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Problem Set: Electro Magnetic Theory
Sem-VI
Unit-I: Maxwell Equations
- A wave travelling in free space, with its E field vector is given by 𝐸 =
𝒐
𝑉/𝑚. Find D , B and H and draw E and H at t = 0. Show that the wave
speed depends only on the properties of medium.
- In a material for which 𝜎 = 0 , 𝜀 = 10 𝜀
𝑜
and 𝜇 = 𝜇
𝑜
displacement current density is
represented by 𝒊
10 cos
2 × 10
8
2
. (a) Use the definition of displacement
current density to find D and E. (b) Using faraday law find B and H. (c) Use Ampere’s
circuital law to find out displacement current density. What must β be?
- Show that current flowing between capacitor terminals is equal to the displacement current
for an air filled parallel plate capacitor which is made of circular plates each of radius 0.8 m.
The spacing between the discs is 0.1 m and a voltage of 10 cos( 10
3
t) Volts has been applied
across the capacitor plates.
- Assume μ = μ o,
show that the ratio of the amplitudes of the conduction current density and
displacement current density (a) is 𝜎 𝜔𝜀
for the applied field 𝑬 = 𝐸
𝑜
𝑐𝑜𝑠𝜔𝑡. (b) What is the
ratio of the amplitudes if the applied field is 𝑬 = 𝐸
𝑜
−𝑡/𝜏
, where 𝜏 is real?
- The electric field for a linearly polarized electromagnetic wave propagating in free space is
given by 𝑬 = 𝑛̂ 𝐸
𝑜
exp[𝑖(𝜔𝑡 − 4 𝑥 + 8 𝑦 − 8 𝑧)]. Where 𝒏̂ = [ 8 𝑥̂ + 2 𝑦̂ − 2 𝑧]
represents the
unit vector along E. Find (a) the wavelength and frequency of wave (b) unit vector along
the direction of propagation and (c) Check if the wave is transverse in nature?
- A wave of frequency 1MHz propagates in free space having electric field vector
100 cos(𝜔𝑡 − 𝛽𝑥)𝑉/𝑚. Find out wave propagation constant, intrinsic
impedance, time domain of field vectors and average power density vector. Also determine
the total power crossing through an area of 100 cm
2
of a plane 2 x + y = 5.
- A radio station radiates power of 10
6
watts uniformly over a hemisphere. Find the
magnitude of Poynting vector and amplitude of electric and magnetic fields at a point of 20
km from radio station.
- On a bright day a square meter of earth surface near equator receives about 1400 W/m of
solar radiation. What are the electric and magnetic field strengths? What is the radiation
pressure on a completely reflecting surface placed normally?
- The field vectors of a wave of frequency 200 MHz propagating in free space are given by
4 𝜋
3
V/m, 𝑯 =
120
120 𝜋
4 𝜋
3
𝑥) 𝑗̂ A/m. Determine the
direction of power flow and the average power crossing the surface bounded by y = 2 m, y =
0, z = 2 m, and z = 0.
- Show that Lorentz force formula in term of electromagnetic potentials is
F = q[−𝛻(ф − 𝑣. 𝐴) −
𝑑𝐴
𝑑𝑡
], where A and ф are electromagnetic potentials, v is the velocity
of charged particle and q is charge on the particle.
UNIT-II: E.M. Wave Propagation in Unbounded Media
- The electric field intensity E associated with a plane electromagnetic wave travelling in a
medium characterized by μ r
= 1, ε r
= 4, σ = 0 is 𝑬
𝒙
= 100 𝐶𝑜𝑠( 2 𝜋 × 10
6
𝑡– 𝛽𝑧)V/m.
Compute (a) wave propagation constant, (b) wavelength, (c) phase velocity (d) intrinsic
impedance, (d) also write the time-domain expression for the field vectors of forward
travelling wave.
- Find the electric and magnetic field intensities for a 100-V/m plane wave of frequency
150 MHz travelling in the positive z direction in a medium having 𝜖
𝑟
𝑟
and σ = 0.
