Page 1
Chapter Eight
ELECTROMAGNETIC
WAVES
8.1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other.
Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field. Is the converse also true? Does an
electric field changing with time give rise to a magnetic field? James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field. While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law. He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency.
Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities. These
equations are known as Maxwell’s equations. Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism.
The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space. The speed
of the waves, according to these equations, turned out to be very close to
2024-25
Page 2
Chapter Eight
ELECTROMAGNETIC
WAVES
8.1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other.
Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field. Is the converse also true? Does an
electric field changing with time give rise to a magnetic field? James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field. While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law. He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency.
Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities. These
equations are known as Maxwell’s equations. Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism.
The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space. The speed
of the waves, according to these equations, turned out to be very close to
2024-25
Physics
202
the speed of light( 3 ×10
8
m/s), obtained from optical
measurements. This led to the remarkable conclusion
that light is an electromagnetic wave. Maxwell’s work
thus unified the domain of electricity, magnetism and
light. Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves. Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today.
In this chapter, we first discuss the need for
displacement current and its consequences. Then we
present a descriptive account of electromagnetic waves.
The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10
–12
m) to long
radio waves (wavelength ~10
6
m) is described.
8.2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it. Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field. This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves.
To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B
.
dl = m
0
i (t) (8.1)
to find magnetic field at a point outside the capacitor.
Figure 8.1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows . Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor. For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig. 8.1(a)]. From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq. (8.1) is B (2p r).
So we have
B (2pr) = m
0
i (t) (8 .2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists of the
nineteenth century. He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain reliable
estimates of molecular
parameters from
measurable quantities
like viscosity, etc.
Maxwell’s greatest
acheivement was the
unification of the laws of
electricity and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations.
From these he arrived at
the most important
conclusion that light is
an electromagnetic
wave. Interestingly,
Maxwell did not agree
with the idea (strongly
suggested by the
Faraday’s laws of
electrolysis) that
electricity was
particulate in nature.
2024-25
Page 3
Chapter Eight
ELECTROMAGNETIC
WAVES
8.1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other.
Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field. Is the converse also true? Does an
electric field changing with time give rise to a magnetic field? James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field. While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law. He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency.
Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities. These
equations are known as Maxwell’s equations. Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism.
The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space. The speed
of the waves, according to these equations, turned out to be very close to
2024-25
Physics
202
the speed of light( 3 ×10
8
m/s), obtained from optical
measurements. This led to the remarkable conclusion
that light is an electromagnetic wave. Maxwell’s work
thus unified the domain of electricity, magnetism and
light. Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves. Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today.
In this chapter, we first discuss the need for
displacement current and its consequences. Then we
present a descriptive account of electromagnetic waves.
The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10
–12
m) to long
radio waves (wavelength ~10
6
m) is described.
8.2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it. Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field. This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves.
To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B
.
dl = m
0
i (t) (8.1)
to find magnetic field at a point outside the capacitor.
Figure 8.1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows . Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor. For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig. 8.1(a)]. From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq. (8.1) is B (2p r).
So we have
B (2pr) = m
0
i (t) (8 .2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists of the
nineteenth century. He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain reliable
estimates of molecular
parameters from
measurable quantities
like viscosity, etc.
Maxwell’s greatest
acheivement was the
unification of the laws of
electricity and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations.
From these he arrived at
the most important
conclusion that light is
an electromagnetic
wave. Interestingly,
Maxwell did not agree
with the idea (strongly
suggested by the
Faraday’s laws of
electrolysis) that
electricity was
particulate in nature.
2024-25
203
Electromagnetic
Waves
Now, consider a different surface, which has the same boundary. This
is a pot like surface [Fig. 8.1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above. Another such surface is shaped like a tiffin box
(without the lid) [Fig. 8.1(c)]. On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq. (8.1) has not changed but the right hand side is zero and not m
0
i,
since no current passes through the surface of Fig. 8.1(b) and (c). So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero.
Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something. The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used.
We can actually guess the missing term by looking carefully at
Fig. 8.1(c). Is there anything passing through the surface S between the
plates of the capacitor? Yes, of course, the electric field! If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/e
0
(see Eq. 2.41). The field is
perpendicular to the surface S of Fig. 8.1(c). It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it. So what
is the electric flux F
E
through the surface S ? Using Gauss’s law, it is
E
0 0
1
= =
Q Q
A A
A
F
e e
= E
(8.3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq. (8.3), we have
d
d
d
d
d
d
F
E
t t
Q Q
t
=
?
