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INTRODUCTION
A brief survey of the electromagnetic waves. We have learnt that an electric current produces a magnetic field. Also a magnetic field changing with time produces an electric field. Can an electric field changing with time produce a magnetic field ? James Clerk Maxwell (1831-1879), argued that this was indeed the case—an electric field changing with time produces a magnetic field. Maxwell noticed that Ampere’s circuital’ law is inconsistent namely, makes non-unique predictions for the magnetic field in situations where electric current changes with time. He showed that consistency requires an additional source of magnetic field, this is called displacement current. This made the laws of electricity and magnetism, symmetrical.
Maxwell formulated a set of four equations, called Maxwell’s equations. With the help of these equations, he predicted that electric and magnetic fields dependent on time and space propagate as transverse waves, called electromagnetic waves. His discovery that electromagnetic waves travel with the speed of light led him to a remarkable conclusion that light is an electromagnetic wave. Heinrich Hert, in 1865, successfully demonstrated the existence of electromagnetic waves. A few years later, Guglidmo Marconi of Italy succeeded in transmitting electromagnetic waves over distances of several kilometres. His experiments brought a revolution in communication which we’are witnessing even today.
MAXWELL’S DISPLACEMENT CURRENT
Inconsistent of Ampere’s circuital law. According to Ampere’s circuital law, the line integral of the magnetic field along any closed loop C is proportional to the current l passing through the closed loop, i.e.,
In 1864, Maxwell showed that equation (1) is logically inconsistent. To prove this inconsistency, we consider a parallel plate capacitor being charged by a battery as shown in Fig. 8.1(a). As the charging
continues, a current I flows through the connecting wires, which of course changes with time. This current produces a magnetic field around the capacitor. Consider two planar loops C2 and C2, C1 just left of the capacitor and C2 in between the capacitor plates, with their planes parallel to these plates.
Now the current I flows across the area bounded by loop C1 because connecting wire passes through it. Hence from Ampere’s law, we have
But the area bounded by C2 lies in the region between the capacitor plates, so no current flows across it.
Imagine the loops C1 and C2 to be infinitesimally close to each other, as shown in Fig. 8.1(b). Then we must have
This result is inconsistent with the equations (2) and (3). So a need for modifying Ampere’s law was felt by Maxwell.
Maxwell’s modification of Ampere’s law : Displacement current. To modify Ampere’s law, Maxwell followed a symmetry consideration. By Faraday’s law, a changing magnetic field induces an electric field, hence a changing electric field must induce a magnetic field. As currents are the usual sources of magnetic fields, a changing electric field must be associated with a current. Maxwell called this current as the displacement current to distinguish it from the usual conduction current caused by the drift of electrons.
Electromagnetic Waves Notes Class 12 Physics
Displacement current is that current which comes into existence, in addition to the conduction current, whenever the electric field and hence the electric flux changes with time.
To maintain the dimensional consistency, the displacement current is given the form :
where fE = electric field × area = EA, is the electric flux across the loop.
∴ Total current across the closed loop
Hence the modified form of the Ampere’s law
is
Unlike the conduction current, the displacement current exists whenever the electric field and hence the electric flux is changing with time. Thus according to Maxwell, the source of a magnetic field is not just the conduction electric current due to flowing charges, but also the time-varying electric field. Hence the total current I is the sum of the conduction current Ic and displacement current ld
Consistency of modified Ampere’s law. For loop Cy there is no electric flux ( = 0). Therefore, from equation (5) we have
For loop C2, conduction current I =0 but Id, ≠ 0, because a time-varying electric field exists in the region between the capacitor plates. Hence
If A be the area of the capacitor plates and q be the charge on the plates at any instant t during the charging process, then the electric field in the gap will be
This agrees with the equation (6), proving the consistency of the Ampere’s modified law (5).
Property of continuity. The sum (Ic + Id) has the important property of continuity along any closed path even when individually Ic and Id may not be continuous. In Fig. 8.1, for example, a current Ic enters one plate and leaves the other plate of the capacitor. The conduction current
is not continuous across the capacitor gap as no charge is transported across this gap. The displacement current Id is zero outside the capacitor plates and in the gap, it has the value
which is exactly the value of the conduction current in the lead wires. Thus the displacement current satisfies the basic condition that the current is continuous.
The sum Ic + has the same value along the entire path (both inside and outside the capacitor plates), although individually the two currents are discontinuous. Clearly, outside the capacitor plates, we have only conduction current Ic = I, and there is no displacement current (Id = 0). While inside the capacitor plates, there is only displacement current (Id = I, and there is no conduction current (Ic = 0). But in any general medium, both Ic and ld are present. However, Ic is larger than Id in a conducting medium while Ic is larger than Id in an insulating medium.
2. Is a displacement current associated with a magnetic field ? Or, can a changing electric flux induce a magnetic field ? Explain it with the help of an example.
Induced magnetic field. A displacement current produces the same physical effects as the conduction current. Like a conduction current, a displacement current is also associated with a magnetic field. Consider, for example, the charging of a parallel plate capacitor by a constant current I in the connecting wires [Fig. 8.2(e)]. This increases the charges on the capacitor plates at a steady rate. Consequently, the electric field between the plates also increases at a steady rate. Between the capacitor plates, there exists a displacement current due to time varying electric field. In such a region, we expect a magnetic field though there is no source of conduction current nearby.
Electromagnetic Waves Notes Class 12 Physics
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