Question

A cylindrical region of radius R contains a uniform magnetic field parallel to its axis. The field is zero outside the cylinder. If the magnitude of the field is changing at the rate dB/dt, the electric field induced at a point 2R from the cylinder axis is:

Answer 1

The magnitude of the induced electric field is (RdB/dt)/4

The induced electric field is gotten from

-∫E.dl = dФ/dt where E = induced electric field, dl = path length vector, Ф = magnetic flux through cylindrical region = AB where A = area of magnetic flux = πR² where R = radius of cylindrical region and B = magnetic field.

So, -∫E.dl = dФ/dt

-∫E.dl = dAB/dt

-∫Edlcos0 = AdB/dt (where E.dl = Edlcos0 = Edl since E and dl are parallel to each other.)

So -∫Edl = πR²dB/dt

-E∫dl = πR²dB/dt (∫dl = 2πr since the integral is the circumference of the path)

-E(2πr) = πR²dB/dt (we integrate dl from r = 0 to 2R)

-E2π(2R - 0) = πR²dB/dt

-E4πR= πR²dB/dt

E = πR²dB/dt ÷ 4πR

E = -(RdB/dt)/4

So, the magnitude of the induced electric field is (RdB/dt)/4

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