- Given the formula
2
𝜔
𝑝
2
𝜔
2
, where, 𝜔
𝑝
2
0
Determine the wavelength at which a certain ionized medium will become transparent if the
number of free electrons per unit volume in the medium is 5.632 × 10
28
m
- Given that the F-layer of the ionosphere has electron density 10
11
/m
3
, what is the maximum
frequency which is reflected from the layer if (i) launched vertically, (ii) launched at an angle
of 7 0° from the horizon?
Unit-III: EM Wave Propagation in Bounded Media
- The half space Z > 0 ia occupied by air with μ o
= π×
H/m and ε o
=9×
F/m. The other
half space Z < 0 is occupied by aferfect dielectric whose parameters are given by μ 2=
μ o
and
ε 2
=3 ε o
. A plane electomagnetic wave eith electric field vector E normal to the plane of
incidence in incident normally on the interface from the half space Z > 0. If the angle of
incidence is 60
o
, find (a) angle of refraction ɵ t
, and (b) the reflection and transmission
coefficients.
- A electromagnetic wave polarised linearly in the plane of incidence is incident at 40
o
on air
glass interface. The refractive index of glass plate is 1.5. Find the ratio of (a) amplitude of
reflected and incident wave and, (b) amplitude of transmitted and incident wave.
- A perpendicularly polarized light propagates from region 1, having parameters
ε r
= 8.5, μ r 1
= 1 and σ 1
= 0 to region 2, (free space) with an angle of incidence of
o
. Given 𝐸
𝑜
𝑖
= 1. 0 × 10
− 6
𝑉/𝑚, find 𝐸
𝑜
𝑟
𝑜
𝑡
𝑜
𝑖
𝑜
𝑟
and 𝐻
𝑜
𝑡
. All symbols have their
usual meaning.
- For a y polarised e.m. wave incident normally at the interface of two media
characteristics by intrinsic impedances η 1
and η 2
. The electric field associated with incident
(E
i
), reflected (E r
) and transmitted (E t
) are given by
𝑖
𝑖𝑜
𝑗
( 𝜔𝑡−𝑘 1
𝑥
)
𝑟
𝑟𝑜
𝑗
( 𝜔𝑡+𝑘 1
𝑥
) ,
𝑡
𝑡𝑜
𝑗
( 𝜔𝑡−𝑘 2
𝑥
)
(a) Obtain the corresponding magnetic field. (b) Show that the ratio of amplitude of (i)
incident and reflected (ii) incident and transmitted waves are given by
𝐸 𝑜𝑟
𝐸
𝑜𝑖
𝜂 2
−𝜂 1
𝜂 2
+𝜂 1
𝐸 𝑜𝑡
𝐸
𝑜𝑖
2 𝜂 2
𝜂 2
+𝜂 1
- Write complete time – domain expressions for incident, reflected, and transmitted electric and
magnetic fields for a plane wave whose electric field vector is given by
𝒊
= 𝒊̂ 300 cos(𝜔𝑡 − 3 𝜋𝑧) a x
passes normally from a material having 𝜇
𝑟 1
𝑟 1
= 4 and
𝜎 = 0 to a material 𝜇
𝑟 2
𝑟 2
= 9 and and 𝜎 = 0.
- A 20 MHz uniform plane wave in vacuum is incident normally on a copper sheet
( σ = 2.9×
7
/m). Most of the power is reflected, but a little propagates into copper sheet.
How far from the surface will the wave penetrated into the copper before 99 % of its power is
dissipated?
- An electromagnetic wave whose electric field is polarized parallel to plane of incidence, is
incident from free space to non-magnetic, non-conducting medium having ε = 3 ε 0
. Here the
wave is not reflected back from the interface. Determine the angle of transmission.
- A uniform plane wave is incident on planar boundary separating regions 1 and 2,
if σ 1
=σ 2
= 0 and μ r
=μ r
= 1. Find the ratio of ε r 2
/ε r 1
, if 20 % of energy in the incident wave is
(a) reflected and (b) transmitted into region. (assume normal incidence)
- A 2 GHz uniform plane wave travelling along z direction is incident from dielectric medium
1, Z < 0, onto other dielectric medium 2, Z > 0. Here boundary between two perfect
nonmagnetic dielectrics is located at z = 0. The wavelengths in the dielectrics are λ 1
= 8 cm
and λ 2
= 6 cm. What percentage of the incident energy on the boundary is (a) reflected and (b)
transmitted?
