and thus 15 cm from the

observer.

102. (a) There are three images. Two are formed by single reflections from each of the

mirrors and the third is formed by successive reflections from both mirrors. The positions

of the images are shown on the two diagrams that follow.

The diagram on the left shows the image *I*1, formed by reflections from the left-hand

mirror. It is the same distance behind the mirror as the object *O* is in front, and lies on the

line perpendicular to the mirror and through the object. Image *I*2 is formed by light that is

reflected from both mirrors. We may consider *I*2 to be the image of *I*1 formed by the

right-hand mirror, extended. *I*2 is the same distance behind the line of the right-hand

mirror as *I*1 is in front and it is on the line that is perpendicular to the line of the mirror.

The diagram on the right shows image *I*3, formed by reflections from the right-hand

mirror. It is the same distance behind the mirror as the object is in front, and lies on the

line perpendicular to the mirror and through the object. As the diagram shows, light that

is first reflected from the right-hand mirror and then from the left-hand mirror forms an

image at *I*2.

(b) For θ = 45°, we have two images in the second mirror caused by the object and its
“first” image, and from these one can construct two new images *I* and *I'* behind the first

mirror plane. Extending the second mirror plane, we can find two further images of *I* and

*I'* which are on equal sides of the extension of the first mirror plane. This circumstance

implies there are no further images, since these final images are each other’s “twins.” We

show this construction in the figure below. Summarizing, we find 1 + 2 + 2 + 2 = 7

images in this case.

(c) For θ = 60°, we have two images in the second mirror caused by the object and its
“first” image, and from these one can construct two new images *I* and *I'* behind the first

mirror plane. The images *I* and *I'* are each other’s “twins” in the sense that they are each

other’s reflections about the extension of the second mirror plane; there are no further

images. Summarizing, we find 1 + 2 + 2 = 5 images in this case.

For θ = 120°, we have two images *I'*1 and *I*2 behind the extension of the second mirror
plane, caused by the object and its “first” image (which we refer to here as *I*1). No further

images can be constructed from *I'*1 and *I*2, since the method indicated above would place

any further possibilities in front of the mirrors. This construction has the disadvantage of

deemphasizing the actual ray-tracing, and thus any dependence on where the observer of

these images is actually placing his or her eyes. It turns out in this case that the number of

images that can be seen ranges from 1 to 3, depending on the locations of both the object

and the observer.

(d) Thus, the smallest number of images that can be seen is 1. For example, if the

observer’s eye is collinear with *I*1 and *I'*1, then the observer can only see one image (*I*1

and not the one behind it). Note that an observer who stands close to the second mirror

would probably be able to see two images, *I*1 and *I*2.

(e) Similarly, the largest number would be 3. This happens if the observer moves further

back from the vertex of the two mirrors. He or she should also be able to see the third

image, *I'*1, which is essentially the “twin” image formed from *I*1 relative to the extension

of the second mirror plane.

103. For a thin lens, (1/*p*) + (1/*i*) = (1/*f *), where *p* is the object distance, *i* is the image

distance, and *f* is the focal length. We solve for *i*:

*i
fp
*

*p f
*=

− .

Let *p = f + x*, where *x* is positive if the object is outside the focal point and negative if it

is inside. Then,

*i
f f x
*

*x
*= +( ) .

Now let *i = f + x'*, where *x'* is positive if the image is outside the focal point and negative

if it is inside. Then,

′ = − = + − =*x i f f f x
x
*

*f
f
*

*x
*

( ) 2

and *xx' = f
*2
.

104. For an object in front of a thin lens, the object distance *p* and the image distance *i* are

related by (1/*p*) + (1/*i*) = (1/*f *), where *f* is the focal length of the lens. For the situation

described by the problem, all quantities are positive, so the distance *x* between the object

and image is *x = p + i*. We substitute *i = x – p* into the thin lens equation and solve for *x*:

*x
p
*

*p f
*=

−

2

.

To find the minimum value of *x*, we set *dx*/*dp* = 0 and solve for *p*. Since

*dx
*

*dp
*

*p p f
*

*p f
*= −

− ( )

( ) ,

2 2

the result is *p* = 2*f*. The minimum distance is

*x
p
*

*p f
*

*f
*

*f f
f*min

( ) .=

− =

− =

2 22

2 4

This is a minimum, rather than a maximum, since the image distance *i* becomes large

without bound as the object approaches the focal point.

105. We place an object far away from the composite lens and find the image distance *i*.

Since the image is at a focal point, *i = f*, where *f* equals the effective focal length of the

composite. The final image is produced by two lenses, with the image of the first lens

being the object for the second. For the first lens, (1/*p*1) + (1/*i*1) = (1/*f*1), where *f*1 is the

focal length of this lens and *i*1 is the image distance for the image it forms. Since *p*1 = ∞,
*i*1 = *f*1. The thin lens equation, applied to the second lens, is (1/*p*2) + (1/*i*2) = (1/*f*2), where

*p*2 is the object distance, *i*2 is the image distance, and *f*2 is the focal length. If the

thicknesses of the lenses can be ignored, the object distance for the second lens is *p*2 = –*i*1.

