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

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 = - (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 = - (d)/(dt)(BA) = -(d)/(dt)(\mu_(o)ni_(o)Asin\omega t)`

`e = - (d)/(dt)(BA) = -(d)/(dt)(\mu_(o)ni_(o)Asin\omega t)`

`e = -\mu_(o)ni_(o)A(d)/(dt)(sin\omega t)`

`e = - \mu_(o)ni_(o)A\omegacos\omega t`