Physics 9702/51 — October/November 2023

Background Concept

A changing magnetic flux through a coil induces an e.m.f. across the coil terminals (Faraday’s law). With two coils placed close together on the same axis, an alternating current in coil C produces a changing magnetic field and hence a changing flux linking coil D. The induced e.m.f. across D depends on how fast the flux changes and how strongly the flux from C links D.

The proposed model is

$$E = \frac{pf^qV}{R}$$

where:

• $E$ is the induced e.m.f. across coil D,
• $f$ is the frequency of the changing flux (set by the driving a.c. frequency in coil C),
• $V$ is the applied potential difference across the series combination (resistor + coil C),
• $R$ is the series resistance,
• $p$ and $q$ are constants to be determined experimentally.

To test such a relationship you must:

1. vary $f$ systematically,
2. measure $E$ reliably,
3. keep other factors constant (especially geometry and the drive amplitude, represented here by $V$ and fixed $R$),
4. use a graph that turns the relationship into a straight line so $p$ and $q$ can be extracted from gradient/intercept.

Understanding the Question

You are asked to plan an experiment (not do a calculation). The key tasks are:

• build a setup where coil C is driven at variable frequency $f$ with a known series resistor $R$,
• measure the induced e.m.f. $E$ across coil D,
• measure $V$ across the supply terminals of (coil C + resistor),
• explain how your data will be analysed to find the constants $p$ and $q$.

The statement “axes on a straight line” is an instruction to keep the coils coaxial; coupling changes strongly if the coils are moved or rotated, so the geometry must be fixed.

Approach

1. Choose equipment that can provide and measure sinusoidal a.c. at variable frequency: a signal generator plus oscilloscope (ideal), or a.c. voltmeters.
2. Make $f$ the only quantity you deliberately change.
3. Keep $V$ constant for every run (because $E$ is predicted to be proportional to $V$ as well as dependent on $f$).
4. For each $f$, measure $E$ and $V$, then compute $ER/V$.
5. Linearise the power law by taking logs so you can plot a straight-line graph and read off $q$ and $p$.

Step-by-Step Reasoning

1) Setting up the apparatus

You need coil C driven by an a.c. source with a fixed resistor $R$ in series (as stated). Coil D should be connected only to a measuring instrument (not a load), otherwise current in D would change the induced e.m.f. you are trying to measure.

Using an oscilloscope is particularly effective:

• It has a high input resistance, so it does not significantly load coil D.
• It can measure frequency directly from the time period, and can measure voltages as peak-to-peak (convert to peak or r.m.s. consistently).

2) Choosing variables and controls

• Independent: $f$ (set by signal generator).
• Dependent: $E$ (measured across D).
• Controls:
  • Coil separation: fix with clamps and measure once with a ruler; do not move coils during the experiment.
  • Alignment: keep coils coaxial and facing each other.
  • $R$: use the same resistor throughout; avoid heating that could change its resistance.
  • $V$: keep constant by adjusting the generator amplitude each time you change $f$.
  • Same waveform (sine) and same coil cores (air core/iron core) throughout.

These controls matter because mutual inductive coupling depends on geometry and magnetic properties. If the coils move or a core is introduced/removed, $E$ can change even if $f$ is unchanged.

3) Collecting data

For each frequency setting:

1. Set $f$ and confirm it (generator readout or oscilloscope measurement).
2. Adjust amplitude so that $V$ across (coil C + resistor) is the same as for every other reading.
3. Record $E$ across coil D.
4. Repeat readings (or take several cycles on the oscilloscope) and average to reduce random uncertainty.

Choose a wide frequency range and at least 6–10 readings so the log-log graph has enough points to show a clear straight-line trend.

4) Analysis to obtain $p$ and $q$

Start from the given relationship:

$$E = \frac{pf^qV}{R}$$

Rearrange to isolate the part depending on $f$:

$$\frac{ER}{V} = pf^q$$

This is a power law in $f$. Taking logs gives a linear form:

$$\ln\left(\frac{ER}{V}\right) = \ln p + q\ln f$$

So if you plot:

• $y = \ln(ER/V)$
• $x = \ln f$

you expect a straight line.

• Gradient $= q$.
• Intercept $= \ln p$.
• Hence $p = e^{\text{intercept}}$.

(You may use $\lg$ instead of $\ln$; then intercept is $\lg p$ and $p = 10^{\text{intercept}}$.)

5) Uncertainties (how to treat them well)

• Measure $V$ and $E$ with the same instrument type and consistent definition (both r.m.s. or both peak).
• Estimate uncertainties from instrument resolution (e.g. oscilloscope scale reading) and repeat measurements.
• Propagate to $ER/V$ approximately using percentage uncertainties: $$\frac{\Delta(ER/V)}{ER/V} \approx \frac{\Delta E}{E} + \frac{\Delta V}{V}$$
• Add error bars on the $\ln(ER/V)$ values if required; a common approximation is $$\Delta(\ln Y) \approx \frac{\Delta Y}{Y}$$

Key Takeaways

• In a planning question, marks come from: a workable setup, correct variable control, valid measurements, and a clear analysis method.
• To determine constants in a power law, use a log-log plot: gradient gives the exponent, intercept gives the multiplicative constant.
• Keeping $V$ constant is essential here because $E$ depends on both $f$ and $V$.

Common Mistakes

• Varying $f$ without keeping $V$ constant, which mixes two effects and prevents a fair test of the $f^q$ dependence.
• Loading coil D (e.g. connecting a low-resistance meter), causing current in D and altering the induced e.m.f.
• Changing the coil separation/alignment between readings, giving inconsistent coupling.
• Plotting $E$ vs $f$ on ordinary axes and trying to guess $p$ and $q$ without linearising.
• Mixing peak, peak-to-peak, and r.m.s. values between $E$ and $V$.

Things to Be Careful About

• Ensure $R$ is the known fixed resistor (not the coil resistance). If $R$ heats up, its value can change; keep currents small.
• State clearly what you mean by “$V$”: it must be the p.d. across the series combination (coil C + resistor) as defined in the question.
• Use the same coil orientation throughout; a small rotation can significantly reduce induced e.m.f.
• Use enough data points and a large enough frequency range so that the log-log graph can reliably give a gradient.
• If using an oscilloscope, set coupling and measurement mode correctly (a.c. coupling if needed; consistent voltage measurement method).