A student is investigating how the discharge of a capacitor through a resistor depends on the capacitance C C\,C of a variable capacitor.
The student charges a variable capacitor of capacitance C C\,C and then discharges it through a resistor of fixed resistance RRR. After a time t=5.0 st = 5.0\text{ s}t=5.0 s, the student records the potential difference V V\,V across the capacitor. This procedure is repeated for different values of CCC.
It is suggested that V V\,V and C C\,C are related by the equation:
V=V0e−tRC V = V_0 \text{e}^{-\frac{t}{RC}} V=V0e−RCtwhere V0 V_0\,V0 is the initial potential difference across the capacitor and t t\,t is the time over which the capacitor has discharged.
Show that if a graph of ln(V/V)\ln(V/\text{V})ln(V/V) on the yyy-axis is plotted against 1C\displaystyle \frac{1}{C}C1 on the xxx-axis, the magnitude of the gradient of the resulting straight line is equal to 5.0R\displaystyle \frac{5.0}{R}R5.0.
For a capacitance of C=22 μFC = 22\text{ }\mu\text{F}C=22 μF, the measured potential difference after 5.0 s is V=3.6±0.3 VV = 3.6 \pm 0.3\text{ V}V=3.6±0.3 V. Calculate the value of ln(V/V)\ln(V/\text{V})ln(V/V) and its absolute uncertainty.
The gradient of the graph of ln(V/V)\ln(V/\text{V})ln(V/V) against 1C\displaystyle \frac{1}{C}C1 is determined to be −2.0×10−5 F-2.0 \times 10^{-5}\text{ F}−2.0×10−5 F.
Determine the value of RRR. Include an appropriate unit.
Determine the value of CCC, in μF\mu\text{F}μF, for which the capacitor discharges to 15% of its original potential difference in 5.0 s.