?
?
?
?
?
=
e e
0 0
1
This implies that for consistency,
e
0
d
d
F
E
t
?
?
?
?
?
?
= i (8.4)
This is the missing term in Ampere’s circuital law. If we generalise
this law by adding to the total current carried by conductors through
the surface, another term which is e
0
times the rate of change of electric
flux through the same surface, the total has the same value of current i
for all surfaces. If this is done, there is no contradiction in the value of B
obtained anywhere using the generalised Ampere’s law. B at the point P
is non-zero no matter which surface is used for calculating it. B at a
point P outside the plates [Fig. 8.1(a)] is the same as at a point M just
inside, as it should be. The current carried by conductors due to flow of
charges is called conduction current. The current, given by Eq. (8.4), is a
new term, and is due to changing electric field (or electric displacement,
an old term still used sometimes). It is, therefore, called displacement
current or Maxwell’s displacement current. Figure 8.2 shows the electric
and magnetic fields inside the parallel plate capacitor discussed above.
The generalisation made by Maxwell then is the following. The source
of a magnetic field is not just the conduction electric current due to flowing
FIGURE 8.1 A
parallel plate
capacitor C, as part of
a circuit through
which a time
dependent current
i (t) flows, (a) a loop of
radius r, to determine
magnetic field at a
point P on the loop;
(b) a pot-shaped
surface passing
through the interior
between the capacitor
plates with the loop
shown in (a) as its
rim; (c) a tiffin-
shaped surface with
the circular loop as
its rim and a flat
circular bottom S
between the capacitor
plates. The arrows
show uniform electric
field between the
capacitor plates.
2024-25
Page 4
Chapter Eight
ELECTROMAGNETIC
WAVES
8.1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other.
Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field. Is the converse also true? Does an
electric field changing with time give rise to a magnetic field? James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field. While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law. He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency.
Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities. These
equations are known as Maxwell’s equations. Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism.
The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space. The speed
of the waves, according to these equations, turned out to be very close to
2024-25
Physics
202
the speed of light( 3 ×10
8
m/s), obtained from optical
measurements. This led to the remarkable conclusion
that light is an electromagnetic wave. Maxwell’s work
thus unified the domain of electricity, magnetism and
light. Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves. Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today.
In this chapter, we first discuss the need for
displacement current and its consequences. Then we
present a descriptive account of electromagnetic waves.
The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10
–12
m) to long
radio waves (wavelength ~10
6
m) is described.
8.2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it. Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field. This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves.
To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B
.
dl = m
0
i (t) (8.1)
to find magnetic field at a point outside the capacitor.
Figure 8.1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows . Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor. For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig. 8.1(a)]. From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq. (8.1) is B (2p r).
So we have
B (2pr) = m
0
i (t) (8 .2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists of the
nineteenth century. He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain reliable
estimates of molecular
parameters from
measurable quantities
like viscosity, etc.
Maxwell’s greatest
acheivement was the
unification of the laws of
electricity and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations.
From these he arrived at
the most important
conclusion that light is
an electromagnetic
wave. Interestingly,
Maxwell did not agree
with the idea (strongly
suggested by the
Faraday’s laws of
electrolysis) that
electricity was
particulate in nature.
2024-25
203
Electromagnetic
Waves
Now, consider a different surface, which has the same boundary. This
is a pot like surface [Fig. 8.1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above. Another such surface is shaped like a tiffin box
(without the lid) [Fig. 8.1(c)]. On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq. (8.1) has not changed but the right hand side is zero and not m
0
i,
since no current passes through the surface of Fig. 8.1(b) and (c). So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero.
Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something. The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used.
We can actually guess the missing term by looking carefully at
Fig. 8.1(c). Is there anything passing through the surface S between the
plates of the capacitor? Yes, of course, the electric field! If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/e
0
(see Eq. 2.41). The field is
perpendicular to the surface S of Fig. 8.1(c). It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it. So what
is the electric flux F
E
through the surface S ? Using Gauss’s law, it is
E
0 0
1
= =
Q Q
A A
A
F
e e
= E
(8.3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq. (8.3), we have
d
d
d
d
d
d
F
E
t t
Q Q
t
=
?
?
?
?
?
?
=
e e
0 0
1
This implies that for consistency,
e
0
d
d
F
E
t
?
?
?
?
?
?
