Physics 9702/51 — October/November 2022

Background Concept

The motion is a damped oscillation. When a conductor moves through a magnetic field, changing magnetic flux through the conductor induces eddy currents. These currents experience magnetic forces (Lenz's law) that oppose the motion, producing a damping force.

The question suggests an exponential decay in the amplitude-like displacement after a fixed number of oscillations:

$$s = s_0 e^{-ABKt}$$

This has the general form $s = s_0 e^{-cA}$ where $c = BKt$ is constant if $B$ and $t$ are kept constant. Exponential relationships are tested most easily by taking logarithms to convert them into a straight-line graph.

Understanding the Question

You must design a practical experiment in which:

• you vary the sheet area $A$;
• you measure the displacement $s$ after exactly five complete oscillations (so the time interval is not fixed, but the number of cycles is);
• you keep $B$ (magnetic flux density) and the thickness $t$ constant;
• you use your results to test whether the exponential dependence on $A$ is correct;
• you then use the straight-line graph gradient (together with measured $B$ and $t$) to calculate the constant $K$.

A key practical difficulty is: how do you release from the same initial displacement $s_0$ each time, and how do you measure $s$ consistently after five oscillations when the sheet is moving?

Approach

1. Make $A$ the only deliberately-changed quantity by preparing multiple copper sheets (or cut-outs) with different areas but the same thickness.
2. Use a magnet arrangement with a reasonably uniform field over the region of motion, and measure $B$ using a Hall probe so that $B$ is known (and can be checked to be constant).
3. Use a fixed stop/jig so the initial displacement $s_0$ is reproducible.
4. Measure $s$ after five oscillations using a pointer and scale (or a travelling microscope) with a consistent reference point on the sheet.
5. Linearise:

$$\ln\left(\frac{s}{s_0}\right) = -BKt\,A$$

so a plot of $\ln(s/s_0)$ against $A$ should be a straight line with gradient $-BKt$, allowing $K$ to be found.

Step-by-Step Reasoning

1. Choosing and measuring $A$

   • Use several rectangular copper sheets with different length and width.
   • Measure $L$ and $W$ using a ruler/vernier calipers and calculate $A = LW$.
   • Using different sheets is usually better than cutting one sheet repeatedly, because cutting can change the suspension hole position and create bending/warping.

2. Controlling thickness $t$

   • Measure thickness using a micrometer at several points and average.
   • Select sheets manufactured to the same thickness. (If $t$ varies, it would change the decay constant and spoil the test.)

3. Producing and controlling $B$

   • Place the sheet between the poles of an electromagnet or between two strong permanent magnets.
   • For an electromagnet, keep the current constant using a stable power supply.
   • Measure $B$ at the sheet position with a calibrated Hall probe and ensure the sheet’s motion stays within a region where $B$ is approximately uniform.

4. Setting the same initial displacement $s_0$

   • Put a rigid stop at a fixed horizontal distance from the equilibrium position.
   • Pull the sheet gently until it just touches the stop, then release without pushing. This ensures $s_0$ is the same for every run.

5. Measuring $s$ after five oscillations

   • Attach a thin pointer to the bottom of the sheet (or mark a reference line on the sheet) and place a scale behind it.
   • Start from the moment of release and count five complete cycles (e.g. from one extreme back to the same extreme five times).
   • At the instant the sheet reaches the extreme position after the 5th oscillation, read the maximum displacement $s$ from the scale.
   • Repeat several times and take a mean because reading an extreme position by eye has reaction-time uncertainty.

6. Linearising and finding $K$
   Starting from

$$s = s_0 e^{-ABKt}$$

divide by $s_0$ and take logs:

$$\ln\left(\frac{s}{s_0}\right) = -ABKt$$

With $B$ and $t$ controlled (constant), this is

$$\ln\left(\frac{s}{s_0}\right) = mA \quad \text{where} \quad m = -BKt$$

So:

• compute $\ln(s/s_0)$ for each sheet;
• plot $\ln(s/s_0)$ (y) vs $A$ (x);
• the gradient $m$ gives

$$K = -\frac{m}{Bt}$$

1. Uncertainty treatment (what to do in practice)
   • Estimate uncertainty in $s$ from the scale resolution and the spread in repeated readings.
   • Propagate to an uncertainty in $\ln(s/s_0)$ (for small uncertainties, fractional uncertainty in $s$ approximates the absolute uncertainty in $\ln s$).
   • Include error bars on $\ln(s/s_0)$ if required and determine uncertainty in gradient using a best-fit and worst-acceptable line.

Key Takeaways

• Exponential relationships are tested by taking logarithms to obtain a straight-line graph.
• Good planning means varying one quantity (here $A$) while keeping others constant ($B$, $t$, $s_0$).
• The gradient of the linear graph is the route to extracting the constant $K$.
• Practical marks often depend on describing how measurements are made repeatably (e.g. a stop for $s_0$, repeats for $s$).

Common Mistakes

• Plotting $s$ against $A$ directly and expecting a straight line (it should be exponential, not linear).
• Forgetting to divide by $s_0$ before taking logs, or not keeping $s_0$ constant.
• Not measuring $B$ (you need $B$ to calculate $K$) or allowing $B$ to vary by moving outside the uniform field.
• Allowing thickness $t$ to change between sheets (this changes the decay factor).
• Measuring $s$ at a random time after five oscillations rather than at the extreme displacement after the 5th cycle.

Things to Be Careful About

• Ensure the oscillation is perpendicular to $\vec{B}$ as stated; changing the angle changes the effective damping.
• Keep the suspension geometry the same: changing hole position changes the oscillation mode and can affect the damping.
• Use enough different areas $A$ (at least 6 values over a wide range) to make a convincing graph.
• If damping is strong, after five oscillations $s$ may be very small and hard to read; choose $A$ values so $s$ remains measurable.
• Safety with magnets (pinch hazard) and with electromagnets (heating/current limits), and sharp sheet edges.