MC3. A light bulb is connected in the circuit shown in the figure with the switch S open and the capacitor uncharged. The battery has no appreciable internal resistance. Which one of the following graphs best describes the brightness B of the bulb as a function of time t after closing the switch?

Background

Topic: RC Circuits and Transient Response

This question tests your understanding of how current changes in an RC circuit after the switch is closed, and how this affects the brightness of the bulb over time.

Key Terms and Formulas:

• RC Circuit: A circuit containing a resistor (R) and a capacitor (C) in series with a voltage source.

• Transient Response: The behavior of the circuit immediately after the switch is closed.

• Current in an RC circuit after closing the switch:

• Brightness of the bulb is proportional to the current flowing through it.

• At , the capacitor is uncharged and acts like a short circuit; as time increases, it charges up and the current decreases.

Step-by-Step Guidance

1. Analyze the circuit at the moment the switch is closed (): The capacitor is uncharged, so it initially allows maximum current to flow through the bulb.

2. As time progresses, the capacitor begins to charge, which reduces the current in the circuit. The charging follows an exponential decay governed by the time constant .

3. Write the expression for current as a function of time: .

4. Relate the current to the brightness of the bulb: Since brightness is proportional to current, the brightness will also decrease exponentially over time.

5. Compare this behavior to the provided graphs and identify which graph shows an exponential decrease in brightness starting from a maximum value at .

Try solving on your own before revealing the answer!