Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

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Electrical Engineering (EE) : Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The document Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Electromagnetic Theory.
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Mutual Inductance 

In the last lecture, we enunciated the Faraday’s law according to which if there is a changing magnetic flux in a circuit, there is an emf induced. Incidentally, the circuit that we are talking about could be an imaginary circuit in space as well.

Consider two circuits, we call them loop 1 and loop 2. Suppose the current in loop 1 is changing with time. This means that the magnetic field produced by the current in loop 1 is also time varying. If loop 2 intercepts the flux of this changing magnetic field, accrding to Faraday’s law an emf would be generated in this circuit.

We can express the magnetic field due to the current I in loop 1, using the Biot Savart’s law,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

where  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev is the position vector of the point of observation (i.e. the point where the field is calculated) with respect to the current element  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev This gives rise to flux in loop 2 which can be written as

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

It can be seen that the flux in the second loop is proprtional to the current in the first loop. We can therefore write,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The proportinaliity constant M21 is known as  the mutual inductance of the two circuits and it depends on geomery of the circuits and their relative positions and orientations.

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The emf in the second circuit due to a change in the current in the first circuit is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

We can show that  the mutual inductance is symmetric, i.e.,  M21 = M12.  To show this, let us reexpress the flux in the second loop in terms of the vector potential of the first loop,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

We have seen that the vector potential of  loop 1 can be expressed as 

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

where r is the distance from the length element  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev to the point where the vector potential is calculated.  If we express  with respect to a fixed origin, then the vector potential at a position  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev on the loop 2 is given by

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Thus the expression from the flux through the second loop is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

This shows that

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Which is manifestly symmetric in the indices 1 and 2, The above relation  is known as Neumann’s formula. 

Self Inductance 

We come to an interesting consequence of the above. Suppose, instead of two circuits, I have just a single loop and we change current in that loop. This change in current changes the magnetic field associated with the current and the flux changes. The given loop uitself will intercept the flux and the changing flux would result in an emf in the circuit itself,  the current induced opposing the changing current. The loop does not care about what caused the changing of flux and an emf develops in the loop as per Faraday’s law. We are talking about a self effect and the emf is known as the “back emf”. The emf is given by

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

where  L is known as the self inductance of the loop.

Example : Self inductance of a solenoid

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Consider a solenoid of length L. Neglecting edge effects, the field of a solenoid is confined to inside of the solenoid and is directed along the axis (z direction) ,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

where n is the number of turns per unit length of the solenoid. This field threads each turn of the loop and the flux “linked” with each turn is given by multiplying the above with the area of the turn,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

If l is the length of the solenoid, there are number of turns in the solenoid,  and the total flux linked is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev The self inductance is thus given by

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Example : Mutual Inductance of two tightly wound solenoids

Consider two solenoids which fit snugly with each other so that all the magnetic field produced by any of the solenoids is intercepted by the other.

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The flux linked with the second loop when the current | flows through the first loop,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

This gives the mutual inductance to be 

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

From the previous  example, we can obtain the self inductances of each of the solenoids,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

One can see that we get, in this case,  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev This is not a general expression  but in general there is such a relationship valid with a proportionality constant known as the “coefficient of coupling” Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev the coefficient of coupling depends on the relative orientation of the two loops.

Example : Two coplanar and concentric loops

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Let us assume R1<<R2 (figure not to scale). The field at the centre  of the bigger loop is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev We can suume that this is the field all over the smaller loop and the flux through the smaller loop is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev The mutual inductance in this case is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Energy of a current distribution

Recall the way we  calculated the energy of a charge distribution. We assumed that all the charges were first at infinity so that there was no electric field in space. We then moved the first charge to its position without any energy cost. The second charge was then moved to its position in the field created by the first charge and so on. We cannot remove all the current distribution to infinity and so we must adopt a new apprach for calculation of energy in this case.

When we establish a current in a circuit, we have seen that a back emf develops because of changing current. Work has to be done to compensate this and establish the value of the steady cUrrent that we wish to develop .

If there is more than one circuit, when current change occurs in any of the circuit, it causes an emf in all others. To maintain the current distribution, work has to be done as  well.

