Question

An electric circuit consists of a charged capacitor \(C\), a resistor \(R\) and a switch \(S\). Initially, the switch is open and all devices are connected in series. A circular loop of wire is placed in the same plane as the circuit. Which one of the following is true about the induced current in the loop after the switch is closed?

• A. it is clockwise and increases
• B. it is counter clockwise and increases
• C. it is clockwise and decreases
• D. it is counter clockwise and decreases

Hint:

In an \(RC\) discharge circuit, current decreases exponentially: \[ I=I_0e^{-t/RC} \] Use Lenz's law to find the direction of induced current.

Answer 1

The Correct Option is D

Concept:
This question is based on electromagnetic induction and Lenz's law. When the switch is closed, the charged capacitor starts discharging through the resistor. During this discharge, current flows in the circuit, but its value decreases with time. For an \(RC\) discharging circuit: \[ I=I_0e^{-t/RC} \] This means current is maximum at the beginning and then decreases exponentially.

Step 1: Understand what happens after closing the switch.
Initially, the capacitor is charged and switch is open. When the switch is closed, the capacitor discharges through the resistor. So current starts flowing in the circuit. But as the capacitor loses charge, the potential difference across it decreases. Therefore, the current also decreases with time.

Step 2: Connect changing current with magnetic field.
A current-carrying circuit produces magnetic field around it. Since the current is decreasing, the magnetic field produced by the circuit also decreases. This decreasing magnetic field changes the magnetic flux through the nearby circular loop.

Step 3: Apply Lenz's law.
According to Lenz's law, the induced current in the circular loop will oppose the decrease in magnetic flux. So the induced current will flow in such a direction that it tries to maintain the original magnetic field. According to the direction shown in the intended figure, the induced current is counter clockwise.

Step 4: Determine whether induced current increases or decreases.
Because the capacitor discharge current decreases exponentially, the changing magnetic flux also becomes weaker with time. Therefore, the induced current in the loop also decreases with time.

Step 5: Check the options.
Option (A) clockwise and increases is incorrect.
Option (B) counter clockwise and increases is incorrect because the discharge current decreases.
Option (C) clockwise and decreases is incorrect because the direction is not clockwise according to the given key.
Option (D) counter clockwise and decreases is correct. Hence, the correct answer is: \[ \boxed{(D)\ \text{it is counter clockwise and decreases}} \]