(i) charging and discharging capacitor through a resistor

The idea

When a capacitor charges through a resistor, the current is initially V/R and falls exponentially as the capacitor voltage rises towards the supply value. Discharging reverses the process: the capacitor drives current back through the resistor, and charge, current and voltage all decay exponentially. The time constant τ = RC is the time for charge or current to fall to 1/e of its starting value, about 37%, and for a discharging capacitor q = Q₀e^(−t/RC). For example, a 100 µF capacitor with a 10 kΩ resistor has τ = 100 × 10⁻⁶ F × 10 × 10³ Ω = 1.0 s. Graphs of charge, current and potential difference against time are exponential curves, and the area under a current-time graph gives charge.

What earns the marks

• States that charging current is largest at the start and falls exponentially as the capacitor voltage rises.
• Applies the time constant τ = RC and interprets it as the time to fall to 1/e of the initial value.
• Uses q = Q₀e^(−t/RC) or I = I₀e^(−t/RC) for discharging.
• Explains that the capacitor voltage rises towards the supply voltage during charging and falls towards zero during discharging.
• Relates the area under a current-time graph to charge transferred.

Common mistakes and how to put them right

• Thinking the current stays constant during charging.

  Put it right: It falls because the potential difference across the resistor decreases as the capacitor voltage rises.

• Confusing the time constant with the time to fully charge or discharge.

  Put it right: After one time constant the value has changed by about 63%, not completely.

• Using τ = R/C instead of τ = RC, which gives the wrong unit and magnitude.

Exam tips

• Check that resistance is in ohms and capacitance in farads before multiplying to find τ.
• Use the initial value and the 1/e point to sketch or identify an exponential decay curve.
• Remember that charging and discharging graphs are mirror images in shape but differ in direction and final value.