We have seen how to write the flux through a given loop when  there is a current in that loop or in another loop. Generalizing this to a large number of loops, we can write the flux through the i-th loop is 

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

where Li is the self inductance of the i-th loop and Mij is the mutual inductance between the ith  and the j-th loops.  The emf through the i-th  loop  when the currents in each loop changes is given by

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The rate of doing work to overcome this emf is 

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Thus, when currents in all the circuits change the rate at which work must be done in maintaining the currents in all the circuits is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The last step followed because we could introduce a factor of ½ and write the second term as a symmetric term in terms of I and j using the fact that  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The total work done is obtained by integrating the above over time,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

This is for a discrete current distribution. For a continuous current distribution, this can be easily generalized to give,  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Displacement Current

We are still left with one issue. We understand the assymmetry in the Gauss’s law of magnetism and electrostatics because of absence of magnetic monopoles. The time dependent phenomena requires little more thought. If a changing magnetic field could induce an electric field, what about the corresponding effect where there is a changinfg electric field exists? The effect was not detected for long because of reasons that would become clear later. Maxwell had thought about this problem and had concluded that such an effect does indeed exist. This was actually Maxwell’s contribution to the set of electromagnetic frield equations which bear his name.

We will illustrate the effect by considering charging of a capacitor plates in a circuit. Assume that we have a circuit which a source and a capacitor, which, for simplicity we take to be a parallel plate capacitor. When the key is closed, current momentarily flows from the battery charging the capacitor plates. We know that there is no current through the gap of the capacitor. Nevertheless, during the operiod when the charging is taking place, there is a changing electric field inside the capacitor plates.

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

During the process of charging a current exists in the external circuit and we can calculate the magnetic field by using Ampere’s law,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Conser the loop to be directed along the magnetic field direction. If we take a disk as a srface defined by this loop, the flux lines pass through this disk and we get  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev as has been done before.

However, we now come to a curious anomaly. Suppose, instead of the disk, we take another surface defined by the same loop as its boundary but a pot slike shape which does not insersect the outsdie wire and passes through the capacitor gap, as shown.

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Since there is no current passes through the surface, we would get  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

This is not understandable as both the surface integral represent a loop integral through the same loop and must give unique answer.

The way Maxwell resolved this apparent anomaly is to postulate that just as there is an induced electric field associated with a changing magnetic flux, there is an induced magnetic field associated with a changing electric flux. This has since been verified experimentally though the effect is much smaller than that of Faraday’s law.

To visualize this, let us assume that the space between the capacitor plate is filled with a dielectric. In an applied electric field the bound charges within the dielectric would be pushed. This would not of course give rise to a current because such charges would never leave the dielectric. However, Maxwell imagined that the effect would provide something akin to a current which would help remove the anomaly talked above.

The electric flux through the second surface is given by 

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Let us calculate the rate of change of this flux.

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The rate of change of free charges is clearly the current flowing in to charge the plates. Maxwell called Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev as the displacement current which is equal to the conduction current flowing in to charge the capacitor. This is the “current” which provides continuity of the current flow and is given by 

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Maxwell then proposed that the Ampere’s law be modified by including this term on the right hand side of the curl equation. In the outside circuit, the electric flux ios zero and only the conduction current exists. Inside the capacitor plates, the conduction current is zero but there is a changing electric flux giving rise to a term which is like a current.

We have then, for Ampere’s law,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

because the displacement current density is obviously given by the second term.

Sincedivergence of a curl is zero, if we take divergence of theabove equation, we get,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

which is just the equation of continuity. Thus all the Maxwell’s equations are in place now. They are

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

We supplement these with two “constitutive relations”

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Tutorial Assignment

1. Calculate the self inductance of a long wire of radius R carrying a current I uniformly distributed over its cross section.

2. A toroidal coil of N turns is tightly wound over a doughnut of inner radius a and outer radius b. A long straight wire passes through the centre of the doughnut and carries a time varying current Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev  Determine the mutual inductance between the coil and the straight wire.

3. Two circuits A and B have a mutual inductance M. At t=0, the current in the coil A is switched on and it increases with times as  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev The emf induced in the coil B as a result is found to change with time as given by the relation ε=ε0 + αt, where ε0  and α are constants. Obtain an expression forElectromagnetic Induction Electrical Engineering (EE) Notes | EduRev

4. A rectangular loop of dimensions αxb lies with its longer sides parallel to a long straight wire, the distance of the wire from the nearer side is d. Calculate the mutual inductance of the pair.