The negative sign must be used since the image formed by the first lens is beyond the

second lens if *i*1 is positive. This means the object for the second lens is virtual and the

object distance is negative. If *i*1 is negative, the image formed by the first lens is in front

of the second lens and *p*2 is positive. In the thin lens equation, we replace *p*2 with –*f*1 and

*i*2 with *f* to obtain

− + =1 1 1

1 2*f f f
*

or

1 1 1

1 2

1 2

1 2*f f f
*

*f f
*

*f f
*= + =

+ .

Thus,

*f
f f
*

*f f
*=

+ 1 2

1 2

.

106. (a) If the object distance is *x*, then the image distance is *D – x* and the thin lens

equation becomes

1 1 1

*x D x f
*+

− = .

We multiply each term in the equation by *fx*(*D – x*) and obtain *x
*2
– *Dx + Df* = 0. Solving

for *x*, we find that the two object distances for which images are formed on the screen are

*x
D D D f
*

*x
D D D f
*

1 2

4

2

4

2 =

− − =

+ −b g b g and .

The distance between the two object positions is

*d x x D D f*= − = −2 1 4b g.

(b) The ratio of the image sizes is the same as the ratio of the lateral magnifications. If the

object is at *p = x*1, the magnitude of the lateral magnification is

*m
i
*

*p
*

*D x
*

*x
*1

1

1

1

1

= = − .

Now *x D d*1
1
2

= −b g, where *d D D f*= −b g, so

*m
D D d
*

*D d
*

*D d
*

*D d
*1

2

2 =

− − −

= + −

b g b g

/

/ .

Similarly, when the object is at *x*2, the magnitude of the lateral magnification is

*m
I
*

*p
*

*D x
*

*x
*

*D D d
*

*D d
*

*D d
*

*D d
*2

2

2

2

2

2

2 = = − =

− + +

= − +

b g b g

/

/ .

The ratio of the magnifications is

*m
*

*m
*

*D d D d
*

*D d D d
*

*D d
*

*D d
*

2

1

2

= − + + −

= − +

F HG

I KJ

b g b g b g b g

/

/ .

107. The sphere (of radius 0.35 m) is a convex mirror with focal length *f* = –0.175 m. We

adopt the approximation that the rays are close enough to the central axis for Eq. 34-4 to

be applicable.

(a) With *p* = 1.0 m, the equation 1/*p* + 1/*i* = 1/*f* yields *i* = –0.15 m, which means the

image is 0.15 m from the front surface, appearing to be *inside* the sphere.

(b) The lateral magnification is *m* = –*i*/*p* which yields *m* = 0.15. Therefore, the image

distance is (0.15)(2.0 m) = 0.30 m.

(c) Since 0*m *> , the image is upright, or not inverted (NI).

108. (a) We use Eq. 34-8 (and Fig. 34-11(b) is useful), with *n*1 = 1 (using the rounded-off

value for air) and *n*2 = 1.5.

1 15 15 1

*p i r
*+ = −. .

Using the sign convention for *r* stated in the paragraph following Eq. 34-8 (so that *r* =

+6.0 cm), we obtain *i* = –90 cm for objects at *p* = 10 cm. Thus, the object and image are

80 cm apart.

(b) The image distance *i* is negative with increasing magnitude as *p* increases from very

small values to some value *p*0 at which point *i*→ −∞. Since 1/(–∞) = 0, the above
equation yields

1 15 1 2

0

0
*p r
*

*p r*= − =. .

Thus, the range for producing virtual images is 0 < *p* ≤ 12 cm.

109. (a) In this case *i* < 0 so *i* =−|*i*|, and Eq. 34-9 becomes 1/*f* = 1/*p* – 1/|*i*|.
We differentiate this with respect to time (*t*) to obtain

*d*|*i*|

*dt
* =

|*i*|

*p
*

2

*dp
*

*dt
* .

As the object is moved toward the lens, p is decreasing, so *dp/dt* < 0. Consequently, the

above expression shows that *d*|*i*|/*dt* < 0; that is, the image moves in from infinity. The

angular magnification *m*θ = θ*' /*θ also increases as the following graph shows (“read” the
graph from left to right since we are considering decreasing *p *from near the focal length

to near 0). To obtain this graph of *m*θ, we chose *f* = 30 cm and *h* = 2 cm.

(b) When the image appears to be at the near point (that is, |*i*| = *Pn*), *m*θ is at its maximum

usable value. The textbook states in section 34-8 that it generally takes *Pn* to be equal to

25 cm (this value, too, was used in making the above graph).

