Capacitor Discharge Equations (CIE A Level Physics)

• 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

• This is represented by the Greek letter tau () and measured in units of seconds (s)
  • It is an easy way to compare the rate of change of similar quantities e.g. charge, current and 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

The Time Constant on Discharging Potential Difference-Time Graphs

The time constant shown on a discharging capacitor for potential difference

• 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: 

• 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  (s)

• This equation shows that the faster the time constant , the quicker the exponential decay of the current, charge or p.d. when discharging
• Also, how big the initial current, charge or p.d. is affects the rate of discharge
  • If X₀ is large, the capacitor will take longer to discharge

• Note: during capacitor discharge, X₀ is always larger than X, this is because the current, charge or p.d. will always be decreasing
• 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:

• Where:
  • I = current (A)
  • I₀ = initial current before discharge (A)

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

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

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

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

The Exponential Function, e

• 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 of its original value

• Where  = 0.3578