Question

A long solenoid of radius R, having N turns per unit length carries a time depend current I(t) = I₀ sin (ωt). The magnitude of induced electric field at a distance R/2 radially from the axis of the solenoid is

1. \(\frac{R}{2}{\mu _0}N\;{I_0}\;\omega \cos \: \left( {\omega t} \right)\)
2. \(\frac{R}{4}{\mu _0}N\;{I_0}\;\omega \cos \:\left( {\omega t} \right)\)
3. \(\frac{R}{2}{\mu _0}N\;{I_0}\;\omega \sin\: \left( {\omega t} \right)\)
4. \(\frac{R}{4}{\mu _0}N\;{I_0}\;\omega \sin \:\left( {\omega t} \right)\)

Answer 1

Correct Answer - Option 2 : \(\frac{R}{4}{\mu _0}N\;{I_0}\;\omega \cos \:\left( {\omega t} \right)\)

Magnetic flux inside solenoid = μ.NI

ϕ = B.A

= μ₀ NT π r², 0 ≤ r ≤ R.

\(\begin{array}{l} - \frac{{d\phi }}{{dt}} = \phi E.dl = E.2\pi r\\ {\mu _0}\;N\pi {r^2}\frac{{dI}}{{dt}} = E.2\pi r\\ E = \frac{{{\mu _0}Nr}}{2}\frac{{dI}}{{dt}}\;\\ I = {I_0}\sin \omega t\\ E = \frac{{{\mu _0}Nr}}{2}{I_0}\;\omega \cos \:\left( {\omega t} \right) \end{array}\)

For \(r = \frac{R}{2},\)

\(E = {\mu _0}\;N{I_0}\;\omega \frac{R}{4}\cos \:\left( {\omega t} \right)\)