(b) time constant of a capacitor–resistor circuit; τ = CR

The idea

The time constant τ of a capacitor–resistor circuit is defined by τ = CR, where C is capacitance in farads (F) and R is resistance in ohms (Ω). Because 1 F × 1 Ω = 1 s, τ has units of seconds. For example, a 470 μF capacitor (470 × 10⁻⁶ F) with a 10 kΩ resistor (10 × 10³ Ω) gives τ = (470 × 10⁻⁶) × (10 × 10³) = 4.7 s. In a discharging circuit, τ is the time for the charge, potential difference or current to fall to 1/e (about 37%) of its initial value. In a charging circuit, τ is the time to reach (1 − 1/e) (about 63%) of the final value. τ is not the time to fully charge or discharge; after each interval of τ the remaining change is reduced by a factor of 1/e, so the process is asymptotic. The time constant depends only on C and R, not on the initial charge or the applied voltage.

What earns the marks

• State the defining equation τ = CR and identify C as capacitance in farads and R as resistance in ohms.
• Show that the product CR has units of seconds, since 1 F × 1 Ω = 1 s.
• Explain that in a discharging circuit τ is the time for charge, potential difference or current to fall to 1/e of its initial value (about 37%).
• Explain that in a charging circuit τ is the time to reach (1 − 1/e) of the final value (about 63%).
• Recognise that τ is independent of the initial charge or applied voltage and that full charge or discharge is approached asymptotically, not reached in a finite multiple of τ.

Common mistakes and how to put them right

• Thinking that τ is the time to fully charge or discharge the capacitor.

  Put it right: τ is the time to reach about 63% of the final value when charging, or to fall to about 37% of the initial value when discharging; the process continues asymptotically.

• Using C in microfarads or R in kilohms without converting to farads and ohms.

  Put it right: Convert all values to base SI units before multiplying, for example 470 μF = 470 × 10⁻⁶ F and 10 kΩ = 10 × 10³ Ω.

• Believing that τ depends on the initial voltage or charge.

  Put it right: τ = CR depends only on the capacitance and resistance; changing the initial conditions changes the starting values but not the time constant.

• Confusing the charging and discharging percentages.

  Put it right: When discharging, the value falls to 1/e (about 37%) after one time constant; when charging, it rises to 1 − 1/e (about 63%) after one time constant.

Exam tips

• Always convert capacitance to farads and resistance to ohms before calculating τ, and state the unit of τ as seconds.
• When asked for the time constant from a graph, identify the time at which the discharging quantity has fallen to about 37% of its initial value, or the charging quantity has risen to about 63% of its final value.
• Remember that τ is a characteristic time, not a total time; avoid saying the capacitor is fully charged or discharged after one, two or five time constants.