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

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 for a discharging capacitor

• 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:

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

• 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

• Also, how big the initial current is affects the rate of discharge

  • If I₀ is large, the capacitor will take longer to discharge

• 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:

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

• 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:

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

• Where:

  • V = p.d across the capacitor (C)

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

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 $\frac{1}{e}$ of its original value

• Where $\frac{1}{e}$ = 0.3679

Capacitor Charge Equation

• When a capacitor is charging, the charge Q and potential difference V increase exponentially

  • Over time, they continue to increase, but at a slower rate

• This means the equation for Q for a charging capacitor is:

$Q=Q_0\left(1-e^{-\frac{t}{RC}}\right)$

• Where:

  • Q = charge on the capacitor plates (C)

  • Q₀ = maximum charge stored on capacitor when fully charged (C)

  • e = the exponential function

  • t = time (s)

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

• Similarly, for V:

$V=V_0\left(1-e^{-\frac{t}{RC}}\right)$

• Where:

  • V = p.d across the capacitor (V)

  • V₀ = maximum potential difference across the capacitor when fully charged (V)

• The charging equation for the current I is the same as its discharging equation, since the current still decreases exponentially

• The key difference with the charging equations is that Q₀ and V₀ are now the final (or maximum) values of Q and V that will be on the plates, rather than the initial values