Capacitor Discharge Equations (Cambridge (CIE) 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 time constant is defined as:

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

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

  • It is a useful way of comparing the rate of change of similar quantities e.g. charge, current or p.d.

• The time constant is defined by the equation:

$\tau$ = RC

• Where:

  • $\tau$ = time constant (s)

  • R = resistance of the resistor (Ω)

  • C = capacitance of the capacitor (F)

• For example, to find the time constant from a voltage-time graph, calculate 0.37V₀ and determine the corresponding time for that value

Time constant on a capacitor discharge graph

The time constant shown on a discharging capacitor for potential difference

Using the capacitor discharge equation

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

  • These can be used to determine the amount of current, charge or p.d. left after a certain amount of time when a capacitor is discharging

• All capacitor discharge equations are of the form: 

$X=X_0{e}^{-\left(\frac{t}{RC}\right)}$

• Where:

  • X = current, charge or potential difference

  • X₀ = initial current, charge or potential difference before discharge

  • e = the exponential function

  • t = time (s)

  • RC = resistance (Ω) × capacitance (F) = the time constant $\tau$ (s)

• This equation shows that

  • the greater the time constant $\tau$, the slower the current, charge or p.d. falls during discharge

  • the greater the initial current, charge or p.d., the greater the rate of discharge, i.e. if X₀ is large, the capacitor will take longer to discharge

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

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

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

• Where:

  • I = current (A)

  • I₀ = initial current before discharge (A)

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

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

• Where:

  • Q = charge on the capacitor plates (C)

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

• The equation for the exponential decay of p.d. on a discharging capacitor is defined by the equation:

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

• Where:

  • V = potential difference on the capacitor plates (V)

  • V₀ = initial potential difference on the capacitor plates (V)

The exponential function

• The symbol e represents the exponential constant, a number which is approximately equal to e = 2.718...

  • On a calculator, it is shown by the button e^x

• The inverse function of e^x is ln(y), known as the natural logarithmic function

  • This is because, if e^x = y, then x = ln(y)

• The 0.37 in the definition of the time constant arises as a result of the exponential constant, the true definition is:

  The time taken for the charge of a capacitor to decrease to $\frac{1}{e}$ of its original value

• Where $\frac{1}{e}$ = 0.3679