Electromagnetic Waves | Physics Optional Notes for UPSC PDF Download

 Page 1


ELECTROMAGNETIC WAVES 
INTRODUCTION 
When an electric field changes, it creates a changing magnetic field, and when a 
magnetic field changes, it creates a changing electric field. This back and forth movement 
gives birth to a sideways-moving wave called an electromagnetic wave. The electric field 
and magnetic field, which change over time, are at right angles to each other and also to 
the direction the wave is traveling.  
Thus the electromagnetic waves consist of sinusoidally time varying electric and 
magnetic field acting at right angles to each other as well as at right angles to the 
direction of propagation.  
 
CONCEPT OF DISPLACEMENT CURRENT 
When we let a capacitor charge up in an electric circuit, electricity flows through the wires 
connecting it. As the capacitor charges, electric charge builds up on its plates, creating a 
changing electric field between them. According to Maxwell's equations, this changing 
electric field behaves like a current flowing through the capacitor, called displacement 
current (I_d). If +q and -q are the charges on the left and right plates of the capacitor at 
any moment, and s is the surface charge density of the capacitor's plates, then the electric 
field between the plates can be calculated as follows: 
 
E=
?? ?? 0
=
q
?? 0
 A
 
If charge on the plates of the capacitor increases by dq in time dt then dq= Idt 
Change in electric field is dE=
dq
?? 0
 A
=
Idt
?? 0
 A
?
dE
dt
=
I
?? 0
 A
 
?? = ?? 0
?? ????
????
= ?? 0
?? ????
(???? )= ?? 0
?? ?? ?? ????
(? ?? ?? = ???? ) ?? ?? = ?? 0
?? ?? ?? ????
 
Page 2


ELECTROMAGNETIC WAVES 
INTRODUCTION 
When an electric field changes, it creates a changing magnetic field, and when a 
magnetic field changes, it creates a changing electric field. This back and forth movement 
gives birth to a sideways-moving wave called an electromagnetic wave. The electric field 
and magnetic field, which change over time, are at right angles to each other and also to 
the direction the wave is traveling.  
Thus the electromagnetic waves consist of sinusoidally time varying electric and 
magnetic field acting at right angles to each other as well as at right angles to the 
direction of propagation.  
 
CONCEPT OF DISPLACEMENT CURRENT 
When we let a capacitor charge up in an electric circuit, electricity flows through the wires 
connecting it. As the capacitor charges, electric charge builds up on its plates, creating a 
changing electric field between them. According to Maxwell's equations, this changing 
electric field behaves like a current flowing through the capacitor, called displacement 
current (I_d). If +q and -q are the charges on the left and right plates of the capacitor at 
any moment, and s is the surface charge density of the capacitor's plates, then the electric 
field between the plates can be calculated as follows: 
 
E=
?? ?? 0
=
q
?? 0
 A
 
If charge on the plates of the capacitor increases by dq in time dt then dq= Idt 
Change in electric field is dE=
dq
?? 0
 A
=
Idt
?? 0
 A
?
dE
dt
=
I
?? 0
 A
 
?? = ?? 0
?? ????
????
= ?? 0
?? ????
(???? )= ?? 0
?? ?? ?? ????
(? ?? ?? = ???? ) ?? ?? = ?? 0
?? ?? ?? ????
 
The conduction current is the current due to the flow of charges in a conductor and is 
denoted as Ic and displacement current is the current due to changing electric field 
between the plate of the capacitor and denoted as ?? ?? so the total current ?? is sum of ?? ?? and 
?? ?? i.e. ?? = ?? ?? + ?? ?? 
Ampere's circuital law can be written as 
?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ I
d
) ? ?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ ?? 0
d?? E
dt
) 
MAXWELL'S EQUATION 
There are four Maxwell' equation given below 
(1) Gauss law in electrostatics: ? ?? ? 
· ????
???? 
=
?? ?? 0
 
(2) Gauss law in magnetism:  ? B
? ? 
· ds
???? 
= 0 
(3) Faraday's law of electromagnetic induction:  ?????? = ? ?? ? 
· ?? 
?? = -
?? ?? ?? ????
 