- A light source is placed at a depth of d=2.5m in a liquid tank and used to illuminate the
surface of the tank. Determine the area of brightness, as viewed from above the surface. The
relative permittivity of liquid at optical frequency is ε r
5 Determine the state of polarization when the x and y components of electric field are given
by the following equations : (a) 𝐸
𝑥
𝑜
cos (𝜔𝑡 + 𝑘𝑧) ,
𝑦
1
√ 2
E
o
cos (𝜔𝑡 + 𝑘𝑧 + 𝜋) (b) 𝐸
𝑥
𝑜
sin(𝜔𝑡 + 𝑘𝑧) , 𝐸
𝑦
= E
o
cos (𝜔𝑡 + 𝑘𝑧) (c) 𝐸
𝑥
𝑜
sin(𝑘𝑧 − 𝜔𝑡), 𝐸
𝑦
= −E
o
sin(𝑘𝑧 − 𝜔𝑡) (d) 𝐸
𝑥
𝑜
sin(𝑘𝑧 − 𝜔𝑡 +
𝜋
4
𝑦
1
√ 2
E
o
sin(𝑘𝑧 − 𝜔𝑡). Plot the rotation of the tip of electric field vector on the plane Z= 0.
6 We have a calcite quarter - wave plate corresponding to λ=4069 Å .For this plate values of
𝑜
𝑒
are 1.78138 and 1.59698 respectively and corresponding to λ=7065 Å the values
are 1.75209 and 1.58861 respectively. A left-circularly polarized beam of λ=7065 Å is
incident on this quarter wave plate. Obtain the state of polarization of the emergent wave
7 A half wave plate (HWP) is introduced between two crossed Polaroids P 1
and P 2
. The
optics axis makes an angle of 15° with the pass axis of P 1
as shown in Fig. 1(a) and (b). If
the unpolarized beam of intensity I o
is normally incident on P 1
and if I 1
, I
2
and I 3
are the
intensities after P 1
, after HWP, and after P 2
, respectively, then calculate I 1
,/ I
0,
I
2
,/ I
0
and I 3
,/ I
0
8 What will be the Brewester angle for a glass slab (n=1.5) immersed in a liquid (n= 1.2).
9 Consider the two crossed polaroid placed in a path of an unpolarised beam of intensity I o
. If
a third polaroid is placed in between the two. (Assume pass axis of third polaroid to be at
o
to the pass axis of either of the polaroids). Calculate the intensity of the transmitted
beam. Assume all polaroids are perfect.
10 A solution of camphor in alcohol in a tube of 25 cm in length containing 50 cm
3
of solution
is found to rotate the plane of vibration of light 10
o
. What is the mass of camphor in unit
volume of solution? The specific rotation of camphor is 66
o
per decimeter for unit
concentration. Calculate the quantity of camphor in the tube contains solution.
11 A length of 15 cm of 5% solution causes an optical rotation of 20
o
. How much length of a
10% solution of the same substance will cause a rotation of 35
o
12 A 20 cm column of cane sugar solution of concentration of 100gm/litre produces rotation of
0
. Find the purity of cane sugar. Given: Specific rotation of pure sugar is 66 dm - 1
g
cm.
13 (a) The refractive indices for quartz (wave length396.8nm), for left-and right-circularly
polarize light, are n L
=1.55821 and n R
=1.55810, respectively? What is the specific rotation of
quartz for this wave length? (b) What thickness of quartz is required to give an optical
rotation of 10
o
for light of 396.8nm?
Unit-V: Wave guide & Optical Fiber
- Determine (a) the length and (b) transit time for the longest and shortest trajectories in a step-
index fiber of length 1km having a core index of 1.46 and a cladding index of 1.
- An optical fiber core diameter and refractive index are 50 μm, 1.6 respectively. Calculate the
number of total internal reflections that a ray incident at 60
o
will suffer in moving through
1m of fiber.
- A light ray enters a glass fiber from air. The refractive index of air ia1. The refractive indices
of core and cladding are 1.5 and 1.47, respectively. Calculate the value of (a) critical angle,
(b) fractional refractive index, (c) numerical aperture and (iv) the acceptance angle.
