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February 23 , 2005 Physics for Scientists&Engineers 2 1
Physics for Scientists &Physics for Scientists &
Engineers Engineers 22
Spring Semester 2005
Lecture 25
February 23 , 2005 Physics for Scientists&Engineers 2 2 InductionInduction ! Last week we learned that a current-carrying loop in a magnetic field experiences a torque ! If we start with a loop with no current in a magnetic field, and force the loop to rotate, we find that a current is induced in the loop ! Further, if we start with a loop with no current and turn on a magnetic field without moving the coil, again a current is induced in the loop ! These effects are described by Faraday’s Law of Induction and are the basis of electric motors and electric power generation from mechanical motion Faraday Faraday’’ss (^) ExperimentsExperiments ! Consider the situation in which we have a wire loop connected to an ammeter so that we can measure current flowing in the loop ! We hold a bar magnet some distance from the loop, pointing the north pole of the magnet toward the loop ! While the magnet is stationary, there is no current flowing in the loop ! What happens if we move the magnet? FaradayFaraday’’ss Experiments (2)Experiments (2)
! When we move the magnet toward the loop, we induce a
positive current in the loop
! Now we turn the magnet around so that the south pole
points toward the loop
! When we move the magnet toward the loop, we induce a
negative current in the loop
February 23 , 2005 Physics for Scientists&Engineers 2 5 FaradayFaraday’’ss Experiments (3)Experiments (3)
! Now let’s point the north pole toward the loop but move
away from the loop
- We get a negative current
! We turn the magnet around so that the south pole points
toward the loop and move away from the loop
- We get a positive current February 23 , 2005 Physics for Scientists&Engineers 2 6 FaradayFaraday’’ss Experiments (4)Experiments (4)
! We can create similar effects by placing a second loop near the
first loop but with a more quantitative result as shown below
! If a constant current is flowing through loop 1, no current will be
induced in loop 2
! If we increase the current in
the loop 1, we observe that a
current is induced in the
loop 2 in the opposite direction
! Thus not only does the
changing current in the first
loop induce a current in the
first loop, the induced current
is in the opposite direction
FaradayFaraday’’ss Experiments (5)Experiments (5) ! Now if we have the current flowing in loop 1 in the same direction as before, and decrease the current as shown below, we induce a current flowing loop 2 in the same direction as the current in loop 1 Law of Induction Law of Induction ! From these observations we see that a changing magnetic field induces a current in a loop ! We can visualize the change in magnetic field as a change in the number of magnetic field lines passing through the loop ! Faraday’s Law of Induction states that:
- An emf is induced in a loop when the number of magnetic
field lines passing through the loop changes with time
! The rate of change of magnetic field lines determines the induced emf
February 23 , 2005 Physics for Scientists&Engineers 2 13 Induction in a Flat Loop - Special CasesInduction in a Flat Loop - Special Cases
! If we leave two of the three variables ( A, B, !) constant,
then we can have the following three special cases
- We leave the area of the loop and its orientation relative to the
magnetic field constant, but vary the magnetic field in time
- We leave the magnetic field as well as the orientation of the loop
relative to the magnetic field constant, but change the area of the
loop that is exposed to the magnetic field
- We leave the magnetic field constant and keep the area of the loop
fixed as well, but allow the angle between the two to change as a
function of time
A ,! constant: Vemf = " A cos! dB
dt
B ,! constant: Vemf = " B cos!
dA
dt
A , B constant: Vemf =! AB sin "
February 23 , 2005 Physics for Scientists&Engineers 2 14 Example: Changing Magnetic FieldExample: Changing Magnetic Field ! A direct current of 600 mA is delivered to an ideal solenoid, resulting in a magnetic field of 0.025 T ! Then the current is increased according to ! Question:
- If a circular loop of radius 3.4 cm with 200 windings is
located inside the solenoid and perpendicular to the
magnetic field, what is the induced voltage at t = 2.0 s in
this loop?
i ( t ) = i 0 ( 1 + 2.4s!^2 t^2 )
Example: Changing Magnetic Field (2) Example: Changing Magnetic Field (2) ! Answer:
- First, the area of the loop is computed (the number of
winding acts as a simple multiplier):
- The magnetic field inside an ideal solenoid is
- Because the magnetic field is linearly proportional to the
current, we obtain the time dependence of the magnetic
field in this case
A = N! R^2 = 200! (0.034 m)^2 = 0.73 m^2
B = μ 0 in
B ( t ) = B 0 ( 1 + 2.4s!^2 t^2 )
Example: Changing Magnetic Field (3) Example: Changing Magnetic Field (3) ! Answer:
- First, the area of the loop is computed (the number of
winding acts as a simple multiplier):
- The magnetic field inside an ideal solenoid is
- Because the magnetic field is linearly proportional to the
current, we obtain the time dependence of the magnetic
field in this case
A = N! R^2 = 200! (0.034 m)^2 = 0.73 m^2
B = μ 0 in
B ( t ) = B 0 ( 1 + 2.4s!^2 t^2 ), B 0 = 0.025 T
February 23 , 2005 Physics for Scientists&Engineers 2 17 Example: Changing Magnetic Field (4) Example: Changing Magnetic Field (4)
! The area of the loop and the angle are kept constant so
! For the induced voltage we then find
A ,! constant: Vemf = " A cos!
dB
dt
Vemf =! A cos "
dB
dt
=! A cos "
d
dt
( B 0 ( 1 + 2.4s!^2 t^2 ))
=! AB 0 cos "( 2 # 2.4s!^2 t )
= !(0.73 m^2 )(0.025 T)(cos 90 °)(4.8s!^2 ) t
at t = 2.0 s, Vemf = !0.17 V
February 23 , 2005 Physics for Scientists&Engineers 2 18 Example: Motion Voltage Example: Motion Voltage
! A rectangular loop of width 3.1 cm and depth 4.8 cm
is pulled out of the gap between two permanent
magnets, with a field of 0.073 T throughout the
gap.
! Question:
- If the loop is removed with a constant velocity of 1.
cm/s, what is the induced voltage in the loop as a
function of time?
! Answer:
- The magnetic field as well as the orientation of the loop relative to the
field remains constant
- What changes is the area of the loop that is exposed to the magnetic
field
Example: Motion Voltage (2)Example: Motion Voltage (2)
! With the narrow gap, there will be very little field
outside the gap
! The effective area of the loop exposed to the field
is A( t) = w"d( t)
! While the entire loop is still inside the gap, no
voltage is produced
! We select the time of arrival of the right edge of
the loop at the right edge of the gap as t = 0
! Then we find
! This formula holds until the left edge of the loop reaches the edge
of the gap, after which the area of the loop is zero
! The left edge arrives at
A ( t ) = w! d ( t ) = w! ( d " vt )
t f = d / v = 4.8 cm / 1.6 cm/s = 3.0 s
Example: Motion Voltage (3) Example: Motion Voltage (3)
! Now we can calculate the induced voltage
! So our result is that during the time interval between 0
and 3 s, a constant voltage of 36 μV is induced, and no
voltage is produced outside this time interval
Vemf =! B cos "
dA
dt
=! B cos "
d
dt
( w # ( d! vt ))
= wvB cos "
= (0.031 m)(0.016 m/s)(0.073 T)
=3.6 # 10 -^5 V
