7.7.3 Charge & Discharge Equations

• 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 for a discharging capacitor

• 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)

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 e^x
• The inverse function of e^x is ln(y), known as the natural logarithmic function
  • This is because, if e^x = 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:

• 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