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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,
where is the position vector of the point of observation (i.e. the point where the field is calculated) with respect to the current element This gives rise to flux in loop 2 which can be written as
It can be seen that the flux in the second loop is proprtional to the current in the first loop. We can therefore write,
The proportinaliity constant M_{21 }is known as the mutual inductance of the two circuits and it depends on geomery of the circuits and their relative positions and orientations.
The emf in the second circuit due to a change in the current in the first circuit is
We can show that the mutual inductance is symmetric, i.e., M_{21 }= M_{12.} To show this, let us reexpress the flux in the second loop in terms of the vector potential of the first loop,
We have seen that the vector potential of loop 1 can be expressed as
where r is the distance from the length element 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 on the loop 2 is given by
Thus the expression from the flux through the second loop is
This shows that
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
where L is known as the self inductance of the loop.
Example : Self inductance of a solenoid
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) ,
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,
If l is the length of the solenoid, there are number of turns in the solenoid, and the total flux linked is The self inductance is thus given by
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.
The flux linked with the second loop when the current  flows through the first loop,
This gives the mutual inductance to be
From the previous example, we can obtain the self inductances of each of the solenoids,
One can see that we get, in this case, 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” the coefficient of coupling depends on the relative orientation of the two loops.
Example : Two coplanar and concentric loops
Let us assume R_{1}<<R_{2} (figure not to scale). The field at the centre of the bigger loop is We can suume that this is the field all over the smaller loop and the flux through the smaller loop is The mutual inductance in this case is
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 ith loop is
where L_{i} is the self inductance of the ith loop and M_{ij} is the mutual inductance between the ith and the jth loops. The emf through the ith loop when the currents in each loop changes is given by
The rate of doing work to overcome this emf is
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
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
The total work done is obtained by integrating the above over time,
This is for a discrete current distribution. For a continuous current distribution, this can be easily generalized to give,
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.
During the process of charging a current exists in the external circuit and we can calculate the magnetic field by using Ampere’s law,
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 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.
Since there is no current passes through the surface, we would get
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
Let us calculate the rate of change of this flux.
The rate of change of free charges is clearly the current flowing in to charge the plates. Maxwell called 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
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,
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,
which is just the equation of continuity. Thus all the Maxwell’s equations are in place now. They are
We supplement these with two “constitutive relations”
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 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 The emf induced in the coil B as a result is found to change with time as given by the relation ε=ε_{0} + αt, where ε_{0 }and α are constants. Obtain an expression for
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
Find the electric field, the magnetic field and show that Ampere’s law remains satisfied if
6. Using the relation for a current distribution, show that the magnetic energy can be expressed as 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 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 so that the flux enclosed by this region per unit length is
The flux “linked” with this strip is this amount multiplied by a contour of radius r, which is so that the total flux linked with a unit length of the wire is
Thus the self inductance is
2. The magnetic field due to the straight conductor is The flux linked with each turn of the toroidal coil is where h is the height of the doughnut. The self inductance is given by
3. The emf in B as a result of change in the current in the circuit A is given by
which gives 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 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,
Thus the mutual inductance is
5. The displacement current density is
which gives The corresponding electric field is
Taking curl of the electric field we get
from which we get,
The H field is then given by
Taking curl of the H field we get
Using we get
6. Use the identity,
Substituting this and using the divergence theorem we can ignore the surface term and get
for linear magnetic material.
For the coaxial cable, with the field being distributed in the inner conductor, the fields are as follows:
The contribution to the energy (per unit length) from the inner conductor is (volume element for unit length is
Contribution from the region between the conductors is
The total energy is
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 where the current is in Amperes and time in seconds. If the radius of cross section of the wire is 50 (mm)^{2 }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 m^{2} is being charged at a uniform rate so that the electric field inside the plate varies with time as 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 so that flux passing through a strip of width dr between r and r+dr and of unit length is The total flux linked is Thus the total self inductance is
2. The magnetic field at the centre of the square (taken in xy) plane) is given by 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 which gives the mutual inductance to be
3. Consider a length L of the wire. The resistance is 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 The displacement current density is
Thus one sees that the displacement current density is very small, the corresponding current density at which keeps on increasing with time
4. The displacement current density
The magnetic field at a distance r from the centre is given by
which gives, substituting r= 0.5 m, B = 2.78 x 10^{5 }T.
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