Capacitor Discharge Equations (Cambridge International A-Level Physics)

The Time Constant

Spec point — 19.3.2

• The time constant of a capacitor discharging through a resistor is a measure of how long it takes for the capacitor to discharge
• The time constant is defined as:

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 ($τ$) and measured in units of seconds(s)

  • It is a useful way of comparing the rate of change of similar quantities e.g. charge, current or p.d.

• The time constant is defined by the equation:

$τ$ = RC

• Where:

  • $τ$ = 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

Time constant on a capacitor discharge graph

The time constant shown on a discharging capacitor for potential difference

Worked Example A capacitor of 7 nF is discharged through a resistor of resistance R. The time constant of the discharge is 5.6 × 10^-3 s.

Calculate the value of R.

Answer:

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

$R=\frac{τ}{C}$

Step 4: Substitute in values and calculate R

$R=\frac{5.6\times10^{-3}}{7\times10^{-9}}=8\times10^{5}Ω=800\mathrm{kΩ}$

Using the 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 when a capacitor is discharging

• All capacitor discharge equations are of the form:

$X=X_{0}e^{-(\frac{t}{RC})}$

• 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 $τ$ (s)

• This equation shows that

  • the greater the time constant $τ$, the slower the current, charge or p.d. falls during discharge
  • the greater the initial current, charge or p.d., the greater the initial rate of discharge, but the time taken to fall to a given fraction of X₀ depends only on the time constant RC

• 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:

$I=I_{0}e^{(-\frac{t}{RC})}$

• Where:

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

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

$Q=Q_{0}e^{(-\frac{t}{RC})}$

• Where:

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

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

$V=V_{0}e^{(-\frac{t}{RC})}$

• Where:

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

The exponential function

• 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

Worked Example The initial current through a circuit with a capacitor of 620 μF is 0.6 A. The capacitor is connected across the terminals of a 450 Ω resistor.

Calculate the time taken for the current to fall to 0.4 A.

Answer:

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

$I=I_{0}e^{(-\frac{t}{RC})}$

Step 3: Calculate the time constant

$τ$ = RC

$τ$ = 450 × (620 × 10^-6) = 0.279 s

Step 4: Substitute into the current equation

$0.4=0.6\times e^{-\frac{t}{0.279}}$

Step 5: Rearrange for the time t

$\frac{0.4}{0.6}=e^{-\frac{t}{0.279}}$

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

$\mathrm{ln}(\frac{0.4}{0.6})=-\frac{t}{0.279}$

$t=-0.279\times\mathrm{ln}(\frac{0.4}{0.6})=0.1131=0.1s$

Exam Hint Make sure you’re confident in rearranging equations with natural logs (ln) and the exponential function (e). To refresh your knowledge of this, have a look at the AS Maths revision notes on Exponentials & Logarithms