Question

The total charge induced in a conducting loop when it is moved in a uniform magnetic field depends on

• A. initial magnetic flux only.
• B. final magnetic flux only.
• C. the total change in magnetic flux.
• D. the rate of change of magnetic flux.

Hint:

Induced emf depends on rate of change of flux, but total induced charge depends on total flux change.

Answer 1

The Correct Option is C

Concept:
Induced emf is: \[ \mathcal{E}=-\frac{d\Phi}{dt} \] Current is: \[ I=\frac{\mathcal{E}}{R} \] So total charge induced is: \[ q=\int I\,dt=\frac{1}{R}\int \mathcal{E}\,dt \] ip

Step 1: Find expression for total charge.
\[ q=\frac{1}{R}\int \left(-\frac{d\Phi}{dt}\right)dt \] \[ q=-\frac{1}{R}\int d\Phi \] \[ q=\frac{\Delta \Phi}{R} \] in magnitude. ip

Step 2: Interpret the result.
Thus total induced charge depends on the net change in magnetic flux, not on how fast the flux changes. ip Hence, the correct answer is:
\[ \boxed{(C)\ \text{the total change in magnetic flux}} \]