A cylindrical region of radius R contains a uniform magnetic field, parallel to its axis, with magnitude that is changing linearly with time. If r is the radial distance from the cylinder axis, the magnitude of the induced electric field inside the cylinder is proportional to: A) R B) r C) r² D) 1/r E) 1/r²

Question

Quizplus · Accepted Answer

According to Faraday's law of induction, the magnitude of the induced electric field is proportional to the rate of change of magnetic flux through a surface. Inside the cylinder, the magnetic field is parallel to the axis, so the flux through any flat surface is zero. However, if we consider a cylindrical surface with height h and radius r, the flux through this surface is given by:

Φ = B(r)hπr^2

Since the magnetic field is changing linearly with time, we have:

dΦ/dt = πr^2dB/dt

By Faraday's law, the induced electric field is:

E = -(1/c)dΦ/dt

Combining the previous equations, we have:

E = -(πr^2/c)(dB/dt)

Since dB/dt is constant inside the cylinder, we see that the induced electric field is proportional to r. Therefore, the answer is choice B.

A Cylindrical Region of Radius R Contains a Uniform Magnetic