= i (8.4)
This is the missing term in Ampere’s circuital law. If we generalise
this law by adding to the total current carried by conductors through
the surface, another term which is e
0
times the rate of change of electric
flux through the same surface, the total has the same value of current i
for all surfaces. If this is done, there is no contradiction in the value of B
obtained anywhere using the generalised Ampere’s law. B at the point P
is non-zero no matter which surface is used for calculating it. B at a
point P outside the plates [Fig. 8.1(a)] is the same as at a point M just
inside, as it should be. The current carried by conductors due to flow of
charges is called conduction current. The current, given by Eq. (8.4), is a
new term, and is due to changing electric field (or electric displacement,
an old term still used sometimes). It is, therefore, called displacement
current or Maxwell’s displacement current. Figure 8.2 shows the electric
and magnetic fields inside the parallel plate capacitor discussed above.
The generalisation made by Maxwell then is the following. The source
of a magnetic field is not just the conduction electric current due to flowing
FIGURE 8.1 A
parallel plate
capacitor C, as part of
a circuit through
which a time
dependent current
i (t) flows, (a) a loop of
radius r, to determine
magnetic field at a
point P on the loop;
(b) a pot-shaped
surface passing
through the interior
between the capacitor
plates with the loop
shown in (a) as its
rim; (c) a tiffin-
shaped surface with
the circular loop as
its rim and a flat
circular bottom S
between the capacitor
plates. The arrows
show uniform electric
field between the
capacitor plates.
2024-25
Physics
204
charges, but also the time rate of change of electric field. More
precisely, the total current i is the sum of the conduction current
denoted by i
c
, and the displacement current denoted by i
d
(= e
0
(dF
E
/
dt)). So we have
0
d
d
E
c d c
i i i i
t
F
e = + = +
(8.5)
In explicit terms, this means that outside the capacitor plates,
we have only conduction current i
c
= i, and no displacement
current, i.e., i
d
= 0. On the other hand, inside the capacitor, there is
no conduction current, i.e., i
c
= 0, and there is only displacement
current, so that i
d
= i.
The generalised (and correct) Ampere’s circuital law has the same
form as Eq. (8.1), with one difference: “the total current passing
through any surface of which the closed loop is the perimeter” is
the sum of the conduction current and the displacement current.
The generalised law is
B l g
Ñ
d =
d
d
0
µ µ e
0 0
i
t
c
E
+
?
F
(8.6)
and is known as Ampere-Maxwell law.
In all respects, the displacement current has the same physical
effects as the conduction current. In some cases, for example, steady
electric fields in a conducting wire, the displacement current may
be zero since the electric field E does not change with time. In other
cases, for example, the charging capacitor above, both conduction
and displacement currents may be present in different regions of
space. In most of the cases, they both may be present in the same
region of space, as there exist no perfectly conducting or perfectly
insulating medium. Most interestingly, there may be large regions
of space where there is no conduction current, but there is only a
displacement current due to time-varying electric fields. In such a
region, we expect a magnetic field, though there is no (conduction)
current source nearby! The prediction of such a displacement current
can be verified experimentally. For example, a magnetic field (say at point
M) between the plates of the capacitor in Fig. 8.2(a) can be measured and
is seen to be the same as that just outside (at P).
The displacement current has (literally) far reaching consequences.
One thing we immediately notice is that the laws of electricity and
magnetism are now more symmetrical*. Faraday’s law of induction states
that there is an induced emf equal to the rate of change of magnetic flux.
Now, since the emf between two points 1 and 2 is the work done per unit
charge in taking it from 1 to 2, the existence of an emf implies the existence
of an electric field. So, we can rephrase Faraday’s law of electromagnetic
induction by saying that a magnetic field, changing with time, gives rise
to an electric field. Then, the fact that an electric field changing with
time gives rise to a magnetic field, is the symmetrical counterpart, and is
FIGURE 8.2 (a) The
electric and magnetic
fields E and B between
the capacitor plates, at
the point M. (b) A cross
sectional view of Fig. (a).
* They are still not perfectly symmetrical; there are no known sources of magnetic
field (magnetic monopoles) analogous to electric charges which are sources of
electric field.
2024-25
Page 5
Chapter Eight
ELECTROMAGNETIC
WAVES
8.1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other.
Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field. Is the converse also true? Does an
electric field changing with time give rise to a magnetic field? James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field. While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law. He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency.
Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities. These
equations are known as Maxwell’s equations. Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism.
