Physics 9702/31 — October/November 2013

Background Concept

For a cylindrical wire, the diameter $d$ is typically very small (fractions of a millimetre), so a micrometer screw gauge is used rather than a ruler. A micrometer has a much smaller least count (often $0.01\ \text{mm}$), allowing more precise measurements.

Because a wire may not be perfectly uniform, measuring at several positions and averaging improves reliability (random errors reduce).

Understanding the Question

You are asked to measure and record the diameter $d$ of a short sample of wire. This value will later be used to calculate the cross-sectional area $A$ in $\text{m}^2$, so the measurement must be careful and recorded with sensible precision.

Approach

1. Use a micrometer to measure the wire diameter.
2. Check for zero error and correct if necessary.
3. Take repeat readings at different positions and average.
4. Record $d$ to the micrometer resolution (typically $0.01\ \text{mm}$).

Step-by-Step Reasoning

• Close the micrometer gently to check if it reads $0.00\ \text{mm}$. If not, note the zero error.
• Place the wire between the anvil and spindle.
• Tighten using the ratchet (so the contact force is consistent).
• Read the main scale and thimble scale; apply any zero correction.
• Repeat at several points along the wire (and/or rotate the wire) to reduce the effect of ovality.
• Calculate the mean and record it, e.g. $d = 0.32\ \text{mm}$.

Key Takeaways

• Small diameters require a micrometer for adequate precision.
• Repeats and averaging improve data quality.
• Record to the instrument resolution.

Common Mistakes

• Not checking/allowing for zero error.
• Overtightening without the ratchet, flattening the wire and giving a diameter that is too small.
• Taking only one reading on a possibly non-uniform wire.
• Recording too many decimal places (false precision) or too few (loss of precision).

Things to Be Careful About

• Keep units clear: micrometers commonly read in $\text{mm}$, but later calculations need $d$ in $\text{m}$.
• Do not include insulation (if any) in the diameter measurement—measure the metal only.

Background Concept

For a circular cross-section, area is

If you measure diameter $d$ instead of radius, then $r = d/2$ and

Because resistivity calculations use SI units, $A$ must be in $\text{m}^2$, so $d$ must be converted to metres before squaring.

Understanding the Question

You are given the formula

and asked to calculate $A$ in $\text{m}^2$ using your measured diameter $d$.

Approach

1. Convert $d$ into metres.
2. Substitute into $A = \pi d^2 / 4$.
3. Round to a sensible number of significant figures consistent with $d$.

Step-by-Step Reasoning

Using an example measurement $d = 0.32\ \text{mm}$:

• Convert to metres:

• Substitute:

• Square the diameter:

• Multiply by $\pi/4$:

Key Takeaways

• Always convert to metres before squaring.
• The area depends on $d^2$, so small percentage errors in $d$ double in $A$.

Common Mistakes

• Squaring $d$ while still in $\text{mm}$ and then writing $\text{m}^2$.
• Using $A = \pi d^2$ (forgetting the factor $1/4$).
• Over-rounding early and losing accuracy.

Things to Be Careful About

• Keep track of powers of ten when squaring.
• Quote $A$ to an appropriate number of significant figures (usually matching the significant figures in $d$).

Background Concept

A potential divider arrangement can be created by having a long resistance wire in the circuit. A voltmeter must always be connected in parallel with the component/section whose potential difference is being measured.

Crocodile clips act as movable contacts that select a particular length of wire; changing the length changes that section's resistance.

Understanding the Question

You must build the circuit in Fig. 1.1 using the resistance wire on the metre rule, one resistor, and one voltmeter. You then place two crocodile clips (K and L) so the separation along the wire is about $30\ \text{cm}$.

Approach

• Put the fixed resistor in series with the supply (so it limits current).
• Use K and L to define the wire section to be measured.
• Connect the voltmeter across K and L (parallel).
• Choose an initial length near $0.30\ \text{m}$ to start your dataset.

Step-by-Step Reasoning

• Connect the power supply, switch, and resistor in series with the resistance wire.
• Attach crocodile clip K to one point on the wire.
• Attach crocodile clip L to a point about $0.30\ \text{m}$ away (use the metre rule scale).
• Connect the voltmeter leads to K and L so it reads the p.d. across the length $KL$.
• Ensure the voltmeter is on an appropriate voltage range before switching on later.

