EMI Q19 DPP CHAMP

Question

Consider a long solenoid of radius R which has n turns per unit length. A time dependent current I = I₀sin ωt flows in solenoid, magnitude of electric field at a perpendicular distance r < R, from the axis of symmetry of solenoid is found to be E = α/4ωµ₀ nI₀r cosωt, where α is pure number and µ₀ is permability of free space. Find α.

Answer 1

Magnetic flux inside solenoid = \(\mu_0ni\)

\(\phi = B.A\)

\(\phi = \mu_0NI\pi r^2\)    \(0\le r\le R\)

\(\frac{-d\phi}{dt} = \phi E.dl\)

\(\mu_o N\pi r^2 \frac{dI}{dt} = E. 2\pi r\)

\(E = \frac{\mu_0Nr}{2} \frac{dI}{dt}\)

\(I = I_0\sin \omega t\)

\(E = \frac{\mu _0Nr}2 I_0\omega \,\cos\omega t\)

For \(r = \frac R2\)

\(E = \mu _0 NI_0 \omega \frac R4 \cos (\omega t)\)