Physics 9702/52 — October/November 2012

Background Concept

When a magnet moves relative to a coil, the magnetic flux linkage $N\Phi$ through the coil changes. Faraday's law gives the induced e.m.f.

$$E = -\frac{\mathrm{d}(N\Phi)}{\mathrm{d}t}$$

The minus sign is Lenz's law (direction), but for this experiment we measure the magnitude (peak value) so we can focus on $|E|$.

If the magnet leaves the coil faster, the flux linkage changes more rapidly, so the peak induced e.m.f. is larger. The suggestion $E \propto v$ means that, for the same coil and magnet, doubling the exit speed should double the peak e.m.f.

Understanding the Question

You must design an experiment to test whether the maximum induced e.m.f. $E$ (specifically when the magnet is leaving the coil) is directly proportional to the speed $v$ at that instant.

So you need:

• A way to measure the peak induced e.m.f. reliably.
• A way to measure the magnet's speed at/just after it leaves the coil.
• A method to vary $v$ over a sensible range.
• Control variables so that changes in $E$ are due to changes in $v$ only.
• A graph-based analysis that clearly tests direct proportionality.

Approach

A practical way is:

1. Vary $v$ by changing the drop height (gravitational potential energy changes the speed at the bottom).
2. Measure $E$ with an oscilloscope/data logger because the induced voltage is a short pulse.
3. Measure $v$ using two light gates near the exit of the coil: speed is then found from $v = d/\Delta t$.
4. Plot $E$ vs $v$. Direct proportionality is shown by a straight line through the origin; the gradient is the constant of proportionality.

Step-by-Step Reasoning

1. Set up the coil and voltage measurement

   • Connect the coil terminals to an oscilloscope or data logger voltage sensor.
   • Set a suitable time base so the voltage pulse is visible, and use a peak measurement/peak-hold if available.
   • This ensures you capture the maximum e.m.f. as the magnet leaves.

2. Measure the speed at the exit

   • Place two light gates a small distance $d$ apart just below the coil exit.
   • When the magnet passes gate 1 then gate 2, the timer gives $\Delta t$.
   • Calculate speed:

$$v = \frac{d}{\Delta t}$$

• Because the gates are close to the coil exit, this $v$ is a good approximation to the speed as it leaves the coil.

1. Vary $v$

   • Release the magnet from rest at different heights above the coil. Higher release height gives larger speed at the coil exit.
   • Use a consistent release method (e.g. a clamp or simple release mechanism) so the magnet is not given an extra push.

2. Take paired readings $(v, E)$

   • For each height, record the peak e.m.f. $E$ from the trace and compute $v$ from the light gates.
   • Repeat at least three times to reduce random error and obtain a mean value.

3. Control variables
   The relationship should only depend on the speed if other factors are fixed:

   • Same magnet and same orientation (reversing the magnet reverses the polarity of the pulse).
   • Same coil (turns $N$ and area) and fixed geometry.
   • Magnet falls through the centre each time (use a guide tube aligned with the coil).
   • Keep gates fixed relative to coil (you must always measure $v$ at the same position).

4. Analyse the data

   • Create a results table with columns such as $h$, $d$, $\Delta t$, $v$, and $E$.
   • Plot $E$ (y-axis) against $v$ (x-axis).
   • If $E \propto v$, then:
     • The plot should be a straight line.
     • It should pass through the origin within uncertainties.
     • The gradient gives $k$ in $E = kv$.
   • Include uncertainty estimates (e.g. $\delta d$ from ruler resolution, $\delta \Delta t$ from timer resolution, $\delta E$ from trace reading) and show error bars; then judge whether the origin is consistent with the best-fit line.

Key Takeaways

• Use Faraday’s law idea: faster change of flux linkage gives larger induced e.m.f.
• To test proportionality, vary one quantity ($v$), measure the other ($E$), and plot a straight-line graph.
• Good planning is mostly about valid measurement methods and control of variables.
• Repeats and uncertainty/error bars strengthen the conclusion.

Common Mistakes

• Measuring speed too far away from the coil (then $v$ is not the “leaving the coil” speed).
• Using a voltmeter instead of an oscilloscope/data logger (a voltmeter may miss the peak because the pulse is brief).
• Forgetting to control magnet orientation or ensuring it passes centrally (changes the flux linkage pattern).
• Saying “straight line” without stating “through the origin” when claiming direct proportionality.
• Not stating how $v$ is calculated (missing $v = d/\Delta t$).

Things to Be Careful About

• Ensure $d$ is measured accurately and gates are aligned so the magnet reliably triggers them.
• Choose a range of drop heights that gives a clear spread in $v$ (not all speeds nearly the same).
• Keep the coil and gates securely clamped; vibrations can change alignment.
• Peak voltage readings from a trace should be taken consistently (same zero level, same scale); quote $E$ to a sensible resolution.
• Safety: a strong magnet can pinch fingers and can damage electronics/data storage; prevent it from hitting people or equipment when it falls.