Time Varying Fields | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

 Page 1


Introduction: 
 
In our study of static fields so far, we have observed that static electric fields are produced by 
electric charges, static magnetic fields are produced by charges in motion or by steady current. 
Further, static electric field is a conservative field and has no curl, the static magnetic field is 
continuous and its divergence is zero. The fundamental relationships for static electric fields 
among the field quantities can be summarized as: 
(1) 
 
(2) 
For a linear and isotropic medium, 
 
(3) 
Similarly for the magnetostatic case 
 
(4) 
 
(5) 
 
(6) 
It can be seen that for static case, the electric field vectors and  and magnetic field  
vectors and form separate pairs. 
In this chapter we will consider the time varying scenario. In the time varying case we 
will observe that a changing magnetic field will produce a changing electric field and vice versa. 
We begin our discussion with Faraday's Law of electromagnetic induction and then 
present the Maxwell's equations which form the foundation for the electromagnetic theory. 
Faraday's Law of electromagnetic Induction: 
Michael Faraday, in 1831 discovered experimentally that a current was induced in a 
conducting loop when the magnetic flux linking the loop changed. In terms of fields, we can say 
that a time varying magnetic field produces an electromotive force (emf) which causes a current 
in a closed circuit. The quantitative relation between the induced emf (the voltage that arises 
from conductors moving in a magnetic field or from changing magnetic fields) and the rate of 
change of flux linkage developed based on experimental observation is known as Faraday's law. 
Mathematically, the induced emf can be written as 
Page 2


Introduction: 
 
In our study of static fields so far, we have observed that static electric fields are produced by 
electric charges, static magnetic fields are produced by charges in motion or by steady current. 
Further, static electric field is a conservative field and has no curl, the static magnetic field is 
continuous and its divergence is zero. The fundamental relationships for static electric fields 
among the field quantities can be summarized as: 
(1) 
 
(2) 
For a linear and isotropic medium, 
 
(3) 
Similarly for the magnetostatic case 
 
(4) 
 
(5) 
 
(6) 
It can be seen that for static case, the electric field vectors and  and magnetic field  
vectors and form separate pairs. 
In this chapter we will consider the time varying scenario. In the time varying case we 
will observe that a changing magnetic field will produce a changing electric field and vice versa. 
We begin our discussion with Faraday's Law of electromagnetic induction and then 
present the Maxwell's equations which form the foundation for the electromagnetic theory. 
Faraday's Law of electromagnetic Induction: 
Michael Faraday, in 1831 discovered experimentally that a current was induced in a 
conducting loop when the magnetic flux linking the loop changed. In terms of fields, we can say 
that a time varying magnetic field produces an electromotive force (emf) which causes a current 
in a closed circuit. The quantitative relation between the induced emf (the voltage that arises 
from conductors moving in a magnetic field or from changing magnetic fields) and the rate of 
change of flux linkage developed based on experimental observation is known as Faraday's law. 
Mathematically, the induced emf can be written as 
 
Emf =  Volts (7) 
where is the flux linkage over the closed path. 
 
A non zero may result due to any of the following: 
(a) time changing flux linkage a stationary closed path. 
(b) relative motion between a steady flux a closed path. 
(c) a combination of the above two cases. 
The negative sign in equation (7) was introduced by Lenz in order to comply with the 
polarity of the induced emf. The negative sign implies that the induced emf will cause a current 
flow in the closed loop in such a direction so as to oppose the change in the linking magnetic 
flux which produces it. (It may be noted that as far as the induced emf is concerned, the closed 
path forming a loop does not necessarily have to be conductive). 
If the closed path is in the form of N tightly wound turns of a coil, the change in the 
magnetic flux linking the coil induces an emf in each turn of the coil and total emf is the sum of 
the induced emfs of the individual turns, i.e., 
 
Emf = Volts (8) 
By defining the total flux linkage as 
(9) 
The emf can be written as 
 
