19.2.2 Capacitor Discharge Equations

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 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 (τ) 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:

τ = RC

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

Worked example: Time constant

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

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
• 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 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:

Worked example: Current capacitor discharge

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:

Exam 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

Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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