Magnetostatics and Ohms Law - Maxwell's Equations

Magnetostatics and Ohms Law - Maxwell's Equations - Electrical Engineering (EE)

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Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Objectives
In this course you will learn the following
Magnetic Field and Magnetic Flux Density.
Conduction Current Density J.
Page 2

Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Objectives
In this course you will learn the following
Magnetic Field and Magnetic Flux Density.
Conduction Current Density J.
Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Magnetic Field and Magnetic Flux Density

The magnetic field can be defined as the force experienced by a unit magnetic monopole. However,
there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law.
Consider an infinitesimally small piece of a wire carrying current  in it. The length of the piece is say
. The current moment is then defined by the product . Since the piece of the wire can be oriented in
any direction the length  is a vector quantity and therefore should be denoted by . The current  is
a scalar quantity making the current moment  is a vector quantity. Without loss of generality let us
assume that the current element is located at the origin of the co-ordinate system as shown in Fig.

According to the Biot - Savart's law the magnetic field  ( field due to infinitesimally small current
element) at a point  is given as
Where  is the unit vector in the direction of the line joining the current element and the observation
point . Since the vectors  and  lie in the plane of the paper, the cross-product of the two has a
direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards,
the direction of  is upwards and the direction of the magnetic field  will be going into the paper as
indicated by  at point .

If we take some other observation point  on the left of the current element, by right hand screw rule,
the magnetic field will come out of the paper as indicated by .

If one streches his imagination a little, he can then see that the magnetic field forms a circular loop
around the axis of the current element.

The magnitude of the magnetic field is directly proportional to the current moment and is inversely
proportional to the square of the distance between the source and the observation point.

Page 3

Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Objectives
In this course you will learn the following
Magnetic Field and Magnetic Flux Density.
Conduction Current Density J.
Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Magnetic Field and Magnetic Flux Density

The magnetic field can be defined as the force experienced by a unit magnetic monopole. However,
there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law.
Consider an infinitesimally small piece of a wire carrying current  in it. The length of the piece is say
. The current moment is then defined by the product . Since the piece of the wire can be oriented in
any direction the length  is a vector quantity and therefore should be denoted by . The current  is
a scalar quantity making the current moment  is a vector quantity. Without loss of generality let us
assume that the current element is located at the origin of the co-ordinate system as shown in Fig.

According to the Biot - Savart's law the magnetic field  ( field due to infinitesimally small current
element) at a point  is given as
Where  is the unit vector in the direction of the line joining the current element and the observation
point . Since the vectors  and  lie in the plane of the paper, the cross-product of the two has a
direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards,
the direction of  is upwards and the direction of the magnetic field  will be going into the paper as
indicated by  at point .

If we take some other observation point  on the left of the current element, by right hand screw rule,
the magnetic field will come out of the paper as indicated by .

If one streches his imagination a little, he can then see that the magnetic field forms a circular loop
around the axis of the current element.

The magnitude of the magnetic field is directly proportional to the current moment and is inversely
proportional to the square of the distance between the source and the observation point.

The cross-product  also suggests that the strength of the magnetic field is maximum when
and  are perpendicular to each other, that is, for  equal to . As  reduces the angle between
and  reduces consequently reducing the strength of the magnetic field reduces.

For extreme case of  the vectors  and  are colinear making cross product and hence the
magnetic field identically zero. The magnetic field therefore is identically zero at the axis of the current
element.

The magnetic field  is related to  through a medium characteristic parameter called permeability of
the medium as
The permeability of vacuum is denoted by  and its value is  Henery/meter. A ratio of
permeability of a medium to that of the vacuum is called the relative permeability  giving

Page 4

Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Objectives
In this course you will learn the following
Magnetic Field and Magnetic Flux Density.
Conduction Current Density J.
Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Magnetic Field and Magnetic Flux Density

The magnetic field can be defined as the force experienced by a unit magnetic monopole. However,
there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law.
Consider an infinitesimally small piece of a wire carrying current  in it. The length of the piece is say
. The current moment is then defined by the product . Since the piece of the wire can be oriented in
any direction the length  is a vector quantity and therefore should be denoted by . The current  is
a scalar quantity making the current moment  is a vector quantity. Without loss of generality let us
assume that the current element is located at the origin of the co-ordinate system as shown in Fig.

According to the Biot - Savart's law the magnetic field  ( field due to infinitesimally small current
element) at a point  is given as
Where  is the unit vector in the direction of the line joining the current element and the observation
point . Since the vectors  and  lie in the plane of the paper, the cross-product of the two has a
direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards,
the direction of  is upwards and the direction of the magnetic field  will be going into the paper as
indicated by  at point .

If we take some other observation point  on the left of the current element, by right hand screw rule,
the magnetic field will come out of the paper as indicated by .

If one streches his imagination a little, he can then see that the magnetic field forms a circular loop
around the axis of the current element.

The magnitude of the magnetic field is directly proportional to the current moment and is inversely
proportional to the square of the distance between the source and the observation point.

The cross-product  also suggests that the strength of the magnetic field is maximum when
and  are perpendicular to each other, that is, for  equal to . As  reduces the angle between
and  reduces consequently reducing the strength of the magnetic field reduces.