(c) In this case,

*p *= *if/*(*i – f*) = |*i|f/*(*|i| + f*)* = Pn f /*(*Pn + f*).

If we use the small angle approximation, we have θ*'*≈ *h'*/|*i|* and θ≈ *h*/*Pn* (note: this
approximation was not used in obtaining the graph, above). We therefore find *m*θ ≈
(*h'*/|*i|*)/(*h*/*Pn*) which (using Eq. 34-7 relating the ratio of heights to the ratio of distances)

becomes

* m*θ ≈ (*h'*/*h*)(*Pn */ |*i|*) = (|*i|*/*p*)(*Pn */ |*i|*) = (*Pn */ *p*) = [*Pn */ (*Pn f /*(*Pn + f*))] = *Pn + f
f
*

which readily simplifies to the desired result.

(d) The linear magnification (Eq. 34-7) is given by (*h'*/*h*) ≈* m*θ (|*i|*/* Pn*) (see the first in the
chain of equalities, above). Once we set |*i| = Pn* (see part (b)) then this shows the

equality in the magnifications.

110. (a) Suppose one end of the object is a distance *p* from the mirror and the other end is

a distance *p + L*. The position *i*1 of the image of the first end is given by

1 1 1

1*p i f
*+ =

where *f* is the focal length of the mirror. Thus,

*i
f
*

*p f
*

*p
*

1 = − .

The image of the other end is located at

*i
f p L
*

*p L f
*2 =

+ + − b g

,

so the length of the image is

′ = − = −

− +

+ − =

− + −
*L i i
*

*fp
*

*p f
*

*f p L
*

*p L f
*

*f L
*

*p f p L f
*1 2

2b g b gb g .

Since the object is short compared to *p – f*, we may neglect the *L* in the denominator and

write

′ = −

F HG

I
KJ*L L
*

*f
*

*p f
*

2

.

(b) The lateral magnification is *m = –i*/*p* and since *i = fp*/(*p – f *), this can be written *m
*

*= –f*/(*p – f *). The longitudinal magnification is

′ = ′ = −

F HG

I
KJ =*m
*

*L
*

*L
*

*f
*

*p f
m
*

2

2 .

111. (a) In this case *m* > +1 and we know that lens 1 is converging (producing a virtual

image), so that our result for focal length should be positive. Since

*|P + i*1*|* = 20 cm and *i*1 = – 2*p*1, we find *p*1 = 20 cm and *i*1 = – 40 cm. Substituting these

into Eq. 34-9,

1 1 1

1 1 1

*p i f
*+ =

leads to *f*1 = +40 cm, which is positive as we expected.

(b) The object distance is *p*1 = 20 cm, as shown in part (a).

(c) In this case 0 < *m* < 1 and we know that lens 2 is diverging (producing a virtual

image), so that our result for focal length should be negative. Since *|p + i*2*|* = 20 cm and

*i*2* = – p*2/2, we find *p*2 = 40 cm and *i*2 = – 20 cm. Substituting these into Eq. 34-9 leads to

*f*2 = – 40 cm, which is negative as we expected.

(d) *p*2 = 40 cm, as shown in part (c).

112. The water is medium 1, so *n*1 = *nw* which we simply write as *n*. The air is medium 2,

for which *n*2 ≈ 1. We refer to points where the light rays strike the water surface as *A* (on
the left side of Fig. 34-52) and *B* (on the right side of the picture). The point midway

between *A* and *B* (the center point in the picture) is *C*. The penny *P* is directly below *C*,

and the location of the “apparent” or Virtual penny is *V*. We note that the angle ∠*CVB *
(the same as ∠*CVA *) is equal to θ2, and the angle ∠*CPB * (the same as ∠*CPA *) is equal to
θ1. The triangles *CVB* and *CPB* share a common side, the horizontal distance from *C* to *B*
(which we refer to as *x*). Therefore,

tan .θ θ2 = =
*x
*

*d
*

*x
*

*d*a
1and tan

Using the small angle approximation (so a ratio of tangents is nearly equal to a ratio of

sines) and the law of refraction, we obtain

2 2 1

1 1 2 a

tan sin

tan sin

*a
*

*x
*

*d n d
n
*

*x n d
d
*

θ θ θ θ

≈ ≈ ≈

which yields the desired relation: *d*a = *d*/*n*.

113. A converging lens has a positive-valued focal length, so *f*1 = +6 cm, *f*2 = +3 cm, and

*f*3 = +3 cm. We use Eq. 34-9 for each lens separately, “bridging the gap” between the

results of one calculation and the next with *p*2 = *d*12 – *i*1 and *p*3 = *d*23 – *i*2. We also use

Eq. 34-7 for each magnification (*m*1 etc), and *m* = *m*1 *m*2 *m*3 (a generalized version of Eq.

