# 7.7.2 The Time Constant

### 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, current or voltage of a discharging capacitor to decrease to 37% of its original value

• Alternatively, for a charging capacitor:

The time taken for the charge or voltage of a charging capacitor to rise to 63% of its maximum value

• 37% is 0.37 or 1 / e (where e is the exponential function) multiplied by the original value (I0, Q0 or V0)
• This is represented by the Greek letter tau, τ,  and measured in units of seconds (s)
• The time constant provides an easy way to compare the rate of change of similar quantities eg. charge, current and p.d.
• It 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
• The time to half, t1/2 (half-life) for a discharging capacitor is:

The time taken for the charge, current or voltage of a discharging capacitor to reach half of its initial value

• This can also be written in terms of the time constant:

t1/2 = 0.69τ = 0.69RC

#### 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.

#### Exam Tip

Note that the time constant is not the same as half-life. Half-life is how long it takes for the current, charge or voltage to halve whilst the time constant is to 37% of its original value (not 50%)

Although the time constant is given on your data sheet, you will be expected to remember the half-life equation t1/2 = 0.69RC

### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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