CIE A Level Physics复习笔记19.2.2 Capacitor Discharge Equations

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 of a capacitor to decrease to 0.37 of its original value

  • The time constant gives an easy way to compare the rate of change of similar quantities eg. charge, current and p.d.
  • The time constant is defined by the equation:
  • Where:
    • R = resistance of the resistor (Ω)
    • C = capacitance of the capacitor (F)

Time Constant Graph, downloadable AS & A Level Physics revision notes

The graph of voltage-time for a discharging capacitor showing the positions of the first three time constants

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.

Step 1: Write out the known quantities

Capacitance, C = 7 nF = 7 × 10-9 F

Time constant, τ = 5.6 × 10-3 s

Step 2: Write down the time constant equation

τ = RC

Step 3: Rearrange for resistance R

2.-The-Time-Constant-Worked-Example-equation-1

Step 4: Substitute in values and calculate

2.-The-Time-Constant-Worked-Example-equation-2

Using the Capacitor Discharge Equation

  • 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 when a capacitor is discharging
  • The exponential decay of current on a discharging capacitor is defined by the equation:

3.-Using-the-Capacitor-Discharge-Equation-equation-1

  • 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 faster 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, this is because the current I will always be decreasing
  • 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 change in p.d and charge on the capacitor:

3.-Using-the-Capacitor-Discharge-Equation-equation-2

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

3.-Using-the-Capacitor-Discharge-Equation-equation-3

  • 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:

3.-Using-the-Capacitor-Discharge-Equation-definition-equation-4

3.-Using-the-Capacitor-Discharge-Equation-equation-5

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.

Step 1: Write out the known quantities

Initial current before discharge, I0 = 0.6 A

Current, I = 0.4 A

Resistance, R = 450 Ω

Capacitance, C = 620 μF = 620 × 10-6 F

Step 2: Write down the equation for the exponential decay of current

3.-Using-the-Capacitor-Discharge-Equation-equation-1

Step 3: Calculate the time constant

τ = RC

τ = 450 × (620 × 10-6) = 0.279 s

Step 4: Substitute into the current equation

3.-Using-the-Capacitor-Discharge-Equation-Worked-Example-equation-3

Step 5: Rearrange for the time t

3.-Using-the-Capacitor-Discharge-Equation-Worked-Example-equation-4

The exponential can be removed by taking the natural log of both sides:

3.-Using-the-Capacitor-Discharge-Equation-Worked-Example-equation-5

3.-Using-the-Capacitor-Discharge-Equation-Worked-Example-equation-6

Exam Tip

Make sure you’re confident in rearranging equations with natural logs (ln) and the exponential function (e). To refresh your knowledge of this, have a look at the AS Maths revision notes on Exponentials & Logarithms

 

 

 

 

 

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