19.2.2 Capacitor Discharge Equations

• 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 of a capacitor to decrease to 0.37 of its original value

• This is represented by the greek letter tau () and measured in units of seconds (s)
• The time constant gives an easy way to compare the rate of change of similar quantities eg. charge, current and p.d.
• The time constant is defined by the equation:

 = RC

• Where:
  •  = time constant (s)
  • R = resistance of the resistor (Ω)
  • C = capacitance of the capacitor (F)

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

Step 1: Write out the known quantities

Capacitance, C = 7 nF = 7 × 10^-9 F

Time constant, τ = 5.6 × 10^-3 s

Step 2: Write down the time constant equation

τ = RC

Step 3: Rearrange for resistance R

Step 4: Substitute in values and calculate

• 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

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

• 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 faster 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, this is because the current I 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

• Therefore, this equation also describes the change in p.d and charge on the capacitor:

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

• 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:

Step 1: Write out the known quantities

Initial current before discharge, I₀ = 0.6 A

Current, I = 0.4 A

Resistance, R = 450 Ω

Capacitance, C = 620 μF = 620 × 10^-6 F

Step 2: Write down the equation for the exponential decay of current

Step 3: Calculate the time constant

τ = RC

τ = 450 × (620 × 10^-6) = 0.279 s

Step 4: Substitute into the current equation

Step 5: Rearrange for the time t

The exponential can be removed by taking the natural log of both sides: