6.2.2 Capacitor Charge & Discharge Equations

• 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, t_1/2 (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:

t_1/2 = ln(2)  ≈ 0.69  = 0.69RC

• 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)

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