The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space. The speed
of the waves, according to these equations, turned out to be very close to
2024-25
Physics
202
the speed of light( 3 ×10
8
m/s), obtained from optical
measurements. This led to the remarkable conclusion
that light is an electromagnetic wave. Maxwell’s work
thus unified the domain of electricity, magnetism and
light. Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves. Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today.
In this chapter, we first discuss the need for
displacement current and its consequences. Then we
present a descriptive account of electromagnetic waves.
The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10
–12
m) to long
radio waves (wavelength ~10
6
m) is described.
8.2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it. Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field. This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves.
To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B
.
dl = m
0
i (t) (8.1)
to find magnetic field at a point outside the capacitor.
Figure 8.1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows . Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor. For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig. 8.1(a)]. From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq. (8.1) is B (2p r).
So we have
B (2pr) = m
0
i (t) (8 .2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists of the
nineteenth century. He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain reliable
estimates of molecular
parameters from
measurable quantities
like viscosity, etc.
Maxwell’s greatest
acheivement was the
unification of the laws of
electricity and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations.
From these he arrived at
the most important
conclusion that light is
an electromagnetic
wave. Interestingly,
Maxwell did not agree
with the idea (strongly
suggested by the
Faraday’s laws of
electrolysis) that
electricity was
particulate in nature.
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Now, consider a different surface, which has the same boundary. This
is a pot like surface [Fig. 8.1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above. Another such surface is shaped like a tiffin box
(without the lid) [Fig. 8.1(c)]. On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq. (8.1) has not changed but the right hand side is zero and not m
0
i,
since no current passes through the surface of Fig. 8.1(b) and (c). So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero.
Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something. The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used.
We can actually guess the missing term by looking carefully at
Fig. 8.1(c). Is there anything passing through the surface S between the
plates of the capacitor? Yes, of course, the electric field! If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/e
0
(see Eq. 2.41). The field is
perpendicular to the surface S of Fig. 8.1(c). It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it. So what
is the electric flux F
E
through the surface S ? Using Gauss’s law, it is
E
0 0
1
= =
Q Q
A A
A
F
e e
= E
(8.3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq. (8.3), we have
d
d
d
d
d
d
F
E
t t
Q Q
t
=
?
?
?
?
?
?
=
e e
0 0
1
This implies that for consistency,
e
0
d
d
F
E
t
?
?
?
?
?
?
= i (8.4)
This is the missing term in Ampere’s circuital law. If we generalise
this law by adding to the total current carried by conductors through
the surface, another term which is e
0
times the rate of change of electric
flux through the same surface, the total has the same value of current i
for all surfaces. If this is done, there is no contradiction in the value of B
obtained anywhere using the generalised Ampere’s law. B at the point P
is non-zero no matter which surface is used for calculating it. B at a
point P outside the plates [Fig. 8.1(a)] is the same as at a point M just
inside, as it should be. The current carried by conductors due to flow of
charges is called conduction current. The current, given by Eq. (8.4), is a
new term, and is due to changing electric field (or electric displacement,
an old term still used sometimes). It is, therefore, called displacement
current or Maxwell’s displacement current. Figure 8.2 shows the electric
and magnetic fields inside the parallel plate capacitor discussed above.
The generalisation made by Maxwell then is the following. The source
of a magnetic field is not just the conduction electric current due to flowing
FIGURE 8.1 A
parallel plate
capacitor C, as part of
a circuit through
which a time
dependent current
i (t) flows, (a) a loop of
radius r, to determine
magnetic field at a
point P on the loop;
(b) a pot-shaped
surface passing
through the interior
between the capacitor
plates with the loop
shown in (a) as its
rim; (c) a tiffin-
shaped surface with
the circular loop as
its rim and a flat
circular bottom S
between the capacitor
plates. The arrows
show uniform electric
field between the
capacitor plates.
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204
charges, but also the time rate of change of electric field. More
precisely, the total current i is the sum of the conduction current
denoted by i
c
, and the displacement current denoted by i
d
(= e
0
(dF
E
/
dt)). So we have
0
d
d
E
c d c
i i i i
t
F
e = + = +
(8.5)
In explicit terms, this means that outside the capacitor plates,
we have only conduction current i
c
= i, and no displacement
current, i.e., i
d
= 0. On the other hand, inside the capacitor, there is
no conduction current, i.e., i
c
= 0, and there is only displacement
current, so that i
d
= i.