(4) Maxwell - Ampere's circuital law: ? B· d
? 
·= ?? 0
[I
?? + ?? 0
d?? E
dt
] 
PROPERTIES OF ELECTROMAGNETIC WAVES 
(c) The electric and magnetic fields satisfy the following wave equations, which can be 
obtained from Maxwell's third and fourth equations. 
?
2
E
?x
2
= ?? 0
?? 0
?
2
E
?t
2
  and  
?
2
 B
?x
2
= ?? 0
?? 0
?
2
 B
?t
2
 
? Electromagnetic waves travel through vacuum with the speed of light c, where 
c =
1
v?? 0
?? 0
= 3 × 10
8
 m/s 
? The electric and magnetic fields of an electromagnetic wave are perpendicular to 
each other and also perpendicular to the direction of wave propagation. Hence, 
these are transverse waves. 
? The instantaneous magnitudes of ?? ? 
 and ?? ? 
 in an electromagnetic wave are related 
by the expression 
?? ?? = ?? 
? Electromagnetic waves carry energy. The rate of flow of energy crossing a unit 
area is described by the Pointing vector ?? 
. Where  ?? 
=
1
?? 0
?? ? 
× ?? ? 
 
Page 3


ELECTROMAGNETIC WAVES 
INTRODUCTION 
When an electric field changes, it creates a changing magnetic field, and when a 
magnetic field changes, it creates a changing electric field. This back and forth movement 
gives birth to a sideways-moving wave called an electromagnetic wave. The electric field 
and magnetic field, which change over time, are at right angles to each other and also to 
the direction the wave is traveling.  
Thus the electromagnetic waves consist of sinusoidally time varying electric and 
magnetic field acting at right angles to each other as well as at right angles to the 
direction of propagation.  
 
CONCEPT OF DISPLACEMENT CURRENT 
When we let a capacitor charge up in an electric circuit, electricity flows through the wires 
connecting it. As the capacitor charges, electric charge builds up on its plates, creating a 
changing electric field between them. According to Maxwell's equations, this changing 
electric field behaves like a current flowing through the capacitor, called displacement 
current (I_d). If +q and -q are the charges on the left and right plates of the capacitor at 
any moment, and s is the surface charge density of the capacitor's plates, then the electric 
field between the plates can be calculated as follows: 
 
E=
?? ?? 0
=
q
?? 0
 A
 
If charge on the plates of the capacitor increases by dq in time dt then dq= Idt 
Change in electric field is dE=
dq
?? 0
 A
=
Idt
?? 0
 A
?
dE
dt
=
I
?? 0
 A
 
?? = ?? 0
?? ????
????
= ?? 0
?? ????
(???? )= ?? 0
?? ?? ?? ????
(? ?? ?? = ???? ) ?? ?? = ?? 0
?? ?? ?? ????
 
The conduction current is the current due to the flow of charges in a conductor and is 
denoted as Ic and displacement current is the current due to changing electric field 
between the plate of the capacitor and denoted as ?? ?? so the total current ?? is sum of ?? ?? and 
?? ?? i.e. ?? = ?? ?? + ?? ?? 
Ampere's circuital law can be written as 
?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ I
d
) ? ?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ ?? 0
d?? E
dt
) 
MAXWELL'S EQUATION 
There are four Maxwell' equation given below 
(1) Gauss law in electrostatics: ? ?? ? 
· ????
???? 
=
?? ?? 0
 
(2) Gauss law in magnetism:  ? B
? ? 
· ds
???? 
= 0 
(3) Faraday's law of electromagnetic induction:  ?????? = ? ?? ? 
· ?? 
?? = -
?? ?? ?? ????
 
