Question

A parallel plate capacitor (of capacitance \( C \)) with circular plates of radius \( r_0 \) located at positions \( \pm \alpha \), is connected in series with a resistor \( R \) and is charged by a battery of voltage \( V \). Consider a circular loop \( L \) of radius \( \frac{r_0}{2} \) parallel to the capacitor plates is located at the center. Which of the following statements is correct?

• A. The charge on the capacitor at time $t$ is $q(t) = CR(1-e^{-t/(CV)})$
• B. The charge on the capacitor at time $t$ is $q(t) = CV(1-e^{-t/(CV)})$
• C. The flux through the loop L is independent of the area enclosed by it
• D. The magnetic field is directed orthogonal to the loop L
• E. The magnetic field is directed along the loop L

Hint:

In charging circuits, always check the exponent. It must have the dimensions of time ($RC$). If you see $CV$ in the exponent, it is a dimensional error.

Answer 1

The Correct Option is B

Concept:
In an RC (Resistor-Capacitor) charging circuit, the buildup of charge is an exponential process governed by the time constant $\tau = RC$.

Step 1: Apply Kirchhoff's Loop Law.
For the series circuit: \[ V - iR - \frac{q}{C} = 0 \] Substituting $i = \frac{dq}{dt}$ and solving the differential equation with the initial condition $q(0) = 0$: \[ q(t) = CV(1 - e^{-t/RC}) \] Note that in the image provided, option (B) is written as $q(t) = CV(1-e^{-t/(CV)})$, which contains a typo in the exponent (it should be $RC$). However, among the provided options, (B) represents the correct functional form for charging.

Step 2: Analyze Displacement Current and Magnetic Fields.
According to the Ampere-Maxwell Law, the changing electric field between the capacitor plates creates a "displacement current." This displacement current generates a magnetic field.
• The electric field lines are straight and directed from one plate to the other (orthogonal to the plates).
• The resulting magnetic field lines form concentric circles around the axis of the capacitor.
• Therefore, at any point on the loop L, the magnetic field is tangent to the loop, meaning the magnetic field is directed along the loop L.