Lecture 17 - Vector Potential - MAGNETIC FIELD Notes | EduRev

: Lecture 17 - Vector Potential - MAGNETIC FIELD Notes | EduRev

 Page 1


 Module 3 : MAGNETIC FIELD
 Lecture 17 : Vector Potential
 
 
 
Objectives
 
In this lecture you will learn the following
Define vector potential for a magnetic field.
Understand why vector potential is defined in a gauge.
Calculate vector potential for simple geometries.
Define electromotive force and state Faraday's law of induction
 
Vector Potential
 
For the electric field case, we had seen that it is possible to define a scalar function called the ``potential"
whose negative gradient is equal to the electric field : . The existence of such a scalar function is a
consequence of the conservative nature of the electric force. It also followed that the electric field is
irrotational, i.e. . 
For the magnetic field, Ampere's law gives a non-zero curl
Since the curl of a gradient is always zero, we cannot express as a gradient of a scalar function as it would
then violate Ampere's law. 
However, we may introduce a vector function such that
 
This would automatically satisfy since divergence of a curl is zero. is known as vector
potential . Recall that a vector field is uniquely determined by specifying its divergence and curl. As is a
physical quantity, curl of is also so. However, the divergence of the vector potential has no physical
meaning and consequently we are at liberty to specify its divergence as per our wish. This freedom to choose
a vector potential whose curl is and whose divergence can be conveniently chosen is called by
mathematicians as a choice of a gauge . If is a scalar function any transformation of the type
gives the same magnetic field as curl of a gradient is identically zero. The transformation above is known as
gauge invariance . (we have a similar freedom for the scalar potential of the electric field in the sense that
Page 2


 Module 3 : MAGNETIC FIELD
 Lecture 17 : Vector Potential
 
 
 
Objectives
 
In this lecture you will learn the following
Define vector potential for a magnetic field.
Understand why vector potential is defined in a gauge.
Calculate vector potential for simple geometries.
Define electromotive force and state Faraday's law of induction
 
Vector Potential
 
For the electric field case, we had seen that it is possible to define a scalar function called the ``potential"
whose negative gradient is equal to the electric field : . The existence of such a scalar function is a
consequence of the conservative nature of the electric force. It also followed that the electric field is
irrotational, i.e. . 
For the magnetic field, Ampere's law gives a non-zero curl
Since the curl of a gradient is always zero, we cannot express as a gradient of a scalar function as it would
then violate Ampere's law. 
However, we may introduce a vector function such that
 
This would automatically satisfy since divergence of a curl is zero. is known as vector
potential . Recall that a vector field is uniquely determined by specifying its divergence and curl. As is a
physical quantity, curl of is also so. However, the divergence of the vector potential has no physical
meaning and consequently we are at liberty to specify its divergence as per our wish. This freedom to choose
a vector potential whose curl is and whose divergence can be conveniently chosen is called by
mathematicians as a choice of a gauge . If is a scalar function any transformation of the type
gives the same magnetic field as curl of a gradient is identically zero. The transformation above is known as
gauge invariance . (we have a similar freedom for the scalar potential of the electric field in the sense that
it is determined up to an additive constant. Our most common choice of is one for which at infinite
distances.) 
A popular gauge choice for is one in which
which is known as the ``Coulomb gauge". It can be shown that such a choice can always be made.
 
Exercise 1
 
Show that a possible choice of the vector potential for a constant magnetic field is given by 
. Can you construct any other ?
(Hint : Take in z-direction, express in component form and take its curl.)
 
Biot-Savart's Law for Vector Potential
 
Biot-Savart's law for magnetic field due to a current element 
may be used to obtain an expression for the vector potential. Since the element does not depend on the
position vector of the point at which the magnetic field is calculated, we can write 
the change in sign is because . 
Thus the contribution to the vector potential from the element is 
The expression is to be integrated over the path of the current to get the vector potential for the system
Example 16   
 
Obtain an expression for the vector potential at a point due to a long current carrying wire.
 
Solution :
 
Take the wire to be along the z-direction, perpendicular to the plane of the page with current flowing in a
direction out of the page. The magnitude of the field at a point P is with its direction being along
the tangential unit vector at P, 
 
The direction of makes an angle with the x direction, where . Thus
Page 3


 Module 3 : MAGNETIC FIELD
 Lecture 17 : Vector Potential
 
 
 
Objectives
 
In this lecture you will learn the following
Define vector potential for a magnetic field.
Understand why vector potential is defined in a gauge.
Calculate vector potential for simple geometries.
Define electromotive force and state Faraday's law of induction
 
Vector Potential
 
For the electric field case, we had seen that it is possible to define a scalar function called the ``potential"
whose negative gradient is equal to the electric field : . The existence of such a scalar function is a
consequence of the conservative nature of the electric force. It also followed that the electric field is
irrotational, i.e. . 
For the magnetic field, Ampere's law gives a non-zero curl
Since the curl of a gradient is always zero, we cannot express as a gradient of a scalar function as it would
then violate Ampere's law. 
However, we may introduce a vector function such that
 