Key Takeaways

• Voltmeter in parallel; resistor and wire in series.
• Clips define the length $l$ of wire under test.

Common Mistakes

• Putting the voltmeter in series (gives incorrect readings and may stop the circuit working properly).
• Measuring $l$ as a straight-line distance rather than along the metre rule scale.
• Poor contact at clips (intermittent readings).

Things to Be Careful About

• Place clips firmly and on clean wire to reduce contact resistance changes.
• Avoid very small $l$ (tiny voltages) and very large $l$ (wire heating, larger resistance causing large p.d. changes).

Background Concept

A metre rule is read by aligning the eye perpendicular to the scale to avoid parallax. The length between two points is found by taking two position readings and subtracting.

Understanding the Question

You must measure the separation $l$ between crocodile clips K and L along the wire and record it in metres.

Approach

• Read the metre-rule positions of K and L.
• Subtract to get $l$.
• Convert to metres and record with suitable precision (typically to the nearest $1\ \text{mm}$, i.e. $0.001\ \text{m}$).

Step-by-Step Reasoning

• Suppose K is at $12.3\ \text{cm}$ and L is at $42.3\ \text{cm}$.
• Then

Key Takeaways

• Use two readings and subtraction to reduce zero/endpoint errors.
• Record in metres for later graphing.

Common Mistakes

• Recording $30.0$ but writing unit $\text{m}$.
• Measuring from the ends of the wire rather than between the clip contact points.
• Parallax error when reading the scale.

Things to Be Careful About

• Ensure both clip contact points correspond to the measured positions.
• Keep the same resolution for all $l$ readings in the table.

Background Concept

Resistive heating increases the temperature of the wire and can change its resistance, introducing systematic error. Switching off between readings limits heating and improves repeatability.

Understanding the Question

After recording $V_1$ and $V_2$, you are instructed to switch off the supply.

Approach

Turn off the supply promptly after measurements.

Step-by-Step Reasoning

• Open the switch / turn off the power supply.
• Allow the wire to cool if it feels warm before moving clips for the next length.

Key Takeaways

• Minimising heating improves data quality.
• Good practice: power off when not actively measuring.

Common Mistakes

• Leaving the circuit powered while adjusting clips (sparks, heating, drifting readings).

Things to Be Careful About

• If the wire has noticeably warmed, wait for it to return to room temperature before the next reading.

Background Concept

If two wire segments have equal length and uniform cross-sectional area, their resistances are proportional to length. Setting $KL = LM = l$ ensures you are comparing like with like; this is essential for the intended relationship between $V_1/V_2$ and $l$.

Understanding the Question

You must place crocodile clip M so that the length from L to M is exactly the same as the measured length $l$ from K to L.

Approach

• Use metre-rule position readings: if L is at position $x_L$, then place M at $x_L + l$ (in the same units).

Step-by-Step Reasoning

• If you measured $l = 0.300\ \text{m}$, then $l = 30.0\ \text{cm}$.
• If L is at $40.0\ \text{cm}$, place M at $70.0\ \text{cm}$.
• Re-check $LM$ by subtracting positions.

Key Takeaways

• Equal lengths are vital for the ratio method.
• Use position subtraction rather than trying to measure between clip bodies.

Common Mistakes

• Setting M approximately rather than carefully matching $l$.
• Measuring from K to M instead of from L to M.

Things to Be Careful About

• Ensure M is placed along the wire in the same direction from L (do not accidentally place it on the other side of L).

Background Concept

When current flows through a resistance wire, it can heat up. Heating changes the wire’s resistance and can affect voltage readings, so the circuit should be powered only when ready to take readings, and readings should be taken promptly.

Understanding the Question

This step instructs you to switch on the supply so that the voltmeters show $V_1$ and $V_2$.

Approach

• Before switching on: confirm correct wiring and appropriate voltmeter range.
• Switch on briefly to take readings.

Step-by-Step Reasoning

• Ensure both voltmeters are connected in parallel to the correct segments.
• Set voltmeters to a range that will not overload.
• Close the switch and allow readings to stabilise.

