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# Lecture 19 - Time Varying Field - MAGNETIC FIELD Notes | EduRev

## : Lecture 19 - Time Varying Field - MAGNETIC FIELD Notes | EduRev

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Module 3 : MAGNETIC FIELD
Lecture 19 : Time Varying Field

Objectives

In this lecture you will learn the following
Relate time varying magnetic field with emf generated.
Define mutual inductance and calculate it in simple cases.
Define self inductance.
Calculate energy stored in a magnetic field.

Time Varying Field

Even where there is no relative motion between an observer and a conductor, an emf (and consequently an
induced current for a closed conducting loop) may be induced if the magnetic field itself is varying with time as
flux change may be effected by change in magnetic field with time. In effect it implies that a changing
magnetic field is equivalent to an electric field in which an electric charge at rest experiences a force.
Consider, for example, a magnetic field whose direction is out of the page but whose magnitude varies
with time. The magnetic field fills a cylindrical region of space of radius . Let the magnetic field be time
varying and be given by
Since does not depend on the axial coordinate as well as the azimuthal angle , the electric field is also
independent of these quantities. Consider a coaxial circular path of radius which encloses a time
varying flux. By symmetry of the problem, the electric field at every point of the cicular path must have the
same magnitude and must be tangential to the circle.
Thus the emf is given by
Page 2

Module 3 : MAGNETIC FIELD
Lecture 19 : Time Varying Field

Objectives

In this lecture you will learn the following
Relate time varying magnetic field with emf generated.
Define mutual inductance and calculate it in simple cases.
Define self inductance.
Calculate energy stored in a magnetic field.

Time Varying Field

Even where there is no relative motion between an observer and a conductor, an emf (and consequently an
induced current for a closed conducting loop) may be induced if the magnetic field itself is varying with time as
flux change may be effected by change in magnetic field with time. In effect it implies that a changing
magnetic field is equivalent to an electric field in which an electric charge at rest experiences a force.
Consider, for example, a magnetic field whose direction is out of the page but whose magnitude varies
with time. The magnetic field fills a cylindrical region of space of radius . Let the magnetic field be time
varying and be given by
Since does not depend on the axial coordinate as well as the azimuthal angle , the electric field is also
independent of these quantities. Consider a coaxial circular path of radius which encloses a time
varying flux. By symmetry of the problem, the electric field at every point of the cicular path must have the
same magnitude and must be tangential to the circle.
Thus the emf is given by

Equating these, we get for   ,
For , the flux is , so that
and the electric field foir  is
Exercise 1

A conducting circle having a radius at time is in a constant magnetic field perpendicular to its
plane. The circle expands with time with its radius becoming at time . Calculate the
emf developed in the circle.
(Ans. )

Mutual Inductance

According to Faraday's law, a changing magnetic flux in a loop causes an emf to be generated in that loop.
Consider two stationary coils carrying current. The first coil has turn and carries a current . The
second coil contains turns. The current in the first coil is the source of a magnetic field in the region
around the coil. The second loop encloses a flux , where is the surface of one
turn of the loop. If the current in the first coil is varied, ,and consequently will vary with time.
The variation of causes an emf  to be developed in the second coil. Since is proportional to , so
is . The emf, which is the rate of change of flux is, therefore, proportional to ,
Page 3

Module 3 : MAGNETIC FIELD
Lecture 19 : Time Varying Field

Objectives

In this lecture you will learn the following
Relate time varying magnetic field with emf generated.
Define mutual inductance and calculate it in simple cases.
Define self inductance.
Calculate energy stored in a magnetic field.

Time Varying Field

Even where there is no relative motion between an observer and a conductor, an emf (and consequently an
induced current for a closed conducting loop) may be induced if the magnetic field itself is varying with time as
flux change may be effected by change in magnetic field with time. In effect it implies that a changing
magnetic field is equivalent to an electric field in which an electric charge at rest experiences a force.
Consider, for example, a magnetic field whose direction is out of the page but whose magnitude varies
with time. The magnetic field fills a cylindrical region of space of radius . Let the magnetic field be time
varying and be given by
Since does not depend on the axial coordinate as well as the azimuthal angle , the electric field is also
independent of these quantities. Consider a coaxial circular path of radius which encloses a time
varying flux. By symmetry of the problem, the electric field at every point of the cicular path must have the
same magnitude and must be tangential to the circle.
Thus the emf is given by

