Physics 9702/53 — May/June 2022

Background Concept

The relationship given links the measured capacitor p.d. $V$ to the plate separation $d$ after charge has been transferred from the initially charged plates to an uncharged capacitor. To test a proposed relationship experimentally, you:

• vary one independent variable (here $d$),
• measure the dependent variable (here $V$),
• keep all other relevant quantities constant (especially any that appear in the equation: $A$ and $C$),
• and choose an analysis method that gives a clear test (usually a straight-line graph).

A key Paper 5 skill is linearisation: rearranging an equation into the form

$$y = mx + c$$

so that plotting $y$ against $x$ should give a straight line if the model is correct. From the gradient $m$ and intercept $c$, you can determine the unknown constants.

Understanding the Question

You are given two parallel plates of fixed area $A$ with adjustable separation $d$. The plates are charged using a power supply, then connected to an initially uncharged capacitor (capacitance $C$). You measure the resulting p.d. $V$ across the capacitor.

You must plan an experiment to test whether $V$ depends on $d$ according to

$$\frac{W}{V} = 1 + \frac{Cd}{KA}$$

and explain how to find numerical values for the constants $K$ and $W$ from your results.

Because $K$ and $W$ are constants, the experiment must ensure that each run starts from the same initial conditions (especially the way the plates are charged) so that any change in $V$ is due only to changing $d$.

Approach

1. Choose variables: vary $d$; measure $V$.
2. Design a reliable switching sequence: charge plates in a controlled way, then isolate from the supply and connect to the capacitor for measurement.
3. Control variables: keep $A$ fixed (same overlap), use the same capacitor $C$, keep charging voltage/time the same, and reduce leakage.
4. Linearise the equation into $y = mx + c$ where you can plot using measured quantities (you can measure $V$ and $d$, but you cannot directly compute $W/V$ because $W$ is unknown).
5. Graph and constants: plot the linear graph; obtain gradient and intercept; compute $W$ and $K$.
6. Uncertainties: repeat readings; plot error bars; use worst-acceptable lines to estimate uncertainties in gradient/intercept and therefore in $K$ and $W$.

Step-by-Step Reasoning

1) Practical set-up and why switching is needed

You need two distinct circuit states:

• Charge state: plates connected to the power supply so they are charged in a repeatable way.
• Measure state: power supply disconnected (so it does not affect the redistribution), and plates connected to the uncharged capacitor, with a high-resistance voltmeter across the capacitor.

A DPDT switch (or two labelled switches used carefully) lets you move cleanly between these states.

2) Measurements to take

For each value of $d$:

• measure $d$ with a micrometer/vernier/feeler gauges (ensure plates remain parallel; measure at more than one point and average if possible),
• charge the plates under the same supply setting and charging time,
• discharge the capacitor before each run,
• connect plates to the capacitor and record $V$ once steady.

Also measure:

• $A$ from plate dimensions (length $\times$ width) and keep overlap constant,
• $C$ (from capacitor label or a capacitance meter) and keep the same capacitor for all trials.

Repeats: repeating $V$ at each $d$ and averaging reduces random uncertainty and gives sensible error bars.

3) Linearising the given relationship

Start with

$$\frac{W}{V} = 1 + \frac{Cd}{KA}$$

You cannot plot $W/V$ directly because $W$ is unknown. Divide the whole equation by $W$:

$$\frac{1}{V} = \frac{1}{W} + \frac{C}{KAW}d$$

This is now in straight-line form $y = mx + c$ with:

• $y = 1/V$ (can be calculated from measured $V$),
• $x = d$ (measured directly),
• intercept $c = 1/W$,
• gradient $m = C/(KAW)$.

So if the model is correct, a plot of $1/V$ against $d$ should be a straight line.

4) Determining $W$ and $K$ from the graph

From the intercept:

$$c = \frac{1}{W} \Rightarrow W = \frac{1}{c}$$

From the gradient:

$$m = \frac{C}{KAW}$$

Substitute $W = 1/c$ into the gradient expression:

$$m = \frac{C}{KA(1/c)} = \frac{Cc}{KA}$$

Rearrange to obtain

$$K = \frac{Cc}{Am}$$

So measuring $m$ and $c$ allows you to calculate both constants.

5) Uncertainty treatment (what to say in a plan)

• Put uncertainty in $d$ from the instrument resolution (and from any lack of parallelism if relevant).
• Put uncertainty in $V$ from the voltmeter resolution and repeat scatter.
• When plotting $1/V$, convert the uncertainty: a small uncertainty in $V$ produces an uncertainty in $1/V$ (you can estimate it using percentage uncertainties, e.g. $\Delta(1/V)/(1/V) \approx \Delta V/V$ for small uncertainties).
• Draw a best-fit line and a worst-acceptable line through the error bars to estimate uncertainties in gradient and intercept, then propagate to $W$ and $K$ (e.g. percentage uncertainties add/subtract appropriately for products/quotients).

6) Control of variables (what earns marks)

• $A$ constant: do not change the overlap area; mount plates so only separation changes.
• $C$ constant: use the same capacitor throughout.
• Initial charging constant: same power-supply setting (and same charging time) each run.
• Leakage: clean/dry insulators, avoid touching charged parts, and take readings promptly.

7) Safety

Even at modest voltages, charged plates/capacitors can give an unpleasant shock and can damage instruments if shorted suddenly.

• Switch off/disconnect before altering $d$.
• Discharge through a resistor before handling.
• Use insulated leads and a high-impedance voltmeter.

Key Takeaways

• A strong plan includes: clear variables, a workable method, control of variables, repeat readings, and a linear graph test.
• When a constant (here $W$) is unknown, rearrange so your plotted quantities use only measurable data (here $1/V$ vs $d$).
• For $y = mx + c$, constants are extracted from gradient/intercept: here $W = 1/c$ and $K = Cc/(Am)$.

Common Mistakes

• Trying to plot $W/V$ vs $d$ (impossible because $W$ is unknown).
• Not ensuring the capacitor is fully discharged before each run (changes the initial condition and ruins the test).
• Forgetting to keep $A$ constant (changing overlap area changes the physics).
• Using a low-resistance voltmeter (loads the circuit and alters $V$).
• Not stating how $K$ and $W$ are obtained from the gradient/intercept.

Things to Be Careful About

• Keep plates parallel; if not, the effective separation is ill-defined (large systematic error).
• Choose a suitable range of $d$ and enough points (typically 6–8) for a convincing straight-line test.
• When plotting $1/V$, ensure units are consistent (e.g. $V$ in $\text{V}$ gives $1/V$ in $\text{V}^{-1}$).
• When finding the gradient, use a large triangle and compute $\Delta y/\Delta x$ (not $y/x$).
• Discharge safely through a resistor rather than directly shorting if the charge could be significant.