(4) Maxwell - Ampere's circuital law: ? B· d
? 
·= ?? 0
[I
?? + ?? 0
d?? E
dt
] 
PROPERTIES OF ELECTROMAGNETIC WAVES 
(c) The electric and magnetic fields satisfy the following wave equations, which can be 
obtained from Maxwell's third and fourth equations. 
?
2
E
?x
2
= ?? 0
?? 0
?
2
E
?t
2
  and  
?
2
 B
?x
2
= ?? 0
?? 0
?
2
 B
?t
2
 
? Electromagnetic waves travel through vacuum with the speed of light c, where 
c =
1
v?? 0
?? 0
= 3 × 10
8
 m/s 
? The electric and magnetic fields of an electromagnetic wave are perpendicular to 
each other and also perpendicular to the direction of wave propagation. Hence, 
these are transverse waves. 
? The instantaneous magnitudes of ?? ? 
 and ?? ? 
 in an electromagnetic wave are related 
by the expression 
?? ?? = ?? 
? Electromagnetic waves carry energy. The rate of flow of energy crossing a unit 
area is described by the Pointing vector ?? 
. Where  ?? 
=
1
?? 0
?? ? 
× ?? ? 
 
? Electromagnetic waves carry momentum and hence can exert pressure (P) on 
surfaces, which is known as radiation pressure. For an electromagnetic wave with 
Pointing vector ?? 
, incident upon a perfectly absorbing surface P =
S
C
 
And if incident upon a perfectly reflecting surface ?? =
2?? ?? 
? The electric and magnetic fields of a sinusoidal plane electromagnetic wave 
propagating in the positive x-direction can also be written as 
?? = ?? ?? sin (???? - ???? )  and  ?? = ?? ?? sin (???? - ???? ) 
Where ?? is the angular frequency of the wave and ?? is wave number which are given by 
?? = 2????  and  ?? =
2?? ?? 
(- The intensity of a sinusoidal plane electro-magnetic wave is defined as the average 
value of the Poynting vector taken over one cycle. ?? ????
=
?? ?? ?? ?? 2?? 0
=
?? ?? 2
2?? 0
?? =
?? 2?? 0
?? ?? 2
 
? The fundamental sources of electromagnetic waves are accelerating electric 
charges. For example radio waves emitted by an antenna arises from the 
continuous oscillations (and hence acceleration) of charges within the antenna 
structure. 
? Electromagnetic waves obey the principle of superposition. 
? The electric vector of an electromagnetic field is responsible for all optical effects. 
For this reason electric vector is also called a light vector. 
TRANSVERSE NATURE OF ELECTROMAGNETIC WAVES 
Maxwell figured out that when an electric field changes, it makes a magnetic field change 
too, and the other way around. This back-and-forth movement of changing electric and 
magnetic fields creates electromagnetic waves. These waves have electric and magnetic 
fields that are at right angles to each other, and they also travel sideways, perpendicular 
to both fields. We call these waves transverse electromagnetic waves. 
ELECTROMAGNETIC WAVES 
1. In EM waves both the fields ?? ? 
 and ?? ? 
 vary with time and space and have same ?? 
and ?? · ?? ? 
 and ?? ? 
 are perpendicular to each other and are also perpendicular to 
direction of propagation of wave. 
[
E
? ? 
= Electric field 
B
? ? 
= Magntic field 
] 
2. E
ˆ
× B
ˆ
= v ˆ 
Page 4


ELECTROMAGNETIC WAVES 
INTRODUCTION 
When an electric field changes, it creates a changing magnetic field, and when a 
magnetic field changes, it creates a changing electric field. This back and forth movement 
gives birth to a sideways-moving wave called an electromagnetic wave. The electric field 
and magnetic field, which change over time, are at right angles to each other and also to 
the direction the wave is traveling.  
Thus the electromagnetic waves consist of sinusoidally time varying electric and 
magnetic field acting at right angles to each other as well as at right angles to the 
direction of propagation.  
 
CONCEPT OF DISPLACEMENT CURRENT 
When we let a capacitor charge up in an electric circuit, electricity flows through the wires 
connecting it. As the capacitor charges, electric charge builds up on its plates, creating a 
changing electric field between them. According to Maxwell's equations, this changing 
electric field behaves like a current flowing through the capacitor, called displacement 
current (I_d). If +q and -q are the charges on the left and right plates of the capacitor at 
any moment, and s is the surface charge density of the capacitor's plates, then the electric 
field between the plates can be calculated as follows: 
 
E=
?? ?? 0
=
q
?? 0
 A
 
If charge on the plates of the capacitor increases by dq in time dt then dq= Idt 
Change in electric field is dE=
dq
?? 0
 A
=
Idt
?? 0
 A
?
dE
dt
=
I
?? 0
 A
 
?? = ?? 0
?? ????
????
= ?? 0
?? ????
(???? )= ?? 0
?? ?? ?? ????
(? ?? ?? = ???? ) ?? ?? = ?? 0
?? ?? ?? ????
 
