6.2.3 Modelling Capacitor Discharge

• 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

• 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:

• Comparing this to the discharge equation for a capacitor:

• 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  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

Step 1: Complete the table

• • Add an extra column ln(V) and calculate this for each p.d.

Step 2: Plot the graph of ln(V) against average time t

• • Make sure the axes are properly labelled and the line of best fit is drawn with a ruler

Step 3: Calculate the gradient of the graph

• • The gradient is calculated by:

Step 4: Calculate the capacitance, C

• 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:

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

• Combining these equations gives:

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

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

• This equation is useful for modelling using spreadsheets

Step 1: Draw a tangent at t = 4

Step 2: Calculate the gradient of the tangent to find the current I