Physics 9702/52 — May/June 2022

Background Concept

The suggested relationship is

$$\frac{E}{I} = \frac{2PL}{A} + \frac{Qx}{y^2}$$

The left-hand side, $E/I$, has units of resistance (ohms), because $R = V/I$. This strongly suggests the experiment is effectively measuring the total resistance between the two ends C, made up of two contributions:

• a term proportional to $L$ (resistance of the two lengths of cylindrical conductor leading to the bar), and
• a constant term for a fixed geometry ($x$ and $y$ are constant), representing the bar/connection contribution.

A key planning idea is: if we can vary only $L$ while keeping $A$, $x$ and $y$ constant, then $E/I$ should change linearly with $L$. A straight-line graph allows us to determine constants from the gradient and intercept.

Understanding the Question

You are asked to plan an experiment (so method, measurements, control variables, analysis, and safety are what get the marks).

Given:

• Two parallel cylindrical conductors, each cross-sectional area $A$.
• A bridging metal bar of square cross-section with side $y$.
• Separation of the conductors is $x$.
• For each conductor, the distance from its end C to the centre of the bar is $L$.
• A power supply is connected to the ends C and current $I$ is measured.

Unknowns to determine: constants $P$ and $Q$.

So we must:

1. change $L$ in a controlled way,
2. measure $E$ and $I$ for each $L$ and calculate $E/I$,
3. use a graph to obtain $P$ and $Q$.

Approach

1. Choose independent and dependent variables:
   • independent: $L$ (set by moving the bar position)
   • dependent: $I$ (measured), and also measure $E$ so you can calculate $E/I$
2. Keep $A$, $x$, $y$ constant by using the same conductors/bar and fixing their spacing.
3. Address practical limitations:
   • heating changes resistivity, so limit current and take readings quickly
   • contact resistance at the bar can vary, so use firm, reproducible contacts and clean metal
4. Linearise: the given equation is already in straight-line form $Y = mX + c$ if you take $Y = E/I$ and $X = L$.
5. Extract constants:

$$\text{gradient } m = \frac{2P}{A}, \quad \text{intercept } c = \frac{Qx}{y^2}$$

So measure $A$, $x$, $y$ to compute $P$ and $Q$.

Step-by-Step Reasoning

1. Set up the geometry

   • Clamp the two conductors straight and parallel on an insulating base so $x$ is fixed.
   • Ensure ends C are clearly defined (marks on the conductors help).

2. Electrical circuit

   • Ammeter in series measures the current $I$ through the whole arrangement.
   • Voltmeter across the ends C measures the potential difference across the conductors+bar arrangement. This is the most practical way to obtain the $E$ value used in $E/I$ (and avoids assuming the supply stays exactly constant under load).

3. Vary $L$ reliably

   • Place the bar so it bridges the two conductors with good contact.
   • Measure $L$ as the distance from end C to the centre of the bar (so you must be consistent about where you measure to).
   • Move the bar to several positions to get a wide spread of $L$ values.

4. Take repeated measurements

   • For each $L$, close the switch briefly, record $I$ and $E$, then open the switch.
   • Repeat and average to reduce random variation (especially due to small contact changes).

5. Measure the fixed dimensions

   • Measure conductor diameter $d$ with a micrometer and compute

$$A = \frac{\pi d^2}{4}$$

• Measure bar side length $y$ with a micrometer.
• Measure $x$ carefully: for example, measure the distance between inner faces with calipers and add one diameter, or measure between outer faces and subtract one diameter. The key is that $x$ is centre-to-centre.

1. Analyse using a straight-line graph

   • For each run compute $E/I$.
   • Plot $E/I$ against $L$. If the model is correct, points should lie close to a straight line.
   • Find the gradient $m$ and intercept $c$ from the best-fit line.

2. Determine $P$ and $Q$

   • From

$$\frac{E}{I} = \frac{2P}{A}L + \frac{Qx}{y^2}$$

 identify

$$m = \frac{2P}{A} \Rightarrow P = \frac{mA}{2}$$ $$c = \frac{Qx}{y^2} \Rightarrow Q = \frac{cy^2}{x}$$

(Units should be checked from these expressions; the graph gives $m$ in $\Omega\ \text{m}^{-1}$ and $c$ in $\Omega$.)

Key Takeaways

• Convert the given relationship directly into a straight-line plot by choosing graph variables so it matches $Y = mX + c$.
• Control variables are crucial in electrical-resistance experiments: geometry and temperature must be kept constant.
• Gradient gives one constant ($P$) and intercept gives the other ($Q$), provided you have measured $A$, $x$, and $y$.

Common Mistakes

• Plotting $I$ against $L$ directly: the relationship is linear in $E/I$, not necessarily in $I$.
• Not measuring $E$ for each reading (assuming it is constant even when current changes).
• Allowing the conductors to heat up significantly, changing resistance and spoiling linearity.
• Inconsistent definition of $L$ (measuring to the bar edge sometimes and to the centre other times).
• Forgetting that $x$ is centre-to-centre separation.

Things to Be Careful About

• Contact resistance: it can dominate the intercept if the bar contact is poor; clean surfaces and use strong, repeatable clamping.
• Temperature: even modest heating changes resistivity; use low voltage/current, switch on briefly, and keep time between readings similar.
• Range and number of readings: use enough $L$ values (at least 6–8) over a wide range for a reliable gradient and intercept.
• Graph quality: use a large triangle for gradient and read intercept carefully; poor graph technique directly affects $P$ and $Q$.
• Dimensional measurements: micrometer readings for $d$ and $y$ should include appropriate precision; small errors in $d$ cause larger fractional errors in $A$ because $A \propto d^2$.