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Question

In electromagnetic induction, line integral of induced field E around a close path is __________, induced electric field is ___________.


A

non-zero

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B

non-conservative

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Solution

Step1: Faraday's Law of Electromagnetic Induction

  1. Induced emf in a circuit is proportional to the rate of change of magnetic flux linked with the circuit.
  2. The direction of induced emf is such that it tries to oppose the cause of its generation, i.e., the variation of magnetic flux producing it.

Step2: Induced EMF

If ϕ be the magnetic flux linked with a circuit at any time t then the laws of electromagnetic induction can be expressed mathematically as

ε=-dϕdt, 1

Where ε is the induced emf. Here it is important to point out that ε does not depend on how the flux ϕ is changed.

If the electric field in space is denoted by E then, by definition, the emf around a closed path C is

ε=CE.dl.

If S is an open surface bounded by the curve C placed in a magnetic field B then the magnetic flux through the surface is

ϕ=SB.dS.

So equation 1 can be written as

CE.dl=-ddtSB.dS.

This is the integral form of Faraday's law.

If the circuit is rigid and stationary in our reference system then neither S nor C depend on t and as a result the time derivative can be taken inside the integral, i.e., CE.dl=-SBt.dS,

where partial derivative has been used beacuse B is a function of position and time.

Using Stokes' theorem we can write,

S×E.dS=-SBt.dS.

Since this must be true for any arbitrary fixed surface S, it follows that

×E=-Bt, 2

This is the differential form of Faraday's law.

It relates the space derivatives of E at a particular point to the time rate of change of B at the same point.

Step3: Vector and Scalar Potential

For static electric field ×E=0 and we can introduce scalar potential ϕ by the relation E=-ϕ.

But for a time-varying field ×E0 and we cannot do so. The magnetic vector potential A was defined by the fact that .B=0, which still holds for the time-varying fields. Hence the relation B=×A holds for time-varying situations.

Using it in Faraday's law in equation 2 we get

×E=-t×A or ×E+At=0 3

Since the curl of the gradient of any scalar is always zero, we can write

E+At=-ϕ or E=-ϕ-At 4

From equation 3 and 4 we can find E and B provided that the potentials A and ϕ are known. Equation 4 suggests that for varying fields E can be considered as the sum of the two terms Eq and Ei where

Eq=-ϕ and Ei=-At.

Eq is the field generated by charges; it is conservative and nonsolenoidal.

Ei is the field generated by a varying magnetic field. It is non-conservative and solenoidal.

Hence, in electromagnetic induction, the line integral of the induced field E around a close path is non-zero, the induced electric field is non-conservative.


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