34-11) for the net magnification of the system. Our intermediate results for image

distances are *i*1 = 9 cm and *i*2 = 6 cm. Our final results are as follows:

(a) *i*3 = +7.5 cm.

(b) *m* = −0.75.

(c) The image is real (R).

(d) The image is inverted (I).

(e) It is on the opposite side of lens 3 from the object (which is expected for a real final

image).

114. A converging lens has a positive-valued focal length, so *f*1 = +6 cm, *f*2 = +6 cm, and

*f*3 = +5 cm. We use Eq. 34-9 for each lens separately, “bridging the gap” between the

results of one calculation and the next with *p*2 = *d*12 – *i*1 and *p*3 = *d*23 – *i*2. We also use

Eq. 34-7 for each magnification (*m*1 etc), and *m* = *m*1 *m*2 *m*3 (a generalized version of Eq.

34-11) for the net magnification of the system. Our intermediate results for image

distances are *i*1 = –3 cm and *i*2 = 9 cm. Our final results are as follows:

(a) *i*3 = +10 cm.

(b) *m* = +0.75.

(c) The image is real (R).

(d) The image is not inverted (NI).

(e) It is on the opposite side of lens 3 from the object (which is expected for a real final

image).

115. A converging lens has a positive-valued focal length, so *f*1 = +8 cm, *f*2 = +6 cm, and

*f*3 = +6 cm. We use Eq. 34-9 for each lens separately, “bridging the gap” between the

results of one calculation and the next with *p*2 = *d*12 – *i*1 and *p*3 = *d*23 – *i*2. We also use

Eq. 34-7 for each magnification (*m*1 etc), and *m* = *m*1 *m*2 *m*3 (a generalized version of Eq.

34-11) for the net magnification of the system. Our intermediate results for image

distances are *i*1 = 24 cm and *i*2 = –12 cm. Our final results are as follows:

(a) *i*3 = +8.6 cm.

(b) *m* = +2.6.

(c) The image is real (R).

(d) The image is not inverted (NI)

(e) It is on the opposite side of lens 3 from the object (which is expected for a real final

image).

116. A converging lens has a positive-valued focal length, and a diverging lens has a

negative-valued focal length. Therefore, *f*1 = – 6 cm, *f*2 = +6 cm, and *f*3 = +4 cm. We

use Eq. 34-9 for each lens separately, “bridging the gap” between the results of one

calculation and the next with *p*2 = *d*12 – *i*1 and *p*3 = *d*23 – *i*2. We also use Eq. 34-7 for

each magnification (*m*1 etc), and *m* = *m*1 *m*2 *m*3 (a generalized version of Eq. 34-11) for

the net magnification of the system. Our intermediate results for image distances are *i*1

= –2.4 cm and *i*2 = 12 cm. Our final results are as follows:

(a) *i*3 = – 4.0 cm.

(b) *m* = −1.2.

(c) The image is virtual (V).

(d) The image is inverted (I).

(e) It is on the same side as the object (relative to lens 3) as expected for a virtual image.

117. A converging lens has a positive-valued focal length, and a diverging lens has a

negative-valued focal length. Therefore, *f*1 = – 8.0 cm, *f*2 = – 16 cm, and *f*3 = +8.0 cm.

We use Eq. 34-9 for each lens separately, “bridging the gap” between the results of one

calculation and the next with *p*2 = *d*12 – *i*1 and *p*3 = *d*23 – *i*2. We also use Eq. 34-7 for

each magnification (*m*1 etc), and *m* = *m*1 *m*2 *m*3 (a generalized version of Eq. 34-11) for

the net magnification of the system. Our intermediate results for image distances are *i*1

= –4.0 cm and *i*2 = –6.86 cm. Our final results are as follows:

(a) *i*3 = +24.2 cm.

(b) *m* = −0.58.

(c) The image is real (R).

(d) The image is inverted (I).

(e) It is on the opposite side of lens 3 from the object (as expected for a real image).

118. A converging lens has a positive-valued focal length, and a diverging lens has a

negative-valued focal length. Therefore, *f*1 = +6 cm, *f*2 = − 4 cm, and *f*3 = −12 cm . We
use Eq. 34-9 for each lens separately, “bridging the gap” between the results of one

calculation and the next with *p*2 = *d*12 – *i*1 and *p*3 = *d*23 – *i*2. We also use Eq. 34-7 for

each magnification (*m*1 etc), and *m* = *m*1 *m*2 *m*3 (a generalized version of Eq. 34-11) for

the net magnification of the system. Our intermediate results for image distances are *i*1

= –12 cm and *i*2 = –3.33 cm. Our final results are as follows:

(a) *i*3 = – 5.15 cm ≈– 5.2 cm .

(b) *m* = +0.285 ≈ +0.29.