This would automatically satisfy since divergence of a curl is zero. is known as vector
potential . Recall that a vector field is uniquely determined by specifying its divergence and curl. As is a
physical quantity, curl of is also so. However, the divergence of the vector potential has no physical
meaning and consequently we are at liberty to specify its divergence as per our wish. This freedom to choose
a vector potential whose curl is and whose divergence can be conveniently chosen is called by
mathematicians as a choice of a gauge . If is a scalar function any transformation of the type
gives the same magnetic field as curl of a gradient is identically zero. The transformation above is known as
gauge invariance . (we have a similar freedom for the scalar potential of the electric field in the sense that
it is determined up to an additive constant. Our most common choice of is one for which at infinite
distances.) 
A popular gauge choice for is one in which
which is known as the ``Coulomb gauge". It can be shown that such a choice can always be made.
 
Exercise 1
 
Show that a possible choice of the vector potential for a constant magnetic field is given by 
. Can you construct any other ?
(Hint : Take in z-direction, express in component form and take its curl.)
 
Biot-Savart's Law for Vector Potential
 
Biot-Savart's law for magnetic field due to a current element 
may be used to obtain an expression for the vector potential. Since the element does not depend on the
position vector of the point at which the magnetic field is calculated, we can write 
the change in sign is because . 
Thus the contribution to the vector potential from the element is 
The expression is to be integrated over the path of the current to get the vector potential for the system
Example 16   
 
Obtain an expression for the vector potential at a point due to a long current carrying wire.
 
Solution :
 
Take the wire to be along the z-direction, perpendicular to the plane of the page with current flowing in a
direction out of the page. The magnitude of the field at a point P is with its direction being along
the tangential unit vector at P, 
 
The direction of makes an angle with the x direction, where . Thus
 
 
Hence we have 
We wish to find a vector function whose curl is given by the above. One can see that the following
function fits the requirement
In the following, we will derive this directly from the expression for Biot-Savart's law. If is the distance of P
from an element of length at of the wire, we have, 
 
Thus  
If the above integral is evaluated from to , it diverges. However, we can eliminate the
infinity in the following manner. Let us take the wire to be of length so that
Page 4


 Module 3 : MAGNETIC FIELD
 Lecture 17 : Vector Potential
 
 
 
Objectives
 
In this lecture you will learn the following
Define vector potential for a magnetic field.
Understand why vector potential is defined in a gauge.
Calculate vector potential for simple geometries.
Define electromotive force and state Faraday's law of induction
 
Vector Potential
 
For the electric field case, we had seen that it is possible to define a scalar function called the ``potential"
whose negative gradient is equal to the electric field : . The existence of such a scalar function is a
consequence of the conservative nature of the electric force. It also followed that the electric field is
irrotational, i.e. . 
For the magnetic field, Ampere's law gives a non-zero curl
Since the curl of a gradient is always zero, we cannot express as a gradient of a scalar function as it would
then violate Ampere's law. 
However, we may introduce a vector function such that
 
This would automatically satisfy since divergence of a curl is zero. is known as vector
potential . Recall that a vector field is uniquely determined by specifying its divergence and curl. As is a
physical quantity, curl of is also so. However, the divergence of the vector potential has no physical
meaning and consequently we are at liberty to specify its divergence as per our wish. This freedom to choose
a vector potential whose curl is and whose divergence can be conveniently chosen is called by
mathematicians as a choice of a gauge . If is a scalar function any transformation of the type
gives the same magnetic field as curl of a gradient is identically zero. The transformation above is known as
gauge invariance . (we have a similar freedom for the scalar potential of the electric field in the sense that
it is determined up to an additive constant. Our most common choice of is one for which at infinite
distances.) 
A popular gauge choice for is one in which
which is known as the ``Coulomb gauge". It can be shown that such a choice can always be made.
 
Exercise 1
 
Show that a possible choice of the vector potential for a constant magnetic field is given by 
. Can you construct any other ?
(Hint : Take in z-direction, express in component form and take its curl.)
 
Biot-Savart's Law for Vector Potential
 
Biot-Savart's law for magnetic field due to a current element 
may be used to obtain an expression for the vector potential. Since the element does not depend on the
position vector of the point at which the magnetic field is calculated, we can write 
the change in sign is because . 
Thus the contribution to the vector potential from the element is 
The expression is to be integrated over the path of the current to get the vector potential for the system
Example 16   
 
Obtain an expression for the vector potential at a point due to a long current carrying wire.
 
