Capacitor Charge & Discharge Equations (OCR A Level Physics): Revision Note

The Time Constant

• The time constant of a capacitor discharging through a resistor is a measure of how long it takes for the capacitor to discharge

• The definition of the time constant is:

The time taken for the charge, current or voltage of a discharging capacitor to decrease to 37% of its original value

• Alternatively, for a charging capacitor:

The time taken for the charge or voltage of a charging capacitor to rise to 63% of its maximum value

• 37% is 0.37 or $\frac{1}{e}$ (where e is the exponential function) multiplied by the original value (I₀, Q₀ or V₀)

  • This is represented by the Greek letter tau, $\mathrm{\tau }$, and measured in units of seconds (s)

• The time constant provides an easy way to compare the rate of change of similar quantities e.g. charge, current and p.d.

• It is defined by the equation:

$\mathbf{\tau }$ = RC

• Where:

  • $\mathrm{\tau }$ = time constant (s)

  • R = resistance of the resistor (Ω)

  • C = capacitance of the capacitor (F)

• The time to half, t_1/2 (half-life) for a discharging capacitor is:

The time taken for the charge, current or voltage of a discharging capacitor to reach half of its initial value

• This can also be written in terms of the time constant, τ:

t_1/2 = ln(2) $\mathbf{\tau }$ ≈ 0.69 $\mathbf{\tau }$ = 0.69RC

Charging & Discharging Equations

• The time constant is used in the exponential decay equations for the current, charge or potential difference (p.d.) for a capacitor charging, or discharging, through a resistor

• These equations can be used to determine:

  • The amount of current, charge or p.d. gained after a certain amount of time for a charging capacitor

  • The amount of current, charge or p.d. remaining after a certain amount of time for a discharging capacitor

Capacitor Discharge Equations

• This exponential decay means that no matter how much charge is initially on the plates, the amount of time it takes for that charge to halve is the same

• The exponential decay of current on a discharging capacitor is defined by the equation:

$I=I_0{\mathrm{e}}^{-\frac{t}{RC}}$

• Where:

  • I = current (A)

  • I₀ = initial current before discharge (A)

  • e = the exponential function

  • t = time (s)

  • RC = resistance (Ω) × capacitance (F) = the time constant τ (s)

• This equation shows that the smaller the time constant τ, the quicker the exponential decay of the current when discharging

• The time taken to fall to a given fraction of I₀ depends only on RC, not on the size of I₀

• Note: during capacitor discharge, I₀ is always larger than I, as the current I will always be decreasing

Values of the capacitor discharge equation on a graph and circuit

• The current at any time is directly proportional to the p.d. across the capacitor and the charge across the parallel plates

• Therefore, this equation also describes the charge on the capacitor after a certain amount of time:

$Q=Q_0{\mathrm{e}}^{-\frac{t}{RC}}$

• Where:

  • Q = charge on the capacitor plates (C)

  • Q₀ = initial charge on the capacitor plates (C)

• As well as the p.d. after a certain amount of time:

$V=V_0{\mathrm{e}}^{-\frac{t}{RC}}$

• Where:

  • V = p.d. across the capacitor (V)

  • V₀ = initial p.d. across the capacitor (V)