The generalised (and correct) Ampere’s circuital law has the same
form as Eq. (8.1), with one difference: “the total current passing
through any surface of which the closed loop is the perimeter” is
the sum of the conduction current and the displacement current.
The generalised law is
B l g
Ñ
d =
d
d
0
µ µ e
0 0
i
t
c
E
+
?
F
(8.6)
and is known as Ampere-Maxwell law.
In all respects, the displacement current has the same physical
effects as the conduction current. In some cases, for example, steady
electric fields in a conducting wire, the displacement current may
be zero since the electric field E does not change with time. In other
cases, for example, the charging capacitor above, both conduction
and displacement currents may be present in different regions of
space. In most of the cases, they both may be present in the same
region of space, as there exist no perfectly conducting or perfectly
insulating medium. Most interestingly, there may be large regions
of space where there is no conduction current, but there is only a
displacement current due to time-varying electric fields. In such a
region, we expect a magnetic field, though there is no (conduction)
current source nearby! The prediction of such a displacement current
can be verified experimentally. For example, a magnetic field (say at point
M) between the plates of the capacitor in Fig. 8.2(a) can be measured and
is seen to be the same as that just outside (at P).
The displacement current has (literally) far reaching consequences.
One thing we immediately notice is that the laws of electricity and
magnetism are now more symmetrical*. Faraday’s law of induction states
that there is an induced emf equal to the rate of change of magnetic flux.
Now, since the emf between two points 1 and 2 is the work done per unit
charge in taking it from 1 to 2, the existence of an emf implies the existence
of an electric field. So, we can rephrase Faraday’s law of electromagnetic
induction by saying that a magnetic field, changing with time, gives rise
to an electric field. Then, the fact that an electric field changing with
time gives rise to a magnetic field, is the symmetrical counterpart, and is
FIGURE 8.2 (a) The
electric and magnetic
fields E and B between
the capacitor plates, at
the point M. (b) A cross
sectional view of Fig. (a).
* They are still not perfectly symmetrical; there are no known sources of magnetic
field (magnetic monopoles) analogous to electric charges which are sources of
electric field.
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a consequence of the displacement current being a source of a magnetic
field. Thus, time- dependent electric and magnetic fields give rise to each
other! Faraday’s law of electromagnetic induction and Ampere-Maxwell
law give a quantitative expression of this statement, with the current
being the total current, as in Eq. (8.5). One very important consequence
of this symmetry is the existence of electromagnetic waves, which we
discuss qualitatively in the next section.
MAXWELL’S EQUATIONS IN VACUUM
1. “E
.
dA = Q/?
0
(Gauss’s Law for electricity)
2. “B
.
dA = 0 (Gauss’s Law for magnetism)
3. “E
.
dl =
–d
d
B
F
t
l=
(Faraday’s Law)
4. “B
.
dl =
=
d
d
0
µ µ e
0 0
i
t
c
E
+
F
(Ampere – Maxwell Law)
8.3 ELECTROMAGNETIC WAVES
8.3.1 Sources of electromagnetic waves
How are electromagnetic waves produced? Neither stationary charges
nor charges in uniform motion (steady currents) can be sources of
electromagnetic waves. The former produces only electrostatic fields, while
the latter produces magnetic fields that, however, do not vary with time.
It is an important result of Maxwell’s theory that accelerated charges
radiate electromagnetic waves. The proof of this basic result is beyond
the scope of this book, but we can accept it on the basis of rough,
qualitative reasoning. Consider a charge oscillating with some frequency.
(An oscillating charge is an example of accelerating charge.) This
produces an oscillating electric field in space, which produces an
oscillating magnetic field, which in turn, is a source of oscillating electric
field, and so on. The oscillating electric and magnetic fields thus
regenerate each other, so to speak, as the wave propagates through the
space. The frequency of the electromagnetic wave naturally equals the
frequency of oscillation of the charge. The energy associated with the
propagating wave comes at the expense of the energy of the source – the
accelerated charge.
From the preceding discussion, it might appear easy to test the
prediction that light is an electromagnetic wave. We might think that all
we needed to do was to set up an ac circuit in which the current oscillate
at the frequency of visible light, say, yellow light. But, alas, that is not
possible. The frequency of yellow light is about 6 × 10
14
Hz, while the
frequency that we get even with modern electronic circuits is hardly about
10
11
Hz. This is why the experimental demonstration of electromagnetic
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