Emf = (10) 
Continuing with equation (3), over a closed contour 'C' we can write 
Emf =                           (11) 
where is the induced electric field on the conductor to sustain the current. 
Further, total flux enclosed by the contour 'C ' is given by 
 
(12) 
Where S is the surface for which 'C' is the contour. 
Page 3


Introduction: 
 
In our study of static fields so far, we have observed that static electric fields are produced by 
electric charges, static magnetic fields are produced by charges in motion or by steady current. 
Further, static electric field is a conservative field and has no curl, the static magnetic field is 
continuous and its divergence is zero. The fundamental relationships for static electric fields 
among the field quantities can be summarized as: 
(1) 
 
(2) 
For a linear and isotropic medium, 
 
(3) 
Similarly for the magnetostatic case 
 
(4) 
 
(5) 
 
(6) 
It can be seen that for static case, the electric field vectors and  and magnetic field  
vectors and form separate pairs. 
In this chapter we will consider the time varying scenario. In the time varying case we 
will observe that a changing magnetic field will produce a changing electric field and vice versa. 
We begin our discussion with Faraday's Law of electromagnetic induction and then 
present the Maxwell's equations which form the foundation for the electromagnetic theory. 
Faraday's Law of electromagnetic Induction: 
Michael Faraday, in 1831 discovered experimentally that a current was induced in a 
conducting loop when the magnetic flux linking the loop changed. In terms of fields, we can say 
that a time varying magnetic field produces an electromotive force (emf) which causes a current 
in a closed circuit. The quantitative relation between the induced emf (the voltage that arises 
from conductors moving in a magnetic field or from changing magnetic fields) and the rate of 
change of flux linkage developed based on experimental observation is known as Faraday's law. 
Mathematically, the induced emf can be written as 
 
Emf =  Volts (7) 
where is the flux linkage over the closed path. 
 
A non zero may result due to any of the following: 
(a) time changing flux linkage a stationary closed path. 
(b) relative motion between a steady flux a closed path. 
(c) a combination of the above two cases. 
The negative sign in equation (7) was introduced by Lenz in order to comply with the 
polarity of the induced emf. The negative sign implies that the induced emf will cause a current 
flow in the closed loop in such a direction so as to oppose the change in the linking magnetic 
flux which produces it. (It may be noted that as far as the induced emf is concerned, the closed 
path forming a loop does not necessarily have to be conductive). 
If the closed path is in the form of N tightly wound turns of a coil, the change in the 
magnetic flux linking the coil induces an emf in each turn of the coil and total emf is the sum of 
the induced emfs of the individual turns, i.e., 
 
Emf = Volts (8) 
By defining the total flux linkage as 
(9) 
The emf can be written as 
 
Emf = (10) 
Continuing with equation (3), over a closed contour 'C' we can write 
Emf =                           (11) 
where is the induced electric field on the conductor to sustain the current. 
Further, total flux enclosed by the contour 'C ' is given by 
 
(12) 
Where S is the surface for which 'C' is the contour. 
From (11) and using (12) in (3) we can write 
 
 
 
By applying stokes theorem 
 
Therefore, we can write 
(13) 
 
 
 
(14) 
 
 
(15) 
which is the Faraday's law in the point form 
 
We have said that non zero can be produced in a several ways. One particular case is when a 
time varying flux linking a stationary closed path induces an emf. The emf induced in a 
stationary closed path by a time varying magnetic field is called a transformer emf . 
Statically and dynamically induced EMFs: 
Motional EMF: 
Let us consider a conductor moving in a steady magnetic field as shown in the fig 2. 
 