The conduction current is the current due to the flow of charges in a conductor and is 
denoted as Ic and displacement current is the current due to changing electric field 
between the plate of the capacitor and denoted as ?? ?? so the total current ?? is sum of ?? ?? and 
?? ?? i.e. ?? = ?? ?? + ?? ?? 
Ampere's circuital law can be written as 
?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ I
d
) ? ?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ ?? 0
d?? E
dt
) 
MAXWELL'S EQUATION 
There are four Maxwell' equation given below 
(1) Gauss law in electrostatics: ? ?? ? 
· ????
???? 
=
?? ?? 0
 
(2) Gauss law in magnetism:  ? B
? ? 
· ds
???? 
= 0 
(3) Faraday's law of electromagnetic induction:  ?????? = ? ?? ? 
· ?? 
?? = -
?? ?? ?? ????
 
(4) Maxwell - Ampere's circuital law: ? B· d
? 
·= ?? 0
[I
?? + ?? 0
d?? E
dt
] 
PROPERTIES OF ELECTROMAGNETIC WAVES 
(c) The electric and magnetic fields satisfy the following wave equations, which can be 
obtained from Maxwell's third and fourth equations. 
?
2
E
?x
2
= ?? 0
?? 0
?
2
E
?t
2
  and  
?
2
 B
?x
2
= ?? 0
?? 0
?
2
 B
?t
2
 
? Electromagnetic waves travel through vacuum with the speed of light c, where 
c =
1
v?? 0
?? 0
= 3 × 10
8
 m/s 
? The electric and magnetic fields of an electromagnetic wave are perpendicular to 
each other and also perpendicular to the direction of wave propagation. Hence, 
these are transverse waves. 
? The instantaneous magnitudes of ?? ? 
 and ?? ? 
 in an electromagnetic wave are related 
by the expression 
?? ?? = ?? 
? Electromagnetic waves carry energy. The rate of flow of energy crossing a unit 
area is described by the Pointing vector ?? 
. Where  ?? 
=
1
?? 0
?? ? 
× ?? ? 
 
? Electromagnetic waves carry momentum and hence can exert pressure (P) on 
surfaces, which is known as radiation pressure. For an electromagnetic wave with 
Pointing vector ?? 
, incident upon a perfectly absorbing surface P =
S
C
 
And if incident upon a perfectly reflecting surface ?? =
2?? ?? 
? The electric and magnetic fields of a sinusoidal plane electromagnetic wave 
propagating in the positive x-direction can also be written as 
?? = ?? ?? sin (???? - ???? )  and  ?? = ?? ?? sin (???? - ???? ) 
Where ?? is the angular frequency of the wave and ?? is wave number which are given by 
?? = 2????  and  ?? =
2?? ?? 
(- The intensity of a sinusoidal plane electro-magnetic wave is defined as the average 
value of the Poynting vector taken over one cycle. ?? ????
=
?? ?? ?? ?? 2?? 0
=
?? ?? 2
2?? 0
?? =
?? 2?? 0
?? ?? 2
 
