• 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 (?) × 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 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:

• 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 (?) × 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