Key Takeaways

• Good practical work includes checking before energising a circuit.
• Minimise heating effects by not leaving the current on unnecessarily.

Common Mistakes

• Leaving the power on for long periods, causing drift in readings.
• Switching on with incorrect meter settings (over-range).

Things to Be Careful About

• If the wire warms, allow it to cool before repeating measurements for consistency.

Background Concept

Voltmeters measure potential difference and should have high resistance, so they do not significantly change the circuit. Readings should be recorded consistently (same decimal places) to reflect instrument resolution.

Understanding the Question

With the circuit connected as in Fig. 1.2, voltmeter 1 gives $V_1$ across K–L and voltmeter 2 gives $V_2$ across L–M. You must write both values down with units.

Approach

• Wait for the readings to settle.
• Read and record each voltmeter value with the correct unit (V) and appropriate resolution.

Step-by-Step Reasoning

• Observe voltmeter 1 across K–L and record $V_1$.
• Observe voltmeter 2 across L–M and record $V_2$.
• Example set: $V_1 = 0.44\ \text{V}$ and $V_2 = 0.40\ \text{V}$.

Key Takeaways

• Record units and consistent decimal places.
• Take readings quickly to reduce heating drift.

Common Mistakes

• Swapping $V_1$ and $V_2$.
• Missing units.
• Inconsistent decimal places between readings.

Things to Be Careful About

• If readings fluctuate, check clip contacts and ensure the wire is not moving.

Background Concept

Resistive heating increases the temperature of the wire and can change its resistance, introducing systematic error. Switching off between readings limits heating and improves repeatability.

Understanding the Question

After recording $V_1$ and $V_2$, you are instructed to switch off the supply.

Approach

Turn off the supply promptly after measurements.

Step-by-Step Reasoning

• Open the switch / turn off the power supply.
• Allow the wire to cool if it feels warm before moving clips for the next length.

Key Takeaways

• Minimising heating improves data quality.
• Good practice: power off when not actively measuring.

Common Mistakes

• Leaving the circuit powered while adjusting clips (sparks, heating, drifting readings).

Things to Be Careful About

• If the wire has noticeably warmed, wait for it to return to room temperature before the next reading.

Background Concept

A good graph makes it easy to see the relationship between two variables and to determine the gradient and intercept. In Cambridge practical papers, marks are awarded for:

• correct choice of axes
• correct labels (quantity and unit)
• sensible scales (not cramped, not awkward)
• accurate plotting

Understanding the Question

You must produce a graph with $y = V_1/V_2$ and $x = l$. This is designed to test the linear relationship

Approach

• Put the independent variable $l$ on the horizontal axis.
• Put the dependent variable $V_1/V_2$ on the vertical axis.
• Select scales so your points fill the graph area.

Step-by-Step Reasoning

• Decide the range of $l$ values from your table and set the $x$-axis scale accordingly.
• Decide the range of $V_1/V_2$ values and set the $y$-axis scale accordingly.
• Label axes clearly: $l / \text{m}$ and $V_1/V_2$.
• Plot each point as a small cross; ensure points are not thick blobs.

Key Takeaways

• Correct axes and scales are as important as the plotted points.
• $V_1/V_2$ is dimensionless, so no unit on the $y$-axis.

Common Mistakes

• Swapping axes (plotting $l$ on $y$).
• Forgetting units on the $x$-axis.
• Using a scale that wastes most of the grid.

Things to Be Careful About

• Keep plotting accuracy high: use a ruler to read coordinates, and plot neat crosses.
• If two points are close, ensure they are still distinguishable.

Background Concept

A best-fit line represents the overall trend of the data. For a linear relationship, it should be a straight line that balances the scatter: roughly equal numbers of points above and below the line.

Understanding the Question

After plotting $V_1/V_2$ against $l$, you must draw the straight line of best fit.

Approach

Use a ruler and draw one thin straight line that best represents the trend.

Step-by-Step Reasoning

• Visually judge the trend.
• Place the ruler so the line passes through the middle of the data scatter.
• Draw a thin straight line across the full span of the plotted data (not just between two points).

Key Takeaways

• Best-fit means balancing scatter, not forcing the line through every point.

Common Mistakes

• Joining dots with a zig-zag line.
• Forcing the line through the origin when the data do not support it.

