2020 A/L Physics Past Paper · Question 29 (English Medium)

Answer: (5) A straight line through the origin

What the question asks

We must find: the graph showing how the e.m.f. $E$ induced in a conducting loop of fixed area varies with the rate of change $R = \dfrac{dB}{dt}$ of the magnetic flux density.

Relevant theory

The magnetic flux across a certain space in a magnetic field is a measurement of the number of lines of force across that space.

Electromagnetic induction can be briefly introduced as a phenomenon where an electromotive force is produced between the ends of a conducting coil or any other conductor when the magnetic flux intersecting is changing.

The magnitude of the electromotive force induced in a conducting coil or a circuit at any instant is directly proportional to the rate of change of the magnetic flux or the flux linkage number intersecting it at that instant.

The direction of the induced electromotive force is such that its current flows in a direction opposing the change of flux causing it.

Required equations

$\phi = BA$

$E = \frac{d\phi}{dt} = A\,\frac{dB}{dt}$

$E = AR$

Reasoning and calculation

Axes: the horizontal axis is the rate of change of the magnetic flux density $R$ ($\text{T s}^{-1}$), the vertical axis is the induced e.m.f. $E$ (V). These are not $B$ and time — they are a rate and its effect.

Starting point: if $R = 0$, however strong the field, it is not changing; since the flux is not changing, $E = 0$. So the graph starts from the origin.

Sections: the loop's area $A$ and its orientation do not change; across the whole range of $R$ the same relation $E = AR$ holds — there is no point where the behaviour changes, just one section.

Behaviour in that section: $E$ is directly proportional to $R$ — if $R$ doubles, $E$ doubles; if $R$ triples, $E$ triples.

Boundary values: at $R = 0$: $E = 0$; e.g. $A = 0.1\ \text{m}^2$, $R = 0.3\ \text{T s}^{-1}$: $E = 0.03\ \text{V}$. At larger $R$ the relation still does not change — no levelling off, no bending.

Slope, curvature, sign: slope $\dfrac{dE}{dR} = A$ — constant, positive; no curvature; for $R > 0$, $E > 0$ (magnitude; Lenz's law gives the direction — the current in the loop opposes the growing field).

Reading the options

Comparing with the options: a straight line starting at the origin, of constant slope — only option (5) satisfies both conditions ($E = 0$ at $R = 0$; $E \propto R$).

Why the other options fail

$E = 0$ for every $R$ — this says induction does not occur in a changing field.

As $R$ increases, the rise in $E$ slows down — a levelling-off curve.

Even at $R = 0$, $E \neq 0$ — an e.m.f. with an unchanging flux.

An upward-bending curve, like $E \propto R^2$.

Conclusion

$E = AR$

$\phi = BA$, with $A$ constant: $E = A\dfrac{dB}{dt} = AR$ — $E \propto R$, a straight line through the origin — the answer is (5).

Assuming the induced e.m.f. depends on the strength of the magnetic field ($B$) itself.

The e.m.f. depends only on the rate at which the flux changes: $E = A\dfrac{dB}{dt}$.