An aerospace engineer is testing an emergency backup power supply for a satellite system. The system uses a high-capacity reservoir capacitor C C\,C which discharges through a variable calibration resistor R R\,R during a diagnostic cycle lasting 15.0 s.
The engineer charges the capacitor and then discharges it through the resistor. After a time t=15.0 st = 15.0\text{ s}t=15.0 s, the potential difference V V\,V across the capacitor is measured. This procedure is repeated for several different resistance values of RRR.
The relationship between the potential difference V V\,V and the resistance R R\,R is given by the equation:
V=Vinite−tCR V = V_{\text{init}} \text{e}^{-\frac{t}{CR}} V=Vinite−CRtwhere VinitV_{\text{init}}Vinit is the initial potential difference across the capacitor and t t\,t is the discharge time.
Show that if a graph of ln(V/V)\ln(V/\text{V})ln(V/V) on the yyy-axis is plotted against 1R\displaystyle \frac{1}{R}R1 on the xxx-axis, the magnitude of the gradient of the resulting straight line is equal to 15C\displaystyle \frac{15}{C}C15.
For a resistor of R=56 kΩR = 56\text{ k}\OmegaR=56 kΩ, the measured potential difference after 15.0 s is V=2.8±0.2 VV = 2.8 \pm 0.2\text{ V}V=2.8±0.2 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 1R\displaystyle \frac{1}{R}R1 is determined to be -7.5 × 104 Ω.
Determine the value of CCC. Include an appropriate unit.
Determine the value of RRR, in kΩ\text{k}\OmegakΩ, for which the capacitor discharges to 18% of its original potential difference in 15.0 s.
