Question

A small loop of area A is inside of, and has its axis in the same direction as, a long solenoid of n turns per unit length and current i. If i = i0 sin(ωt), find the emf in the loop. (Use the following as necessary: A, n, i0, ω, t, and μ0.)

Answer 1

Answer:

EMF, $e = - \mu_{o}ni_{o}A\omegacos\omega t$

Solution:

As per the question:

Area of the loop is A

No. of turns in the solenoid is n

Current, $i = i_{o}sin\omega t$

Now, we know that by Faraday's law:

EMF, $e = - \frac{d\phi}{dt}$                   (1)

where

$\phi$ = flux linkage

Now,

$\phi = BA$

where

B = Magnetic Flux

A = Cross-sectional Area

$B = \mu_{o}ni = \mu_{o}ni_{o}sin\omega t$

$e = - \frac{d}{dt}(BA) = -\frac{d}{dt}(\mu_{o}ni_{o}Asin\omega t)$      

$e = - \frac{d}{dt}(BA) = -\frac{d}{dt}(\mu_{o}ni_{o}Asin\omega t)$      

$e = -\mu_{o}ni_{o}A\frac{d}{dt}(sin\omega t)$      

$e = - \mu_{o}ni_{o}A\omegacos\omega t$