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First exams 2025

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Capacitor Discharge Equations (CIE A Level Physics)

Revision Note

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Ann H

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The time constant

  • 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 (tau) 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:

tau = RC

  • Where:
    • tau = 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

19-3-2-time-constant-on-graph--cie-new

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, tau = 5.6 × 10-3 s

Step 2: Write down the time constant equation

tau = RC

Step 3: Rearrange for resistance R

R space equals space tau over C

Step 4: Substitute in values and calculate R

R space equals space fraction numerator 5.6 space cross times space 10 to the power of negative 3 end exponent over denominator 7 space cross times space 10 to the power of negative 9 end exponent end fraction space equals space 8 space cross times space 10 to the power of 5 space straight capital omega space equals space 800 space straight k straight capital omega

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 space equals space X subscript 0 e to the power of negative stretchy left parenthesis fraction numerator t over denominator R C end fraction stretchy right parenthesis end exponent

  • Where:
    • = 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 tau (s)
  • This equation shows that
    • the greater the time constant tau, the faster the current, charge or p.d. falls during discharge
    • the greater the initial current, charge or p.d., the greater the rate of discharge, i.e. if X0 is large, the capacitor will take longer to discharge
  • 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 space equals space I subscript 0 e to the power of stretchy left parenthesis negative fraction numerator t over denominator R C end fraction stretchy right parenthesis end exponent

  • Where:
    • I = current (A)
    • I0 = initial current before discharge (A)
  • The equation for exponential decay of charge on a discharging capacitor is defined by the equation:

Q space equals space Q subscript 0 e to the power of stretchy left parenthesis negative fraction numerator t over denominator R C end fraction stretchy right parenthesis end exponent

  • Where:
    • Q = charge on the capacitor plates (C)
    • Q0 = initial charge on the capacitor plates (C)
  • The equation for exponential decay of p.d. on a discharging capacitor is defined by the equation:

V space equals space V subscript 0 e to the power of stretchy left parenthesis negative fraction numerator t over denominator R C end fraction stretchy right parenthesis end exponent

  • 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 ex
  • The inverse function of ex is ln(y), known as the natural logarithmic function
    • This is because, if ex = 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 1 over e of its original value

  • Where 1 over e = 0.3578

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 space equals space I subscript 0 e to the power of stretchy left parenthesis negative fraction numerator t over denominator R C end fraction stretchy right parenthesis end exponent 

Step 3: Calculate the time constant

tau = RC

tau = 450 × (620 × 10-6) = 0.279 s

Step 4: Substitute into the current equation

0.4 space equals space 0.6 space cross times space e to the power of negative fraction numerator t over denominator 0.279 end fraction end exponent

Step 5: Rearrange for the time t

fraction numerator 0.4 over denominator 0.6 end fraction space equals space e to the power of negative fraction numerator t over denominator 0.279 end fraction end exponent

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

ln stretchy left parenthesis fraction numerator 0.4 over denominator 0.6 end fraction stretchy right parenthesis space equals space minus fraction numerator t over denominator 0.279 end fraction

t space equals space minus 0.279 space cross times space ln stretchy left parenthesis fraction numerator 0.4 over denominator 0.6 end fraction stretchy right parenthesis space equals space 0.1131 space equals space 0.1 space s

Examiner Tip

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

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Ann H

Author: Ann H

Expertise: Physics

Ann obtained her Maths and Physics degree from the University of Bath before completing her PGCE in Science and Maths teaching. She spent ten years teaching Maths and Physics to wonderful students from all around the world whilst living in China, Ethiopia and Nepal. Now based in beautiful Devon she is thrilled to be creating awesome Physics resources to make Physics more accessible and understandable for all students no matter their schooling or background.