Things to Be Careful About

• Use a sharp pencil and ruler so the line thickness does not introduce reading error for gradient/intercept.

Background Concept

For a straight-line graph of the form

• the gradient is

• the $y$-intercept is the value of $y$ when $x=0$.

Here $y = V_1/V_2$ (dimensionless) and $x = l$ (in m), so the gradient must have units $\text{m}^{-1}$.

Understanding the Question

You must find the gradient and the $y$-intercept of your best-fit line on the graph of $V_1/V_2$ against $l$.

Approach

• Choose two points far apart on the best-fit line (not necessarily data points).
• Calculate gradient using $\Delta y/\Delta x$.
• Read the intercept by extending the line to $l=0$.

Step-by-Step Reasoning

• Select two convenient points on the line, e.g. $(l, V_1/V_2) = (0.10, 0.70)$ and $(0.50, 1.50)$.
• Compute changes:

• Gradient:

• Intercept: extend the line back to $l=0$ and read $V_1/V_2$ there, giving (example) $0.50$.

Key Takeaways

• Use a large triangle to reduce percentage uncertainty in the gradient.
• Gradient units come from $y$ units divided by $x$ units.

Common Mistakes

• Using $\Delta x/\Delta y$ instead of $\Delta y/\Delta x$.
• Using two nearby points, causing a large gradient uncertainty.
• Reading the intercept from a data point rather than from the best-fit line.

Things to Be Careful About

• Read from the line, not from plotted crosses.
• Keep enough significant figures in gradient and intercept consistent with graph-reading precision.

Background Concept

Any straight-line graph follows

Comparing with

we identify:

• $y \leftrightarrow V_1/V_2$
• $x \leftrightarrow l$
• $m \leftrightarrow P$
• $c \leftrightarrow Q$

Since $V_1/V_2$ is dimensionless, $P$ must have units of $1/\text{m}$ so that $Pl$ is dimensionless.

Understanding the Question

You have already obtained the gradient and intercept from the graph in part (e)(iii). You now convert those into the constants $P$ and $Q$.

Approach

• Set $P =$ gradient.
• Set $Q =$ $y$-intercept.

Step-by-Step Reasoning

If (example) your graph gave:

• gradient $= 2.0\ \text{m}^{-1}$
• intercept $= 0.50$

Then

Key Takeaways

• Comparing equations is a fast way to identify constants.
• Check units to confirm you have matched correctly.

Common Mistakes

• Swapping $P$ and $Q$.
• Giving $Q$ a unit (it is dimensionless here).

Things to Be Careful About

• Ensure the gradient unit is written as $\text{m}^{-1}$ because the $x$-axis was in metres.

Background Concept

Resistivity $\rho$ is a material property that links resistance $R$ to geometry:

In this practical, a rearranged relationship is provided:

where $P$ comes from the graph, $A$ is the cross-sectional area from the diameter, and $R$ is given.

Unit check:

• $P$ in $\text{m}^{-1}$
• $A$ in $\text{m}^2$
• $R$ in $\Omega$

So $PAR$ has units $\text{m}^{-1} \times \text{m}^2 \times \Omega = \Omega\ \text{m}$, as required.

Understanding the Question

You must use your value of $A$ from (a)(ii) and your value of $P$ from (f)(i), together with $R = 10\ \Omega$, to calculate the resistivity $\rho$.

Approach

• Substitute directly into $\rho = PAR$.
• Keep everything in SI units.
• Quote the result to sensible significant figures.

Step-by-Step Reasoning

Using example values:

Substitute:

Multiply the numbers and powers of ten:

Key Takeaways

• Resistivity calculations depend strongly on accurate diameter because $A \propto d^2$.
• Always check units: $\rho$ must be $\Omega\ \text{m}$.

Common Mistakes

• Using $A$ in $\text{mm}^2$ or $\text{cm}^2$ instead of $\text{m}^2$.
• Forgetting $R = 10\ \Omega$ or using the wrong resistor value.
• Using $Q$ instead of $P$ in the resistivity formula.

Things to Be Careful About

• Use your own measured $A$ and graph-derived $P$ (your final $\rho$ is student-dependent).
• Keep appropriate significant figures; don’t claim more precision than your measurements justify.