Fig 2 
If a charge Q moves in a magnetic field , it experiences a force 
 
(16) 
This force will cause the electrons in the conductor to drift towards one end and leave the other 
end positively charged, thus creating a field and charge separation continuous until electric and 
magnetic forces balance and an equilibrium is reached very quickly, the net force on the moving 
conductor is zero. 
Page 4


Introduction: 
 
In our study of static fields so far, we have observed that static electric fields are produced by 
electric charges, static magnetic fields are produced by charges in motion or by steady current. 
Further, static electric field is a conservative field and has no curl, the static magnetic field is 
continuous and its divergence is zero. The fundamental relationships for static electric fields 
among the field quantities can be summarized as: 
(1) 
 
(2) 
For a linear and isotropic medium, 
 
(3) 
Similarly for the magnetostatic case 
 
(4) 
 
(5) 
 
(6) 
It can be seen that for static case, the electric field vectors and  and magnetic field  
vectors and form separate pairs. 
In this chapter we will consider the time varying scenario. In the time varying case we 
will observe that a changing magnetic field will produce a changing electric field and vice versa. 
We begin our discussion with Faraday's Law of electromagnetic induction and then 
present the Maxwell's equations which form the foundation for the electromagnetic theory. 
Faraday's Law of electromagnetic Induction: 
Michael Faraday, in 1831 discovered experimentally that a current was induced in a 
conducting loop when the magnetic flux linking the loop changed. In terms of fields, we can say 
that a time varying magnetic field produces an electromotive force (emf) which causes a current 
in a closed circuit. The quantitative relation between the induced emf (the voltage that arises 
from conductors moving in a magnetic field or from changing magnetic fields) and the rate of 
change of flux linkage developed based on experimental observation is known as Faraday's law. 
Mathematically, the induced emf can be written as 
 
Emf =  Volts (7) 
where is the flux linkage over the closed path. 
 
A non zero may result due to any of the following: 
(a) time changing flux linkage a stationary closed path. 
(b) relative motion between a steady flux a closed path. 
(c) a combination of the above two cases. 
The negative sign in equation (7) was introduced by Lenz in order to comply with the 
polarity of the induced emf. The negative sign implies that the induced emf will cause a current 
flow in the closed loop in such a direction so as to oppose the change in the linking magnetic 
flux which produces it. (It may be noted that as far as the induced emf is concerned, the closed 
path forming a loop does not necessarily have to be conductive). 
If the closed path is in the form of N tightly wound turns of a coil, the change in the 
magnetic flux linking the coil induces an emf in each turn of the coil and total emf is the sum of 
the induced emfs of the individual turns, i.e., 
 
Emf = Volts (8) 
By defining the total flux linkage as 
(9) 
The emf can be written as 
 
Emf = (10) 
Continuing with equation (3), over a closed contour 'C' we can write 
Emf =                           (11) 
where is the induced electric field on the conductor to sustain the current. 
Further, total flux enclosed by the contour 'C ' is given by 
 
(12) 
Where S is the surface for which 'C' is the contour. 
From (11) and using (12) in (3) we can write 
 
 
 
By applying stokes theorem 
 
Therefore, we can write 
(13) 
 
 
 
(14) 
 
 
(15) 
which is the Faraday's law in the point form 
 
We have said that non zero can be produced in a several ways. One particular case is when a 
time varying flux linking a stationary closed path induces an emf. The emf induced in a 
stationary closed path by a time varying magnetic field is called a transformer emf . 
Statically and dynamically induced EMFs: 
Motional EMF: 
Let us consider a conductor moving in a steady magnetic field as shown in the fig 2. 
 
Fig 2 
If a charge Q moves in a magnetic field , it experiences a force 
 
(16) 
This force will cause the electrons in the conductor to drift towards one end and leave the other 
end positively charged, thus creating a field and charge separation continuous until electric and 
magnetic forces balance and an equilibrium is reached very quickly, the net force on the moving 
conductor is zero. 
 
