Physics 9702/53 — May/June 2021

Background Concept

When a d.c. supply is suddenly connected to a series combination of a coil (inductor) and a resistor, the current does not jump instantly to its final value. The inductor produces an induced emf that opposes changes in current, so the current rises over a short time.

In a simple series $RL$ circuit, the rise of current after a step in voltage is governed by an exponential approach to a steady value, with a characteristic timescale called the time constant. In general, the larger the resistance, the faster the current reaches its steady value.

The question suggests a specific relationship:

$$t = \frac{KN^2A}{LR}$$

Here $N$, $A$, and $L$ are properties of the coil, and $K$ is a constant. If the coil is unchanged, then $N$, $A$, and $L$ are constant, so the equation predicts that $t \propto 1/R$.

Understanding the Question

You are asked to design an experiment (Paper 5 planning style) to test whether $t$ depends on $R$ in the way suggested, when a switch connects a high-voltage d.c. supply to a series coil + resistor.

So you must:

• vary $R$ (independent variable),
• measure the time $t$ for the current to reach its maximum/steady value (dependent variable),
• keep coil geometry ($N$, $A$, $L$) and other conditions constant,
• decide how to measure a current that changes over a few milliseconds,
• analyse results using a graph to check the relationship and extract $K$.

Because the change occurs in a few ms, a meter is too slow; an oscilloscope or data logger is appropriate.

Approach

1. Measure current indirectly: place a small known shunt resistor in series and measure the voltage across it; since $V = Ir$, the measured voltage is proportional to current.
2. Use single-shot recording: trigger the oscilloscope at switch closure to capture the transient.
3. Repeat for multiple $R$ values and average $t$ to reduce random error.
4. Linearise the given relationship: since $t \propto 1/R$, plot $t$ against $1/R$.
5. Use gradient to find $K$ because the gradient contains $K$ multiplied/divided by known coil parameters.

Step-by-Step Reasoning

1) Choosing variables

• You can control $R$ easily by swapping resistors.
• The coil parameters $N$, $A$, $L$ must remain fixed: therefore use the same coil for all runs.
• The time $t$ is what you are trying to see change, so it is the dependent variable.

2) Capturing a millisecond transient

A standard ammeter typically cannot resolve the current change over a few milliseconds. Instead:

• Put a small resistor $r_s$ in series.
• Measure $V_s(t)$ across $r_s$ with an oscilloscope.
• Convert to current using:

$$I(t) = \frac{V_s(t)}{r_s}.$$

This method has two big advantages:

• the oscilloscope has ms (or better) time resolution,
• measuring a voltage is easy and fast.

3) Defining and measuring $t$

The question states “time for the current in the resistor to reach a maximum value”. In reality, the current approaches a steady value; it does not usually become perfectly constant instantly.

So you must adopt an operational definition of “reaches maximum” and keep it consistent, for example:

• measure the time from switch closure to the point where the trace first becomes flat within the oscilloscope resolution, or
• measure time to reach a stated fraction (e.g. $0.99I_{\max}$).

Either is acceptable in a plan as long as you state what you will do and apply it consistently.

4) Controlling variables (what could accidentally change)

• Supply voltage: keep the power supply setting fixed.
• Coil temperature: repeated large currents can heat the coil and change its resistance and inductive behaviour. Minimise by using short switching times and cooling intervals, and/or choose $R$ values that limit current.
• Extra resistances (leads, shunt, coil resistance): keep the same shunt and wiring each time. If necessary, choose $R$ values much larger than coil resistance so the tested $R$ dominates.

5) Analysis to test the relationship

Start with the given model:

$$t = \frac{KN^2A}{LR}$$

Treat $N^2A/L$ as constant (because the coil is unchanged). Then:

$$t = \left(\frac{KN^2A}{L}\right)\left(\frac{1}{R}\right)$$

This is of the form $y = mx$ with $y=t$ and $x=1/R$.

So:

• make a table of $R$, $1/R$, and measured mean $t$,
• plot $t$ vs $1/R$,
• check for a straight line passing through (0,0) within uncertainties.

6) Finding $K$ from the gradient

If the graph is straight, its gradient $m$ is:

$$m = \frac{KN^2A}{L}.$$

So:

$$K = \frac{mL}{N^2A}.$$

You then need values for $N$, $A$ and $L$:

• $L$ measured with a ruler (coil length),
• $A$ from $A=\pi r^2$ using radius $r$ measured with calipers,
• $N$ from specification or counting turns.

7) Uncertainties (what you would write to gain evaluation marks)

• Repeat traces for each $R$ to estimate random uncertainty in $t$.
• Read $t$ using oscilloscope cursors; uncertainty is related to time-base resolution and how clearly the plateau is defined.
• Measure $R$ with a multimeter; record its uncertainty from meter resolution.
• Use error bars on $t$ (and possibly on $1/R$), draw a best-fit line and a worst acceptable line to estimate uncertainty in gradient, then propagate to $K$.

8) Safety

High-voltage d.c. supplies can be dangerous. Sensible precautions:

• power off before changing resistors or wiring,
• use insulated leads and a properly rated switch,
• limit current to prevent overheating of resistors/coil (check power ratings),
• keep switch closed only briefly to avoid excessive heating.

Key Takeaways

• In a planning question, marks come from: clear variables, workable measurement method, control of variables, correct linearising graph, and extracting constants from gradient/intercept.
• For fast transients, use an oscilloscope and a shunt resistor to measure current indirectly.
• To test $t \propto 1/R$, plot $t$ against $1/R$ and use the gradient to find $K$.

Common Mistakes

• Trying to measure ms-scale changes with an ammeter (too slow).
• Not stating how $t$ is defined/measured from the trace (needs a clear criterion).
• Plotting $t$ against $R$ instead of $1/R$ (won’t be linear for this model).
• Forgetting to measure the actual resistor values (unmarked resistors cannot be assumed correct).
• Failing to mention control of coil temperature/current (heating changes behaviour).

Things to Be Careful About

• Keep the definition of “maximum current reached” consistent across all runs.
• Ensure the shunt resistor is small and of suitable power rating; otherwise it alters the circuit significantly or overheats.
• Choose a suitable range of $R$ so that $t$ values spread out enough to see a trend (not all clustered within timing resolution).
• When extracting gradient, use a large triangle and correct units: $t$ in $\text{s}$ and $1/R$ in $\Omega^{-1}$.
• When calculating $A=\pi r^2$, use $r$ in metres if you want SI consistency, and quote $K$ with appropriate significant figures based on measurement precision.