(c) The image is virtual (V).

(d) The image is not inverted (NI).

(e) It is on the same side as the object (relative to lens 3) as expected for a virtual image.

119. (a) The discussion in the textbook of the refracting telescope (a subsection of §34-8)

applies to the Newtonian arrangement if we replace the objective lens of Fig. 34-19 with

an objective mirror (with the light incident on it from the right). This might suggest that

the incident light would be blocked by the person’s head in Fig. 34-19, which is why

Newton added the mirror *M'* in his design (to move the head and eyepiece out of the way

of the incoming light). The beauty of the idea of characterizing both lenses and mirrors

by focal lengths is that it is easy, in a case like this, to simply carry over the results of the

objective-lens telescope to the objective-mirror telescope, so long as we replace a positive

*f* device with another positive *f* device. Thus, the converging lens serving as the objective

of Fig. 34-19 must be replaced (as Newton has done in Fig. 34-54) with a concave mirror.

With this change of language, the discussion in the textbook leading up to Eq. 34-15

applies equally as well to the Newtonian telescope: *m*θ* = – f*ob/*f*ey.

(b) A meter stick (held perpendicular to the line of sight) at a distance of 2000 m subtends

an angle of

θ stick m

2000 m rad.≈ =1 0 0005.

multiplying this by the mirror focal length gives (16.8 m) (0.0005) = 8.4 mm for the size

of the image.

(c) With *r* = 10 m, Eq. 34-3 gives *f*ob = 5 m. Plugging this into (the absolute value of) Eq.

34-15 leads to *f*ey = 5/200 = 2.5 cm.

120. Consider a single ray from the source to the mirror and let θ be the angle of incidence. The angle of reflection is also θ and the reflected ray makes an angle of 2θ with the incident ray.

Now we rotate the mirror through the angle α so that the angle of incidence increases to θ + α. The reflected ray now makes an angle of 2(θ + α) with the incident ray. The reflected ray has been rotated through an angle of 2α. If the mirror is rotated so the angle of incidence is decreased by α, then the reflected ray makes an angle of 2(θ – α) with the incident ray. Again it has been rotated through 2α. The diagrams below show the situation for α = 45°. The ray from the object to the mirror is the same in both cases and the reflected rays are 90° apart.

121. (a) If we let *p* → ∞ in Eq. 34-8, we get *i* = *n*2 *r* /(*n*2 – *n*1). If we set *n*1 = 1 (for air)
and restrict *n*2 so that 1 < *n*2 < 2, then this suggests that *i* > 2*r* (so this image does form

before the rays strike the opposite side of the sphere). We can still consider this as a sort

of “virtual” object for the second imaging event, where this “virtual” object distance is

2*r* – *i* = (*n* – 2) *r* /(*n* – 1), where we have simplified the notation by writing *n*2 = *n*.

Putting this in for *p *in Eq. 34-8 and being careful with the sign convention for *r* in that

equation, we arrive at the final image location: *i*′ = (0.5)(2 – *n*)*r*/(*n* – 1).

(b) The image is to the right of the right side of the sphere.

122. Setting *n*air = 1, *n*water = *n*, and *p *= *r*/2 in Eq. 34-8 (and being careful with the sign

convention for *r* in that equation), we obtain *i* = –*r*/(1 + *n*), or |*i*| = *r*/(1 + *n*). Then we use

similar triangles (where *h* is the size of the fish and *h*′ is that of the “virtual fish”) to set
up the ratio

*h*′
*r* – |*i*|

=
*h
*

*r/*2
.

Using our previous result for |*i*|, this gives *h*′*/h = *2(1 – 1/(1 + *n*)) = 1.14.

123. (a) Our first step is to form the image from the first lens. With *p*1 = 10 cm and

1 15 cm*f *= − , Eq. 34-9 leads to

1

1 1 1

1 1 1
6.0cm.*i
*

*p i f
*+ = = −

The corresponding magnification is *m*1 = –*i*1/*p*1 = 0.60. This image serves the role of

“object” for the second lens, with *p*2 = 12 + 6.0 = 18 cm, and *f*2 = 12 cm. Now, Eq. 34-9

leads to

2

2 2 2

1 1 1
36 cm*i
*

*p i f
*+ = = .

(b) The corresponding magnification is *m*2 = –*i*2/*p*2 = –2.0, which results in a net

magnification of *m = m*1*m*2 = –1.2. The height of the final image is (in absolute value)

(1.2)(1.0 cm) = 1.2 cm.

(c) The fact that *i*2 is positive means that the final image is real.

(d) The fact that *m* is negative means that the orientation of the final image is inverted

with respect to the (original) object.