Solution :
 
Take the wire to be along the z-direction, perpendicular to the plane of the page with current flowing in a
direction out of the page. The magnitude of the field at a point P is with its direction being along
the tangential unit vector at P, 
 
The direction of makes an angle with the x direction, where . Thus
 
 
Hence we have 
We wish to find a vector function whose curl is given by the above. One can see that the following
function fits the requirement
In the following, we will derive this directly from the expression for Biot-Savart's law. If is the distance of P
from an element of length at of the wire, we have, 
 
Thus  
If the above integral is evaluated from to , it diverges. However, we can eliminate the
infinity in the following manner. Let us take the wire to be of length so that
 
The integral is evaluated by substituting , so that . We get
where . 
In terms of and , we have 
Thus to leading order in , 
As expected, for , the expression diverges. However, since itself is not physical while curl of 
is, the constant term (which diverges in the limit of ) is of no consequence and is
given by
which is the same as Eqn. (1)
 
Example 17
 
Obtain an expression for the vector potential of a solenoid.
 
Solution :
We had seen that for a solenoid, the field is parallel to the axis for points inside the solenoid and is zero
Page 5


 Module 3 : MAGNETIC FIELD
 Lecture 17 : Vector Potential
 
 
 
Objectives
 
In this lecture you will learn the following
Define vector potential for a magnetic field.
Understand why vector potential is defined in a gauge.
Calculate vector potential for simple geometries.
Define electromotive force and state Faraday's law of induction
 
Vector Potential
 
For the electric field case, we had seen that it is possible to define a scalar function called the ``potential"
whose negative gradient is equal to the electric field : . The existence of such a scalar function is a
consequence of the conservative nature of the electric force. It also followed that the electric field is
irrotational, i.e. . 
For the magnetic field, Ampere's law gives a non-zero curl
Since the curl of a gradient is always zero, we cannot express as a gradient of a scalar function as it would
then violate Ampere's law. 
However, we may introduce a vector function such that
 
This would automatically satisfy since divergence of a curl is zero. is known as vector
potential . Recall that a vector field is uniquely determined by specifying its divergence and curl. As is a
physical quantity, curl of is also so. However, the divergence of the vector potential has no physical
meaning and consequently we are at liberty to specify its divergence as per our wish. This freedom to choose
a vector potential whose curl is and whose divergence can be conveniently chosen is called by
mathematicians as a choice of a gauge . If is a scalar function any transformation of the type
gives the same magnetic field as curl of a gradient is identically zero. The transformation above is known as
gauge invariance . (we have a similar freedom for the scalar potential of the electric field in the sense that
it is determined up to an additive constant. Our most common choice of is one for which at infinite
distances.) 
A popular gauge choice for is one in which
which is known as the ``Coulomb gauge". It can be shown that such a choice can always be made.
 
Exercise 1
 
Show that a possible choice of the vector potential for a constant magnetic field is given by 
. Can you construct any other ?
(Hint : Take in z-direction, express in component form and take its curl.)
 
Biot-Savart's Law for Vector Potential
 
Biot-Savart's law for magnetic field due to a current element 
may be used to obtain an expression for the vector potential. Since the element does not depend on the
position vector of the point at which the magnetic field is calculated, we can write 
the change in sign is because . 
Thus the contribution to the vector potential from the element is 
The expression is to be integrated over the path of the current to get the vector potential for the system
Example 16   
 
Obtain an expression for the vector potential at a point due to a long current carrying wire.
 
Solution :
 
Take the wire to be along the z-direction, perpendicular to the plane of the page with current flowing in a
direction out of the page. The magnitude of the field at a point P is with its direction being along
the tangential unit vector at P, 
 
The direction of makes an angle with the x direction, where . Thus
 
 
Hence we have 
We wish to find a vector function whose curl is given by the above. One can see that the following
function fits the requirement
In the following, we will derive this directly from the expression for Biot-Savart's law. If is the distance of P
from an element of length at of the wire, we have, 
 
Thus  
If the above integral is evaluated from to , it diverges. However, we can eliminate the
infinity in the following manner. Let us take the wire to be of length so that
 
The integral is evaluated by substituting , so that . We get
where . 
In terms of and , we have 
Thus to leading order in , 
As expected, for , the expression diverges. However, since itself is not physical while curl of 
is, the constant term (which diverges in the limit of ) is of no consequence and is
given by
which is the same as Eqn. (1)
 
Example 17
 
Obtain an expression for the vector potential of a solenoid.
 
Solution :
We had seen that for a solenoid, the field is parallel to the axis for points inside the solenoid and is zero
 
outside.
 
 Take a circle of radius perpendicular to the axis of the solenoid. The flux of the magnetic field is
 
 
Since is axial, is directed tangentially to the circle. Further, from symmetry, the magnitude of 
is constant on the circumference of the circle.
Use of Stoke's theorem gives
Thus
where is the unit vector along the tangential direction.
 
Exercise 2
 
Obtain an expression for the vector potential inside a cylindrical wire of radius carrying a current .
 (Ans. )
The existence of a vector potential whose curl gives the magnetic field directly gives
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