? The fundamental sources of electromagnetic waves are accelerating electric 
charges. For example radio waves emitted by an antenna arises from the 
continuous oscillations (and hence acceleration) of charges within the antenna 
structure. 
? Electromagnetic waves obey the principle of superposition. 
? The electric vector of an electromagnetic field is responsible for all optical effects. 
For this reason electric vector is also called a light vector. 
TRANSVERSE NATURE OF ELECTROMAGNETIC WAVES 
Maxwell figured out that when an electric field changes, it makes a magnetic field change 
too, and the other way around. This back-and-forth movement of changing electric and 
magnetic fields creates electromagnetic waves. These waves have electric and magnetic 
fields that are at right angles to each other, and they also travel sideways, perpendicular 
to both fields. We call these waves transverse electromagnetic waves. 
ELECTROMAGNETIC WAVES 
1. In EM waves both the fields ?? ? 
 and ?? ? 
 vary with time and space and have same ?? 
and ?? · ?? ? 
 and ?? ? 
 are perpendicular to each other and are also perpendicular to 
direction of propagation of wave. 
[
E
? ? 
= Electric field 
B
? ? 
= Magntic field 
] 
2. E
ˆ
× B
ˆ
= v ˆ 
V
? ? 
× E
? ? 
C
2
= B
? ? 
 
3. Energy is equally divided in electric and magnetic form. 
Energy density =
1
2
?? 0
?? rns
2
=
1
2
?? ?? s
2
?? 2
?? 0
 
=
1
2
?? rms
2
?? 0
= Energy density due to ?? 
 
Total energy density = E
0
E
rms
2
=
?? rms
2
?? 0
 
4. Total intensity =
 Power 
 Area 
=
ED× volume 
t× area 
= ED×
Al
At
 
Total intensity = E
0
E
mms
2
c 
5. When this light gets absorbed 
 Force applied =
?? ?? Momentum transfer =
?? × ?? ?? =
 Energy 
?? 
6. EM waves can be polarized. 
7. Accelerated charges (eg. oscillating charge) is a source of EM waves. 
Example. A plane electromagnetic wave of frequency 25MHz travels in free space along 
the ?? -direction. At a particular point in space and time, ?? = 6.3j ˆV/m . What is ?? at this 
point? 
Solution: Using Eq. ?? 0
=
?? 0
?? , the magnitude of ?? is 
?? =
?? ?? =
6.3 V/m
3× 10
8
 m/s
= 2.1× 10
-8
 T 
To find the direction, we note that ?? is along ?? -direction and the wave propagates along 
?? -axis. Therefore, ?? should be in a direction perpendicular to both ?? - and ?? -axes. Using 
vector algebra. 
?? × ?? Should be along ?? -direction. Since, (+??ˆ)× (+k
ˆ
)= i,?? is along the ?? -direction. 
Thus, ?? = 2.1× 10
-8
k
ˆ
T 
Example. The magnetic field in a plane electromagnetic wave is given by 
B
?? = 2× 10
-7
sin (0.5× 10
3
x+ 1.5 × 10
11
t)T 
(a) What is the wavelength and frequency of the wave? 
Page 5


ELECTROMAGNETIC WAVES 
INTRODUCTION 
When an electric field changes, it creates a changing magnetic field, and when a 
magnetic field changes, it creates a changing electric field. This back and forth movement 
gives birth to a sideways-moving wave called an electromagnetic wave. The electric field 
and magnetic field, which change over time, are at right angles to each other and also to 
the direction the wave is traveling.  
Thus the electromagnetic waves consist of sinusoidally time varying electric and 
magnetic field acting at right angles to each other as well as at right angles to the 
direction of propagation.  
 
CONCEPT OF DISPLACEMENT CURRENT 
When we let a capacitor charge up in an electric circuit, electricity flows through the wires 
connecting it. As the capacitor charges, electric charge builds up on its plates, creating a 
changing electric field between them. According to Maxwell's equations, this changing 
electric field behaves like a current flowing through the capacitor, called displacement 
current (I_d). If +q and -q are the charges on the left and right plates of the capacitor at 
any moment, and s is the surface charge density of the capacitor's plates, then the electric 
field between the plates can be calculated as follows: 
 
E=
?? ?? 0
=
q
?? 0
 A
 
If charge on the plates of the capacitor increases by dq in time dt then dq= Idt 
Change in electric field is dE=
dq
?? 0
 A
=
Idt
?? 0
 A
?
dE
dt
=
I
?? 0
 A
 
?? = ?? 0
?? ????
????
= ?? 0
?? ????
(???? )= ?? 0
?? ?? ?? ????
(? ?? ?? = ???? ) ?? ?? = ?? 0
?? ?? ?? ????
 