124. (a) Without the diverging lens (lens 2), the real image formed by the converging lens

(lens 1) is located at a distance

*i
f p
*

1

1 1

1 1

1 1 1

20

1

40 40= −

F HG

I KJ = −

F HG

I KJ =

− −

cm cm cm

to the right of lens 1. This image now serves as an object for lens 2, with *p*2 = –(40 cm –

10 cm) = –30 cm. So

*i
f p
*

2

2 2

1 1

1 1 1

15

1

30 30= −

F HG

I KJ = − − −

F HG

I KJ = −

− −

cm cm cm.

Thus, the image formed by lens 2 is located 30 cm to the left of lens 2.

(b) The magnification is *m* = (–*i*1/*p*1) × (–*i*2/*p*2) = +1.0 > 0, so the image is not inverted.

(c) The image is virtual since *i*2 < 0.

(d) The magnification is *m* = (–*i*1/*p*1) × (–*i*2/*p*2) = +1.0, so the image has the same size as
the object.

125. (a) For the image formed by the first lens

*i
f p
*

1

1 1

1 1

1 1 1

10

1

20 20= −

F HG

I KJ = −

F HG

I KJ =

− −

cm cm cm.

For the subsequent image formed by the second lens *p*2 = 30 cm – 20 cm = 10 cm, so

*i
f p
*

2

2 2

1 1

1 1 1

12 5

1

10 50= −

F HG

I KJ = −

F HG

I KJ = −

− −

. cm cm cm.

Thus, the final image is 50 cm to the left of the second lens, which means that it coincides

with the object.

(b) The magnification is

*m
i
*

*p
*

*i
*

*p
*=
F
HG
I
KJ
F
HG
I
KJ =
F
HG

I KJ

−F HG

I KJ = −

1

1

2

2

20 50 50

cm

20cm

cm

10cm . ,

which means that the final image is five times larger than the original object.

(c) The image is virtual since *i*2 < 0.

(d) The image is inverted since *m* < 0.

126. (a) We solve Eq. 34-9 for the image distance *i: i = pf*/(*p – f *). The lens is diverging,

so its focal length is *f* = –30 cm. The object distance is *p* = 20 cm. Thus,

*i *=
−

− − = −

20 30

20 30 12

cm cm

cm cm cm.

b gb g b g b g

The negative sign indicates that the image is virtual and is on the same side of the lens as

the object.

(b) The ray diagram, drawn to scale, is shown below.

127. We set up an *xyz* coordinate system where the individual planes (*xy, yz, xz*) serve as

the mirror surfaces. Suppose an incident ray of light *A* first strikes the mirror in the *xy*

plane. If the unit vector denoting the direction of *A* is given by

cos(α)i^ + cos(β)j^ + cos(γ)k^

where α, β, γ are the angles *A* makes with the axes, then after reflection off the *xy* plane
the unit vector becomes cos(α)i^ + cos(β)j^ – cos(γ)k^ (one way to rationalize this is to
think of the reflection as causing the angle γ to become π − γ). Next suppose it strikes
the mirror in the *xz* plane. The unit vector of the reflected ray is now cos(α)i^ – cos(β)j^ –
cos(γ)k^ . Finally as it reflects off the mirror in the *yz* plane α becomes π − α, so the unit
vector in the direction of the reflected ray is given by – cos(α)i^ – cos(β)j^ – cos(γ)k^ ,
exactly reversed from *A*’s original direction. A further observation may be made: this

argument would fail if the ray could strike any given surface twice and some

consideration (perhaps an illustration) should convince the student that such an

occurrence is not possible.

128. The fact that it is inverted implies *m* < 0. Therefore, with *m* = –1/2, we have *i = p*/2,

which we substitute into Eq. 34-4:

1 1 1

1 2 1

3

30 0

1

*p i f
*

*p p f
*

*f
*

+ =

+ =

= .

with the unit cm understood. Consequently, we find *f* = 30/3 = 10.0 cm. The fact that *f* >

0 implies the mirror is concave.

129. Since *m* = –2 and *p* = 4.00 cm, then *i* = 8.00 cm (and is real). Eq. 34-9 is

1 1 1

*p i f
*+ =

and leads to *f* = 2.67 cm (which is positive, as it must be for a converging lens).

130. (a) The mirror has focal length *f* = 12.0 cm. With *m* = +3, we have *i* = –3*p*. We

substitute this into Eq. 34-4:

1 1 1

1 1

3

1

12

2

3

1

12

*p i f
*

*p p
*

*p
*

+ =

+ −

=

=

with the unit cm understood. Consequently, we find *p* = 2(12)/3 = 8.0 cm.

(b) With *m* = –3, we have *i* = +3*p*, which we substitute into Eq. 34-4:

1 1 1

1 1

3

1

12

4

3

1

12

*p i f
*

*p p
*

*p
*

+ =

+ =

=

with the unit cm understood. Consequently, we find *p* = 4(12)/3 = 16 cm.

