Question

figure shows three regions of magnetic field each of area A and in each region of magnitude of magnetic field decreases at a constant rate à if E is induced electric field then value of line integral if Edr along given loop is equal to

Answer 1

Integral of E.dr in a given loop is related as

$\int E.dr = \Delta V = EMF$

so this relation shows the EMF induced

as per faraday's law induced EMF is equal to rate of change in magnetic flux

so we can say magnetic flux is given as

$\phi = B.A$

now the rate of change in magnetic flux will be

$EMF = \frac{d\phi}{dt}$

$EMF = \frac{d}{dt}(B.A)$

here area of loop is constant and magnetic field is changing with time

$EMF = A\frac{dB}{dt}$

now we can plug in its value

$EMF = A\alpha$

now by above first equation

$\int E.dr = EMF = A\alpha$

so above is the relation between them