5. An electromagnetic wave travels in a medium of relative permeability 4 and dielectric constant 5. The displacement current density associated with the electromagnetic field varies with space and time as

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev Find the electric field, the magnetic field and show that Ampere’s law remains satisfied if  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

6. Using the relation  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev for a current distribution, show that the magnetic energy can be expressed as  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev A coaxial cable consists of an inner conductor of radius a and an outer conductor of radius b. The current in the inner conductor is uniformly distributed while the outer conductor has negligible thickness and provides a return path to the current.Determine the energy of the system.

Solutions to Tutorial Assignments

1. There are two ways of doing this, the first (and by far the, easiest) is to calculate the energy contained in the magnetic field and equate it to  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev In doing so one has to be careful and discard a self energy contribution from the field outside the wire which gives a divergent term. Consider a strip of unit length and width dr lying between r and r+dr. The field strength on this strip is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev so that the flux enclosed by this region per unit length is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The flux “linked” with this strip is this amount multiplied by a contour of radius r, which is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev so that the total flux linked with a unit length of the wire is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Thus the self inductance is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

2. The magnetic field due to the straight conductor is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev The flux linked with each turn of the toroidal coil is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev where h is the height of the doughnut. The self inductance is given by  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

3. The emf in B as a result of change in the current in the circuit A is given by

 Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

which gives  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev As the current is zero at t=0, the constant of integration is zero.

4. When a current I passes through the long wire, a field  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev gets established at a distance r from the wire, which is perpendicular to the loop. The flux through the loop can      be calculated by considering a strip of width dr at a distance r from the wire,

 Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Thus the mutual inductance is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

5. The displacement current density is 

 Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

 which gives  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev The corresponding electric field is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Taking curl of the electric field we get

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

from which we get,

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The H- field is then given by

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Taking curl of the H field we get

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Using  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev we get

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

6. Use the identity, 

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Substituting this and using the divergence theorem we can ignore the surface term and get

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

for linear magnetic material.

For the coaxial cable, with the field being distributed in the inner conductor, the fields are as follows:

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The contribution to the energy (per unit length) from the inner conductor is (volume element for unit length is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Contribution from the region between the conductors is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

The total energy is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Self Assessment Questions

1. A coaxial cable consists of an inner conductor of radius a and an outer conductor of radius b. The current in the inner conductor is uniformly distributed while the outer conductor has negligible thickness and provides a return path to the current. Calculate the self inductance of the cable.

2. Find the mutual inductance of two coplanar squares with a common centre assuming that the square located inside is much smaller in dimension b than the bigger square which has a side a.

3. The current through a straight wire varies with time as  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev where the current is in Amperes and time in seconds. If the radius of cross section of the wire is 50 (mm)and the resistivity 10-8 Ωm ,estimate the displacement current density and compare it with the conduction current density.

4. A capacitor plate of area 0.3 m2 is being charged at a uniform rate so that the electric field inside the plate varies with time as  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev Calculate the displacement current and estimate the magnetic field strength at a distance 5 cm from the centre of the capacitor plate along a line parallel to both the plates.

Solutions to Self Assessment Questions

1. The self inductance of the inner conductor was calculated in Problem 1. Because the thickness of the outer conductor is negligible, the flux linked is zero and there is no self      inductance contribution from the outer shell. There is however, flux linked between the two conductors.The field in this region is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev so that flux passing through a strip of        width dr between r and r+dr and of unit length is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev The total flux linked is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev Thus the total self inductance is

   Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

2. The magnetic field at the centre of the square (taken in xy) plane) is given by  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev Since the dimension of the second coil is much smaller than that of the first, it can        be assumed that this field exists over the entire square of side b, as a result of which the flux through the second loop is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev which gives the mutual inductance to           be  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

3. Consider a length L of the wire. The resistance is  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev If E is the electric field in the wire, the potential difference between the ends of the wire      is V=EL  so that the current  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev The displacement current density is

Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

Thus one sees that the displacement current density is very small, the corresponding current density at  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev which keeps on increasing with time

4. The displacement current density  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

  The magnetic field at a distance r from the centre is given by

  Electromagnetic Induction Electrical Engineering (EE) Notes | EduRev

 which gives, substituting   r= 0.5 m, B = 2.78 x 10-5 T. 

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