Equating these, we get for   ,
For , the flux is , so that
and the electric field foir  is
Exercise 1

A conducting circle having a radius at time is in a constant magnetic field perpendicular to its
plane. The circle expands with time with its radius becoming at time . Calculate the
emf developed in the circle.
(Ans. )

Mutual Inductance

According to Faraday's law, a changing magnetic flux in a loop causes an emf to be generated in that loop.
Consider two stationary coils carrying current. The first coil has turn and carries a current . The
second coil contains turns. The current in the first coil is the source of a magnetic field in the region
around the coil. The second loop encloses a flux , where is the surface of one
turn of the loop. If the current in the first coil is varied, ,and consequently will vary with time.
The variation of causes an emf  to be developed in the second coil. Since is proportional to , so
is . The emf, which is the rate of change of flux is, therefore, proportional to ,
where is a constant, called the mutual inductance of the two coils, which depends on geometrical factors
of the two loops, their relative orientation and the number of turns in each coil.

Analogously, we can argue that if the second loop carries a current which is varied with time, it generates
an induced emf in the first coil given by
For instance, consider two concentric solenoids, the outer one having turns per unit length and inner one
with turns per unit length. The solenoids are wound over coaxial cylinders of length each. If the current
in the outer solenoid is , the field due to it is , which is confined within the solenoid. The
flux enclosed by the inner cylinder is
If the current in the outer solenoid varies with time, the emf in the inner solenoid is
so that
Page 4

Module 3 : MAGNETIC FIELD
Lecture 19 : Time Varying Field

Objectives

In this lecture you will learn the following
Relate time varying magnetic field with emf generated.
Define mutual inductance and calculate it in simple cases.
Define self inductance.
Calculate energy stored in a magnetic field.

Time Varying Field

Even where there is no relative motion between an observer and a conductor, an emf (and consequently an
induced current for a closed conducting loop) may be induced if the magnetic field itself is varying with time as
flux change may be effected by change in magnetic field with time. In effect it implies that a changing
magnetic field is equivalent to an electric field in which an electric charge at rest experiences a force.
Consider, for example, a magnetic field whose direction is out of the page but whose magnitude varies
with time. The magnetic field fills a cylindrical region of space of radius . Let the magnetic field be time
varying and be given by
Since does not depend on the axial coordinate as well as the azimuthal angle , the electric field is also
independent of these quantities. Consider a coaxial circular path of radius which encloses a time
varying flux. By symmetry of the problem, the electric field at every point of the cicular path must have the
same magnitude and must be tangential to the circle.
Thus the emf is given by

Equating these, we get for   ,
For , the flux is , so that
and the electric field foir  is
Exercise 1

A conducting circle having a radius at time is in a constant magnetic field perpendicular to its
plane. The circle expands with time with its radius becoming at time . Calculate the
emf developed in the circle.
(Ans. )

Mutual Inductance

According to Faraday's law, a changing magnetic flux in a loop causes an emf to be generated in that loop.
Consider two stationary coils carrying current. The first coil has turn and carries a current . The
second coil contains turns. The current in the first coil is the source of a magnetic field in the region
around the coil. The second loop encloses a flux , where is the surface of one
turn of the loop. If the current in the first coil is varied, ,and consequently will vary with time.
The variation of causes an emf  to be developed in the second coil. Since is proportional to , so
is . The emf, which is the rate of change of flux is, therefore, proportional to ,
where is a constant, called the mutual inductance of the two coils, which depends on geometrical factors
of the two loops, their relative orientation and the number of turns in each coil.

Analogously, we can argue that if the second loop carries a current which is varied with time, it generates
an induced emf in the first coil given by
For instance, consider two concentric solenoids, the outer one having turns per unit length and inner one
with turns per unit length. The solenoids are wound over coaxial cylinders of length each. If the current
in the outer solenoid is , the field due to it is , which is confined within the solenoid. The
flux enclosed by the inner cylinder is
If the current in the outer solenoid varies with time, the emf in the inner solenoid is
so that

If, on the other hand, the current in the inner solenoid is varied, the field due to it which
is non-zero only within the inner solenoid. The flux enclosed by the outer solenoid is, therefore,
If is varied, the emf in the outer solenoid is    giving
One can see that   .

