Question

A conducting loop of resistance 'R' is moved in a magnetic field, the total induced charge depends upon

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

Hint:

While induced EMF and induced current depend heavily on how fast you move the loop ($e \propto \frac{1}{dt}$), the total transferred charge is independent of time! Moving a coil slowly or snapping it quickly through the same flux difference moves the exact same quantity of electrons.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks to identify the physical parameters that determine the total induced charge ($q$) that flows through a closed conducting circuit when it moves through a magnetic field region.

Step 2: Key Formula or Approach:
According to Faraday's Law of Induction, the magnitude of the induced electromotive force ($e$) is proportional to the rate of change of magnetic flux:
$$e = \frac{d\phi}{dt}$$ By Ohm's law, the induced current $i$ in a loop of resistance $R$ is:
$$i = \frac{e}{R} = \frac{1}{R}\frac{d\phi}{dt}$$ Since electric current is defined as the rate of flow of charge ($i = \frac{dq}{dt}$), we can equate the expressions to isolate the differential charge elements.

Step 3: Detailed Explanation:
Equating the current relations gives:
$$\frac{dq}{dt} = \frac{1}{R}\frac{d\phi}{dt}$$ We can eliminate the time differential element $dt$ from both sides of the expression:
$$dq = \frac{d\phi}{R}$$ Integrating both sides over the full duration of the motion:
$$\Delta q = \frac{\Delta \phi}{R} = \frac{\phi_{\text{final}} - \phi_{\text{initial}}}{R}$$ This mathematical result proves that the total induced charge $\Delta q$ depends strictly on the total change in magnetic flux ($\Delta \phi$) and the loop's electrical resistance ($R$). Notably, it is independent of the time taken or the specific rate of that change.

Step 4: Final Answer:
The total induced charge depends upon the total change in magnetic flux and $R$, matching option (C).