Modelling Capacitor Discharge (OCR A Level Physics): Revision Note

Exponential Decay Graph for Capacitors

• To verify if potential difference, V, or charge, Q, on a capacitor decreases exponentially:

  • Constant ratio method: Plot a V-t graph and check the time constant is constant, or check if the time to halve from its initial value is constant

  • Logarithmic graph method: Plot a graph of ln V against t and check if a straight line graph is obtained

Constant Ratio Method

• A general form of the exponential decay question is given by

$\mathit{x}\mathbf{=}{\mathit{x}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{A}\mathbf{t}}$

• Where A is a constant

• This equation shows that when t = A⁻¹ the value of x will have decreased to approximately 37% of its original value, x₀:

$x=x_0{e}^{-\frac{A}{A}}=x_0{e}^{-1}=x_0\left(\frac{1}{e}\right)\approx 0.37x_0$

• Comparing this to the discharge equation for a capacitor:

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

• Therefore, for a discharging capacitor, when t = τ the potential difference on the capacitor will have decreased to approximately 37% of its original value

• This means that equal intervals of time give equal fractional changes of $\frac{1}{e}$ in potential difference

The graph of voltage-time for a discharging capacitor showing the positions of the first three time constants

• Hence, to validate if potential difference across a capacitor decreases exponentially:

  The time constant, or the time taken for the potential difference to decrease to 37% of its original value, will be constant

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

The time constant shown on a charging and discharging capacitor

Logarithmic Graph Method

• The potential difference (p.d) across the capacitance is defined by the equation:

• Where:

  • V = p.d. across the capacitor (V)

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

  • t = time (s)

  • e = exponential function

  • R = resistance of the resistor (Ω)

  • C = capacitance of the capacitor (F)

• Rearranging this equation for ln(V) by taking the natural log (ln) of both sides:

• Comparing this to the equation of a straight line: y = mx + c

  • y = ln (V)

  • x = t

  • gradient = −1/RC

  • c = ln (V₀)

A straight-line logarithmic graph of ln V against t can be used to verify an exponential relationship

Modelling the Discharge of a Capacitor

• From electricity, the charge is defined as:

ΔQ = IΔt

• Where:

  • I = current (A)

  • ΔQ = change in charge (C)

  • Δt = change in time (s)

• This means that the area under a current-time graph for a charging (or discharging) capacitor is the charge stored for a certain time interval

The area under the I-t graph is the total charge stored in the capacitor in the time interval Δt

• Rearranging for the current:

• This means that the gradient of the charge-time graph is the current at that time

The gradient of a discharging and charging Q-t graph is the current

• In the discharging graph, this is the discharging current at that time

• In the charging graph, this is the charging current at that time

  • To calculate the gradient of a curve, draw a tangent at that point and calculate the gradient of that tangent

• As a capacitor charges or discharges, the current at any time can be found from Ohm's law:

$I=\frac{V}{R}$

• From the definition of capacitance, the value of potential difference at any time is given by:

$V=\frac{Q}{C}$

• Combining these equations gives:

$I=\frac{1}{R}\left(\frac{Q}{C}\right)=\frac{Q}{RC}$

• For a discharging capacitor, the current decreases with time, hence:

$I=-\frac{∆Q}{∆t}$

• This leads to an expression which can be used to solve for the time constant of a discharging capacitor:

$\frac{∆Q}{∆t}=-\frac{Q}{RC}$

• This equation is useful for modelling using spreadsheets