(c) With *m* = –1/3, we have *i = p*/3. Thus, Eq. 34-4 leads to

1 1 1

1 3 1

12

4 1

12

*p i f
*

*p p
*

*p
*

+ =

+ =

=

with the unit cm understood. Consequently, we find *p* = 4(12) = 48 cm.

131. (a) Since *m* = +0.200, we have *i* = –0.2*p* which indicates that the image is virtual (as

well as being diminished in size). We conclude from this that the mirror is convex (and

that *f* = –40.0 cm).

(b) Substituting *i = –p*/5 into Eq. 34-4 produces

1 5 4 1

*p p p f
*− = − = .

Therefore, we find *p* = 160 cm.

132. Since 0 < *m* < 1, we conclude the lens is of the diverging type (so *f* = –40 cm). Thus,

substituting *i* = –3*p*/10 into Eq. 34-9 produces

1 10

3

7

3

1

*p p p f
*− = − = .

Therefore, we find *p* = 93.3 cm and *i* = –28.0 cm, or | *i* | = 28.0 cm.

133. (a) Our first step is to form the image from the first lens. With *p*1 = 3.00 cm and *f*1 =

+4.00 cm, Eq. 34-9 leads to

1

1 1 1

1 1 1
12.0cm.*i
*

*p i f
*+ = = −

The corresponding magnification is *m*1 = –*i*1/*p*1 = 4. This image serves the role of

“object” for the second lens, with *p*2 = 8.00 + 12.0 = 20.0 cm, and *f*2 = –4.00 cm. Now,

Eq. 34-9 leads to

2

2 2 2

1 1 1
3.33 cm*i
*

*p i f
*+ = = − .

(b) The fact that *i*2 is negative means that the final image is virtual (and therefore to the

left of the second lens).

(c) The image is virtual.

(d) With *m*2 = –*i*2/*p*2 = 1/6, the net magnification is *m = m*1*m*2 = 2/3 > 0. The fact that *m* is

positive means that the orientation of the final image is the same as the (original) object.

Therefore, the image is not inverted.

134. (a) Our first step is to form the image from the first lens. With *p*1 = 4.00 cm and *f*1

= –4.00 cm, Eq. 34-9 leads to

1

1 1 1

1 1 1
2.00cm.*i
*

*p i f
*+ = = −

The corresponding magnification is *m*1 = –*i*1/*p*1 = 1/2. This image serves the role of

“object” for the second lens, with *p*2 = 10.0 + 2.00 = 12.0 cm, and *f*2 = –4.00 cm. Now,

Eq. 34-9 leads to

2

2 2 2

1 1 1
3.00 cm*i
*

*p i f
*+ = = − ,

or 2| | 3.00 cm*i *= .

(b) The fact that *i*2 is negative means that the final image is virtual (and therefore to the

left of the second lens).

(c) The image is virtual.

(d) With *m*2 = –*i*2/*p*2 = 1/4, the net magnification is *m = m*1*m*2 = 1/8 > 0. The fact that *m* is

positive means that the orientation of the final image is the same as the (original) object.

Therefore, the image is not inverted.

135. Of course, the shortest possible path between *A* and *B* is the straight line path which

does not go to the mirror at all. In this problem, we are concerned with only those paths

which do strike the mirror. The problem statement suggests that we turn our attention to

the mirror-image point of *A* (call it *A' *) and requests that we construct a proof without

calculus. We can see that the length of any line segment *AP* drawn from *A* to the mirror

(at point *P* on the mirror surface) is the same as the length of its “mirror segment” *A'P*

drawn from *A'* to that point *P*. Thus, the total length of the light path from *A* to *P* to *B* is

the same as the total length of segments drawn from *A'* to *P* to *B*. Now, we dismissed (in

the first sentence of this solution) the possibility of a straight line path directly from *A* to

*B* because it does not strike the mirror. However, we *can* construct a straight line path

from *A'* to *B* which does intersect the mirror surface! Any other pair of segments (*A'P* and

*PB*) would give greater total length than the straight path (with *A'P* and *PB* collinear), so

if the straight path *A'B* obeys the law of reflection, then we have our proof. Now, since

*A'P* is the mirror-twin of *AP*, then they both approach the mirror surface with the same

angle α (one from the front side and the other from the back side). And since *A'P* is
collinear with *PB*, then *PB* also makes the same angle α with respect to the mirror surface
(by vertex angles). If *AP* and *PB* are each α degrees away from the front of the mirror,
then they are each θ degrees (where θ is the complement of α) measured from the normal
axis. Thus, the law of reflection is consistent with the concept of the shortest light path.

136. (a) Since a beam of parallel light will be focused at a distance *f* from the

(converging) lens, then the shorter the focal length *f* the greater the ability for the lens to

bend the light. A window pane is an example of a “lens” with *f* = ∞, yet it has essentially
zero bending ability. Therefore, *P=1/f *is a reasonable definition.

