An infinite solenoid had radius R and n turns post unit leng

An infinite solenoid had radius R and n turns post unit length. the current grows linearly with time: I(t)=ct. Find the electric field at radius r, both inside and outside the solenoid.

Solution

Here,

I(t) = c * t

magnetic field due to solenoid is B

B = u0 * n * I

dB/dt = u0 * n * d/dt(c * t)

dB/dt = u0 * n * c

Now , inside the solenoid

2pi * r * E = pi * r^2 * dB/dt

E = r * u0 * n * c/2

the electric field inside the solenoid is r * u0 * n * c/2

Now , outside the solenoid

E * 2pi * r = pi * R^2 * dB/dt

E * 2pi * r = pi * R^2 * u0 * n * c

solving for E

E = R^2 * u0 * n * c/(2*r)

the electric field outside is R^2 * u0 * n * c/(2*r)