# Capacitor Charge & Discharge Equations(OCR A Level Physics)

Author

Katie M

Expertise

Physics

## 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  (where e is the exponential function) multiplied by the original value (I0, Q0 or V0)
• This is represented by the Greek letter tau, , 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:

= RC

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

• 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, τ:

t1/2 = ln(2)  ≈ 0.69  = 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 the datasheet, you will be expected to remember the half-life equation t1/2 = 0.69RC

## Charging & Discharging Equations

• The time constant is used in the exponential decay equations for the current, charge or potential difference (p.d.) for a capacitor charging, or discharging, through a resistor
• These equations can be used to determine:
• The amount of current, charge or p.d. gained after a certain amount of time for a charging capacitor
• The amount of current, charge or p.d. remaining after a certain amount of time for a discharging capacitor

#### Capacitor Discharge Equations

• This exponential decay means that no matter how much charge is initially on the plates, the amount of time it takes for that charge to halve is the same
• 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 smaller 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, as the current I will always be decreasing

Values of the capacitor discharge equation on a graph and circuit

• 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 charge on the capacitor after a certain amount of time:

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

• As well as the p.d. after a certain amount of time:

• Where:
• V = p.d. across the capacitor (C)
• V0 = initial p.d. across the capacitor (C)

#### 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.

#### Capacitor Charge Equations

• When a capacitor is charging, the way the charge Q and potential difference V increases stills shows exponential decay
• Over time, they continue to increase but at a slower rate
• This means the equation for Q for a charging capacitor is:

• Where:
• Q = charge on the capacitor plates (C)
• Q0 = maximum charge stored on capacitor when fully charged (C)
• e = the exponential function
• t = time (s)
• RC = resistance (Ω) × capacitance (F) = the time constant τ (s)
• Similarly, for V:

• Where:
• V = p.d. across the capacitor (V)
• V0 = maximum potential difference across the capacitor when fully charged (V)
• The charging equation for the current I is the same as its discharging equation since the current still decreases exponentially
• The key difference with the charging equations is that Q0 and V0 are now the final (or maximum) values of Q and V that will be on the plates, rather than the initial values

#### Worked example

A capacitor is to be charged to a maximum potential difference of 12 V between its plate.

Calculate how long it takes to reach a potential difference 10 V given that it has a time constant of 0.5 s.

#### Exam Tip

Knowledge of the exponential constant, a number which is approximately equal to e = 2.718..., is crucial for mastering charge and discharge equations, so make sure you know how to use it:

• 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, so make sure you know that the true definition is:

Time constant = The time taken for the charge of a capacitor to decrease to of its original value

Where = 0.3678

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### Author:Katie M

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.