This equality can be proved quite generally from Biot-Savart's law. Consider two circuits shown in the figure.
The field at , due to current in the loop (called the primary ) is
where . We have seen that can be expressed in terms of a vector potential , where
, by Biot-Savart's law
The flux enclosed by the second loop, (called the secondary ) is
Page 5

Module 3 : MAGNETIC FIELD
Lecture 19 : Time Varying Field

Objectives

In this lecture you will learn the following
Relate time varying magnetic field with emf generated.
Define mutual inductance and calculate it in simple cases.
Define self inductance.
Calculate energy stored in a magnetic field.

Time Varying Field

Even where there is no relative motion between an observer and a conductor, an emf (and consequently an
induced current for a closed conducting loop) may be induced if the magnetic field itself is varying with time as
flux change may be effected by change in magnetic field with time. In effect it implies that a changing
magnetic field is equivalent to an electric field in which an electric charge at rest experiences a force.
Consider, for example, a magnetic field whose direction is out of the page but whose magnitude varies
with time. The magnetic field fills a cylindrical region of space of radius . Let the magnetic field be time
varying and be given by
Since does not depend on the axial coordinate as well as the azimuthal angle , the electric field is also
independent of these quantities. Consider a coaxial circular path of radius which encloses a time
varying flux. By symmetry of the problem, the electric field at every point of the cicular path must have the
same magnitude and must be tangential to the circle.
Thus the emf is given by

Equating these, we get for   ,
For , the flux is , so that
and the electric field foir  is
Exercise 1

A conducting circle having a radius at time is in a constant magnetic field perpendicular to its
plane. The circle expands with time with its radius becoming at time . Calculate the
emf developed in the circle.
(Ans. )

Mutual Inductance

According to Faraday's law, a changing magnetic flux in a loop causes an emf to be generated in that loop.
Consider two stationary coils carrying current. The first coil has turn and carries a current . The
second coil contains turns. The current in the first coil is the source of a magnetic field in the region
around the coil. The second loop encloses a flux , where is the surface of one
turn of the loop. If the current in the first coil is varied, ,and consequently will vary with time.
The variation of causes an emf  to be developed in the second coil. Since is proportional to , so
is . The emf, which is the rate of change of flux is, therefore, proportional to ,
where is a constant, called the mutual inductance of the two coils, which depends on geometrical factors
of the two loops, their relative orientation and the number of turns in each coil.

Analogously, we can argue that if the second loop carries a current which is varied with time, it generates
an induced emf in the first coil given by
For instance, consider two concentric solenoids, the outer one having turns per unit length and inner one
with turns per unit length. The solenoids are wound over coaxial cylinders of length each. If the current
in the outer solenoid is , the field due to it is , which is confined within the solenoid. The
flux enclosed by the inner cylinder is
If the current in the outer solenoid varies with time, the emf in the inner solenoid is
so that

If, on the other hand, the current in the inner solenoid is varied, the field due to it which
is non-zero only within the inner solenoid. The flux enclosed by the outer solenoid is, therefore,
If is varied, the emf in the outer solenoid is    giving
One can see that   .

This equality can be proved quite generally from Biot-Savart's law. Consider two circuits shown in the figure.
The field at , due to current in the loop (called the primary ) is
where . We have seen that can be expressed in terms of a vector potential , where
, by Biot-Savart's law
The flux enclosed by the second loop, (called the secondary ) is

Clearly,
It can be seen that the expression is symmetric between two loops. Hence we would get an identical
expression for . This expression is, however, of no significant use in obtaining the mutual inductance
because of rather difficult double integral.
Thus a knowledge of mutual inductance enables us to determine, how large should be the change in the
current (or voltage) in a primary circuit to obtain a desired value of current (or voltage) in the secondary
circuit. Since , we represent mutual inductance by the symbol . The emf in the
secondary circuit is given by   , where is the variable current in the primary circuit.
Units of is that of Volt-sec/Ampere which is known as Henry (h)

Example 22

Consider two parallel rings and with radii and respectively with a separation between their
centres. Radius of is much smaller than that of , so that the field experienced by due to a current
in may be taken to be uniform over its area. Find the mutual inductance of the rings.

Solution :

The field experienced by the smaller ring may be taken to be given by the expression for the magnetic field of
a ring along its axis. We had earlier shown that at a distance from the centre of the ring, the field along the
axis is given by

The flux enclosed by is

By Faraday's law the emf in is
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