The conduction current is the current due to the flow of charges in a conductor and is 
denoted as Ic and displacement current is the current due to changing electric field 
between the plate of the capacitor and denoted as ?? ?? so the total current ?? is sum of ?? ?? and 
?? ?? i.e. ?? = ?? ?? + ?? ?? 
Ampere's circuital law can be written as 
?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ I
d
) ? ?  B
? ? 
· dl
???? 
= ?? 0
(I
c
+ ?? 0
d?? E
dt
) 
MAXWELL'S EQUATION 
There are four Maxwell' equation given below 
(1) Gauss law in electrostatics: ? ?? ? 
· ????
???? 
=
?? ?? 0
 
(2) Gauss law in magnetism:  ? B
? ? 
· ds
???? 
= 0 
(3) Faraday's law of electromagnetic induction:  ?????? = ? ?? ? 
· ?? 
?? = -
?? ?? ?? ????
 
(4) Maxwell - Ampere's circuital law: ? B· d
? 
·= ?? 0
[I
?? + ?? 0
d?? E
dt
] 
PROPERTIES OF ELECTROMAGNETIC WAVES 
(c) The electric and magnetic fields satisfy the following wave equations, which can be 
obtained from Maxwell's third and fourth equations. 
?
2
E
?x
2
= ?? 0
?? 0
?
2
E
?t
2
  and  
?
2
 B
?x
2
= ?? 0
?? 0
?
2
 B
?t
2
 
? Electromagnetic waves travel through vacuum with the speed of light c, where 
c =
1
v?? 0
?? 0
= 3 × 10
8
 m/s 
? The electric and magnetic fields of an electromagnetic wave are perpendicular to 
each other and also perpendicular to the direction of wave propagation. Hence, 
these are transverse waves. 
? The instantaneous magnitudes of ?? ? 
 and ?? ? 
 in an electromagnetic wave are related 
by the expression 
?? ?? = ?? 
? Electromagnetic waves carry energy. The rate of flow of energy crossing a unit 
area is described by the Pointing vector ?? 
. Where  ?? 
=
1
?? 0
?? ? 
× ?? ? 
 
? Electromagnetic waves carry momentum and hence can exert pressure (P) on 
surfaces, which is known as radiation pressure. For an electromagnetic wave with 
Pointing vector ?? 
, incident upon a perfectly absorbing surface P =
S
C
 
And if incident upon a perfectly reflecting surface ?? =
2?? ?? 
? The electric and magnetic fields of a sinusoidal plane electromagnetic wave 
propagating in the positive x-direction can also be written as 
?? = ?? ?? sin (???? - ???? )  and  ?? = ?? ?? sin (???? - ???? ) 
Where ?? is the angular frequency of the wave and ?? is wave number which are given by 
?? = 2????  and  ?? =
2?? ?? 
(- The intensity of a sinusoidal plane electro-magnetic wave is defined as the average 
value of the Poynting vector taken over one cycle. ?? ????
=
?? ?? ?? ?? 2?? 0
=
?? ?? 2
2?? 0
?? =
?? 2?? 0
?? ?? 2
 
? The fundamental sources of electromagnetic waves are accelerating electric 
charges. For example radio waves emitted by an antenna arises from the 
continuous oscillations (and hence acceleration) of charges within the antenna 
structure. 
? Electromagnetic waves obey the principle of superposition. 
? The electric vector of an electromagnetic field is responsible for all optical effects. 
For this reason electric vector is also called a light vector. 
TRANSVERSE NATURE OF ELECTROMAGNETIC WAVES 
Maxwell figured out that when an electric field changes, it makes a magnetic field change 
too, and the other way around. This back-and-forth movement of changing electric and 
magnetic fields creates electromagnetic waves. These waves have electric and magnetic 
fields that are at right angles to each other, and they also travel sideways, perpendicular 
to both fields. We call these waves transverse electromagnetic waves. 
ELECTROMAGNETIC WAVES 
1. In EM waves both the fields ?? ? 
 and ?? ? 
 vary with time and space and have same ?? 
and ?? · ?? ? 
 and ?? ? 
 are perpendicular to each other and are also perpendicular to 
direction of propagation of wave. 
[
E
? ? 
= Electric field 
B
? ? 
= Magntic field 
] 
2. E
ˆ
× B
ˆ
= v ˆ 
V
? ? 
× E
? ? 
C
2
= B
? ? 
 
