CIE AS & A Level Physics 9702: Topic 19: Capacitance- Unit : 19.3 Discharging a capacitor Study Notes

Step 1: $\quad$ Write out the known quantities

$
\begin{aligned}
& \text { Capacitance, } \mathrm{C}=7 \mathrm{nF}=7 \times 10^{-9} \mathrm{~F} \\
& \text { Time constant, } \mathrm{\tau}=5.6 \times 10^{-3} \mathrm{~s}
\end{aligned}
$

Step 2: $\quad$ Write down the time constant equation
$
\mathbf{T}=\mathbf{R C}
$

Step 3: Rearrange for resistance $R$

$
\mathrm{R}=\frac{\tau}{C}
$
Step 4: Substitute in values and calculate

$
R=\frac{5.6 \times 10^{-3}}{7 \times 10^{-9}}=8 \times 10^5 \Omega=800 \mathbf{k} \Omega
$

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:

$
\mathrm{I}=\mathrm{I}_0 e^{-\frac{t}{R C}}
$

– Where:

•  $\mathrm{I}=$ current $(\mathrm{A})$
• $\mathrm{I}_0=$ initial current before discharge (A)
• $\mathrm{e}=$ the exponential function
• $0 \mathrm{t}=$ time $(\mathrm{s})$
• $\mathrm{RC}=$ resistance $(\Omega) \times$ capacitance $(F)=$ the time constant $\tau$ (s)

•  This equation shows that the faster the time constant $\tau$, the quicker the exponential decay of the current when discharging
•  Also, how big the initial current is affects the rate of discharge

• • If $\mathrm{I}_0$ is large, the capacitor will take longer to discharge
•  Note: during capacitor discharge, $I_0$ 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:

$
Q=Q_0 e^{-\frac{t}{R C}}
$$

•  Where:
• •  $\mathrm{Q}=$ charge on the capacitor plates (C)
  •  $\mathrm{Q}_0=$ initial charge on the capacitor plates (C)

$
\mathrm{V}=\mathrm{V}_0 e^{-\frac{t}{R C}}
$

• Where:

• •  $V=$ p.d across the capacitor (C)
  •  $V_0=$ 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 $\mathrm{e}=2.718 \ldots$
•  On a calculator it is shown by the button $\mathrm{e}^{\mathrm{x}}$
• The inverse function of $\mathrm{e}^{\mathrm{x}}$ is $\ln (\mathrm{y})$, 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:
  The time taken for the charge of a capacitor to decrease to $\frac{1}{e}$ of its original value
•  Where $\frac{1}{e}=0.3678 \ldots$