Physics 9702 · AS & A Level · Characteristics of alternating currents

Characteristics of alternating currents — practice question

An electron in a metal rod undergoes random motion around a mean position. If an alternating voltage is connected across the ends of the rod, the mean position may be treated as moving with simple harmonic motion along the rod’s axis. Fig. 4.1 shows how the displacement $x$ of the mean position from a fixed point on the rod’s axis changes with time $t$.
(a(i))[1]

Find the amplitude of the oscillations.

(a(ii))[1]

Find the angular frequency for the oscillations.

(a(iii))[2]

Using your results from (a)(i) and (a)(ii), show that the electron’s maximum drift speed $v_0$ is $1.1 \times 10^{-7}\,\text{m s}^{-1}$.

(b(i))[2]

The rod has cross-sectional area $4.3\,\text{cm}^2$ and a conduction-electron number density (charge carriers) of $8.5 \times 10^{28}\,\text{m}^{-3}$. It may be assumed that every conduction electron in the rod oscillates in phase with, and with the same amplitude as, the motion shown in Fig. 4.1. Use the information in a(iii) to find the size $I_0$ of the maximum current in the rod.

(b(ii))[2]

On Fig. 4.2, draw the way the current $I$ in the rod changes with time $t$ for $t = 0$ to $t = 0.40\,\mu\text{s}$.

(b(iii))[2]

Use your answers in a(ii) and b(i) to derive an expression for $I$ in terms of $t$, with $I$ in A and $t$ in s.

(b(iv))[1]

Find the root-mean-square (r.m.s.) current in the rod.

Worked solution & mark scheme

This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: Amplitude equals half the peak-to-peak displacement

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