(b) First we must consider the two-lens situation in the limit that *d* (their separation)

becomes vanishingly small. We place an object far away from the composite lens and

find the image distance *i*. Since the image is at a focal point *i = f*, the effective focal

length of the composite can be determined in this way. The final image is produced by

two lenses, with the image of the first lens being the object for the second. For the first

lens we have 1/*p*1 + 1/*i*1 = 1/*f*1 , where *f*1 is the focal length of the first lens. Since *p*1 =

∞, we find *i*1 = *f*1. The thin lens equation, applied to the second lens, gives

*i*2 = *i* = *p*2* f*2 / (*p*2−* f*2),

where *p*2= *d* − *i*1* = *−*f*1 in this situation. Therefore, *i* (thought of as *f* for the equivalent
single lens [equivalent to the 2 lens system] as explained above) is equal to

−*f*1* f*2 / (−*f*1−* f*2) or
1

*f
* =

1

*f*1
+

1

*f*2
.

Next, using the definition for *P*, we readily get the desired result.

137. (a) Suppose that the lens is placed to the left of the mirror. The image formed by the

converging lens is located at a distance

*i
f p
*

= − F HG

I KJ = −

F HG

I KJ =

− − 1 1 1

0 50

1

10 10

1 1

. . .

m m m

to the right of the lens, or 2.0 m – 1.0 m = 1.0 m in front of the mirror. The image formed

by the mirror for this real image is then at 1.0 m to the right of the the mirror, or 2.0 m +

1.0 m = 3.0 m to the right of the lens. This image then results in another image formed by

the lens, located at a distance

11

1 1 1 1 0.60m

0.50m 3.0m
*i
*

*f p
*

−−

′ = − = − = ′

to the left of the lens (that is, 2.6 cm from the mirror).

(b) The lateral magnification is

*m
i
*

*p
*

*i
*

*p
*= −
F
HG
I
KJ −

′ ′

F HG

I KJ = − F HG

I KJ − F HG

I KJ = +

10

10

0 60

30 0 20

.

.

.

. . .

m

m

m

m

(c) The final image is real since *i'* > 0.

(d) The image is to the left of the lens.

(e) It also has the same orientation as the object since *m* > 0. Therefore, the image is not

inverted.

138. (a) Since *m* = +0.250, we have *i* = – 0.25*p* which indicates that the image is virtual

(as well as being diminished in size). We conclude from this that the mirror is convex and

that *f* < 0; in fact, *f* = – 2.00 cm. Substituting *i = – p*/4 into Eq. 34-4 produces

1 4 3 1

*p p p f
*− = − =

Therefore, we find *p* = 6.00 cm and *i* = – 1.50 cm, or | | 1.50 cm*i *= .

(b) The focal length is negative.

(c) As shown in (a), the image is virtual.

139. First, we note that — *relative to the water* — the index of refraction of the carbon

tetrachloride should be thought of as *n* = 1.46/1.33 = 1.1 (this notation is chosen to be

consistent with problem 15). Now, if the observer were in the water, directly above the 40

mm deep carbon tetrachloride layer, then the apparent depth of the penny as measured

below the surface of the carbon tetrachloride is *d*a = 40 mm/1.1 = 36.4 mm. This

“apparent penny” serves as an “object” for the rays propagating upward through the 20

mm layer of water, where this “object” should be thought of as being 20 mm + 36.4 mm

= 56.4 mm from the top surface. Using the result of problem 15 again, we find the

perceived location of the penny, for a person at the normal viewing position above the

water, to be 56.4 mm/1.33 = 42 mm below the water surface.

140. (a) We show the α = 0.500 rad, *r* =12 cm, *p *= 20 cm calculation in detail. The
understood length unit is the centimeter:

The distance from the object to point *x*: *d* = *p – r + x* = 8 + *x*

*y* = *d *tan α = 4.3704 + 0.54630*x*

From the solution of *x
*2
+ *y
*

2
= *r
*

2
we get *x* = 8.1398.

β = tan−1(*y/x*) = 0.8253 rad
γ = 2 β − α = 1.151 rad

From the solution of tan(γ)= *y/*(*x + i *−* r*) we get *i* = 7.799 . The other results are shown
without the intermediate steps:

For α = 0.100 rad, we get *i* = 8.544 cm; for α = 0.0100 rad, we get *i* = 8.571 cm. Eq. 34-
3 and Eq. 34-4 (the mirror equation) yield *i* = 8.571 cm.

(b) Here the results are: (α = 0.500 rad, *i* = −13.56 cm), (α = 0.100 rad, *i* = −12.05 cm),
(α = 0.0100 rad, *i* = −12.00 cm). The mirror equation gives *i* = −12.00 cm.