For extreme case of  the vectors  and  are colinear making cross product and hence the
magnetic field identically zero. The magnetic field therefore is identically zero at the axis of the current
element.

The magnetic field  is related to  through a medium characteristic parameter called permeability of
the medium as
The permeability of vacuum is denoted by  and its value is  Henery/meter. A ratio of
permeability of a medium to that of the vacuum is called the relative permeability  giving

Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law

Conduction Current Density J
In situations where there is a variation in the current magnitude it is rather inadequate to define just the total current flow
in the system. One can then use the current density as the primary parameter which is defined as the current flow per
unit area.
Consider a cylinder made of a conducting material with an arbitrary cross-section . Let us assume that the current
density  exists in the cylinder over a length ` '. If the conductivity of the cylinder is  its resistivity is . The
resistance  of the cylinder between its two ends A and B is then given by
For simplicity assuming that the current density is constant all over the cross-section, the total current in the conductor is
Due to the current flow in the conductor, there is a voltage drop  between point A and B. If we assume the conducting
cylinder to be of infinitesimal length we can assume the electric field associated with the voltage drop to be constant
between A and B. Then one can write the voltage  to be
Now from Ohm's law we have
Substituting for ,  and  we get
The magnitude of the conduction current density  is proportional to the electric field strength E.

From the Ohm's law we also know that the current flows in the direction in which the potential drop is maximum. That is,
the current flows in the direction in which the potential  has maximum change. This direction is nothing but the direction
of the gradient of . The direction of gradient of V is same as that of the electric field . Therefore we can conclude
that not only the magnitude of  is proportional to  but its direction is also same as that of . The relation then can be
written for vector  and  as
This is the general form of the Ohms law.
Page 5

Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Objectives
In this course you will learn the following
Magnetic Field and Magnetic Flux Density.
Conduction Current Density J.
Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law
Magnetic Field and Magnetic Flux Density

The magnetic field can be defined as the force experienced by a unit magnetic monopole. However,
there are no magnetic monoples. The field due to a current element is given by the Biot-Savart law.
Consider an infinitesimally small piece of a wire carrying current  in it. The length of the piece is say
. The current moment is then defined by the product . Since the piece of the wire can be oriented in
any direction the length  is a vector quantity and therefore should be denoted by . The current  is
a scalar quantity making the current moment  is a vector quantity. Without loss of generality let us
assume that the current element is located at the origin of the co-ordinate system as shown in Fig.

According to the Biot - Savart's law the magnetic field  ( field due to infinitesimally small current
element) at a point  is given as
Where  is the unit vector in the direction of the line joining the current element and the observation
point . Since the vectors  and  lie in the plane of the paper, the cross-product of the two has a
direction prependicular to the plane of the paper. Since the direction of the current is assumed upwards,
the direction of  is upwards and the direction of the magnetic field  will be going into the paper as
indicated by  at point .

If we take some other observation point  on the left of the current element, by right hand screw rule,
the magnetic field will come out of the paper as indicated by .

If one streches his imagination a little, he can then see that the magnetic field forms a circular loop
around the axis of the current element.

The magnitude of the magnetic field is directly proportional to the current moment and is inversely
proportional to the square of the distance between the source and the observation point.

The cross-product  also suggests that the strength of the magnetic field is maximum when
and  are perpendicular to each other, that is, for  equal to . As  reduces the angle between
and  reduces consequently reducing the strength of the magnetic field reduces.

For extreme case of  the vectors  and  are colinear making cross product and hence the
magnetic field identically zero. The magnetic field therefore is identically zero at the axis of the current
element.

The magnetic field  is related to  through a medium characteristic parameter called permeability of
the medium as
The permeability of vacuum is denoted by  and its value is  Henery/meter. A ratio of
permeability of a medium to that of the vacuum is called the relative permeability  giving

Module 3 : Maxwell's Equations
Lecture 20 : Magnetostatics and Ohms Law

Conduction Current Density J
In situations where there is a variation in the current magnitude it is rather inadequate to define just the total current flow
in the system. One can then use the current density as the primary parameter which is defined as the current flow per
unit area.
Consider a cylinder made of a conducting material with an arbitrary cross-section . Let us assume that the current
density  exists in the cylinder over a length ` '. If the conductivity of the cylinder is  its resistivity is . The
resistance  of the cylinder between its two ends A and B is then given by
For simplicity assuming that the current density is constant all over the cross-section, the total current in the conductor is
Due to the current flow in the conductor, there is a voltage drop  between point A and B. If we assume the conducting
cylinder to be of infinitesimal length we can assume the electric field associated with the voltage drop to be constant
between A and B. Then one can write the voltage  to be
Now from Ohm's law we have
Substituting for ,  and  we get
The magnitude of the conduction current density  is proportional to the electric field strength E.

From the Ohm's law we also know that the current flows in the direction in which the potential drop is maximum. That is,
the current flows in the direction in which the potential  has maximum change. This direction is nothing but the direction
of the gradient of . The direction of gradient of V is same as that of the electric field . Therefore we can conclude
that not only the magnitude of  is proportional to  but its direction is also same as that of . The relation then can be
written for vector  and  as
This is the general form of the Ohms law.
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