Question

Magnetic field at a distance \(r\) from z axis is \( B = B_0 r \, \text{kt} \) present in the region. \( B_0 \) is constant and \(t\) is time. The magnitude of induced electric field at a distance \(r\) from z-axis is.

• A. \( \frac{B_0 r^3}{3} \)
• B. \( \frac{2 \pi B_0 r}{3} \)
• C. \( \frac{B_0 r^2}{2 \pi} \)
• D. \( \frac{B_0 r^2}{3} \)

Hint:

Induced electric fields in magnetic fields are directly related to the rate of change of magnetic flux through a given area.

Answer 1

The Correct Option is D

The magnetic field at a distance \( r \) from the z-axis is given by \( B = B_0 r \). According to Faraday's law of induction, the induced electric field is related to the rate of change of magnetic flux. The induced electric field \( E \) is given by: \[ E = -\frac{1}{c} \frac{d\Phi_B}{dt} \] Where \( \Phi_B = B \cdot A = B_0 r \cdot A \) is the magnetic flux. Since \( A = \pi r^2 \), we get: \[ E = \frac{B_0 r^2}{3} \] Thus, the induced electric field at a distance \( r \) from the z-axis is \( \frac{B_0 r^2}{3} \).