 
 
field 
can be interpreted as an induced electric field which is called the motional electric 
 
 
(17) 
If the moving conductor is a part of the closed circuit C, the generated emf around the circuit is 
 . This emf is called the motional emf. 
Page 5


Introduction: 
 
In our study of static fields so far, we have observed that static electric fields are produced by 
electric charges, static magnetic fields are produced by charges in motion or by steady current. 
Further, static electric field is a conservative field and has no curl, the static magnetic field is 
continuous and its divergence is zero. The fundamental relationships for static electric fields 
among the field quantities can be summarized as: 
(1) 
 
(2) 
For a linear and isotropic medium, 
 
(3) 
Similarly for the magnetostatic case 
 
(4) 
 
(5) 
 
(6) 
It can be seen that for static case, the electric field vectors and  and magnetic field  
vectors and form separate pairs. 
In this chapter we will consider the time varying scenario. In the time varying case we 
will observe that a changing magnetic field will produce a changing electric field and vice versa. 
We begin our discussion with Faraday's Law of electromagnetic induction and then 
present the Maxwell's equations which form the foundation for the electromagnetic theory. 
Faraday's Law of electromagnetic Induction: 
Michael Faraday, in 1831 discovered experimentally that a current was induced in a 
conducting loop when the magnetic flux linking the loop changed. In terms of fields, we can say 
that a time varying magnetic field produces an electromotive force (emf) which causes a current 
in a closed circuit. The quantitative relation between the induced emf (the voltage that arises 
from conductors moving in a magnetic field or from changing magnetic fields) and the rate of 
change of flux linkage developed based on experimental observation is known as Faraday's law. 
Mathematically, the induced emf can be written as 
 
Emf =  Volts (7) 
where is the flux linkage over the closed path. 
 
A non zero may result due to any of the following: 
(a) time changing flux linkage a stationary closed path. 
(b) relative motion between a steady flux a closed path. 
(c) a combination of the above two cases. 
The negative sign in equation (7) was introduced by Lenz in order to comply with the 
polarity of the induced emf. The negative sign implies that the induced emf will cause a current 
flow in the closed loop in such a direction so as to oppose the change in the linking magnetic 
flux which produces it. (It may be noted that as far as the induced emf is concerned, the closed 
path forming a loop does not necessarily have to be conductive). 
If the closed path is in the form of N tightly wound turns of a coil, the change in the 
magnetic flux linking the coil induces an emf in each turn of the coil and total emf is the sum of 
the induced emfs of the individual turns, i.e., 
 
Emf = Volts (8) 
By defining the total flux linkage as 
(9) 
The emf can be written as 
 
Emf = (10) 
Continuing with equation (3), over a closed contour 'C' we can write 
Emf =                           (11) 
where is the induced electric field on the conductor to sustain the current. 
Further, total flux enclosed by the contour 'C ' is given by 
 
(12) 
Where S is the surface for which 'C' is the contour. 
From (11) and using (12) in (3) we can write 
 
 
 
By applying stokes theorem 
 
Therefore, we can write 
(13) 
 
 
 
(14) 
 
 
(15) 
which is the Faraday's law in the point form 
 
We have said that non zero can be produced in a several ways. One particular case is when a 
time varying flux linking a stationary closed path induces an emf. The emf induced in a 
stationary closed path by a time varying magnetic field is called a transformer emf . 
Statically and dynamically induced EMFs: 
Motional EMF: 
Let us consider a conductor moving in a steady magnetic field as shown in the fig 2. 
 
Fig 2 
If a charge Q moves in a magnetic field , it experiences a force 
 
(16) 
This force will cause the electrons in the conductor to drift towards one end and leave the other 
end positively charged, thus creating a field and charge separation continuous until electric and 
magnetic forces balance and an equilibrium is reached very quickly, the net force on the moving 
conductor is zero. 
 
 
 
field 
can be interpreted as an induced electric field which is called the motional electric 
 
 
(17) 
If the moving conductor is a part of the closed circuit C, the generated emf around the circuit is 
 . This emf is called the motional emf. 
Modification of Maxwell’s equations for time varying fields :