3. Energy is equally divided in electric and magnetic form. 
Energy density =
1
2
?? 0
?? rns
2
=
1
2
?? ?? s
2
?? 2
?? 0
 
=
1
2
?? rms
2
?? 0
= Energy density due to ?? 
 
Total energy density = E
0
E
rms
2
=
?? rms
2
?? 0
 
4. Total intensity =
 Power 
 Area 
=
ED× volume 
t× area 
= ED×
Al
At
 
Total intensity = E
0
E
mms
2
c 
5. When this light gets absorbed 
 Force applied =
?? ?? Momentum transfer =
?? × ?? ?? =
 Energy 
?? 
6. EM waves can be polarized. 
7. Accelerated charges (eg. oscillating charge) is a source of EM waves. 
Example. A plane electromagnetic wave of frequency 25MHz travels in free space along 
the ?? -direction. At a particular point in space and time, ?? = 6.3j ˆV/m . What is ?? at this 
point? 
Solution: Using Eq. ?? 0
=
?? 0
?? , the magnitude of ?? is 
?? =
?? ?? =
6.3 V/m
3× 10
8
 m/s
= 2.1× 10
-8
 T 
To find the direction, we note that ?? is along ?? -direction and the wave propagates along 
?? -axis. Therefore, ?? should be in a direction perpendicular to both ?? - and ?? -axes. Using 
vector algebra. 
?? × ?? Should be along ?? -direction. Since, (+??ˆ)× (+k
ˆ
)= i,?? is along the ?? -direction. 
Thus, ?? = 2.1× 10
-8
k
ˆ
T 
Example. The magnetic field in a plane electromagnetic wave is given by 
B
?? = 2× 10
-7
sin (0.5× 10
3
x+ 1.5 × 10
11
t)T 
(a) What is the wavelength and frequency of the wave? 
(b) Write an expression for the electric field. 
Solution: (a) Comparing the given equation with 
?? ?? = ?? 0
sin [2?? (
?? ?? +
?? ?? )] 
We get,  ?? =
2?? 0.5×10
3
 m= 1.26 cm 
And  
1
 T
= v =
(1.5×10
11
)
2?? = 23.9GHz 
(b) E
0
= B
0
c = 2× 10
-7
 T× 3× 10
8
 m/s= 6 × 10
1
 V/m 
The electric field component is perpendicular to the direction of propagation and the 
direction of magnetic field. Therefore, the electric field component along the ?? -axis is 
obtained as 
?? ?? = 60sin (0.5× 10
3
?? + 1.5 × 10
11
?? )?? /?? 
Example. Light with an energy flux of 18 W/cm
2
 falls on a nonreflecting surface at 
normal incidence. If the surface has an area of 20 cm
2
, find the average force exerted on 
the surface during 30 minutes. 
Solution: The total energy falling on the surface is 
U = (18 W/cm
2
)× (20 cm
2
)× (30× 60)= 6.48× 10
5
 J 
Therefore, the total momentum delivered (for complete absorption) is 
p =
U
c
=
6.48× 10
5
 J
3× 10
8
 m/s
= 2.16× 10
-3
 kg m/s 
The average force exerted on the surface is 
?? =
?? ?? =
2.16× 10
-3
0.18× 10
4
= 1.2× 10
-6
 N 
How will your result be modified if the surface is a perfect reflector? 
Example. Calculate the electric and magnetic fields produced by the radiation coming 
from a 100 W bulb at a distance of 3 m . Assume that the efficiency of the bulb is 2.5% 
and it is a point source. 
Solution: The bulb, as a point source, radiates light in all directions uniformly. At a 
distance of 3 m , the surface area of the surrounding sphere is 
A = 4?? r
2
= 4?? (3)
2
= 113 m
2
 
The intensity at this distance is