The diameter of a wire XY varies linearly with distance along the wire as shown in Fig. 7.1.

Question 14

(a) The current I in a metal wire is given by the expression

I = Anve.

State what is meant by the symbols A and n. [2]

(b) The diameter of a wire XY varies linearly with distance along the wire as shown in Fig. 7.1.

Fig. 7.1

There is a current I in the wire. At end X of the wire, the diameter is d and the average drift speed of the free electrons is vx. At end Y of the wire, the diameter is d / 2.

On Fig. 7.2, sketch a graph to show the variation of the average drift speed with position

along the wire between X and Y.

Fig. 7.2

[2]

[Total: 4]

Reference: Past Exam Paper – November 2018 Paper 22 Q7

Solution:

(a)

A: (cross-sectional) area (of wire)

n: number of free electrons per unit volume or number density of free electrons

(b)

line drawn between (X, vx) and (Y, 4vx)

line has increasing gradient

{I = Anve

Cross-sectional area A = πd² / 2

I = (πd² / 2) nve

As we move from X to Y, the cross-sectional area (and hence, the diameter) decreases linearly. So, we need to understand the relationship between drift velocity v_x and the diameter.

I = (πd² / 2) nve

v = 2I / πd²e

So, v ∝ 1 / d²

The drift velocity v is inversely proportional to the square of the diameter. If the diameter decreases, the drift velocity increases.

Initially at X, drift velocity = v_x. On the graph, we have the point (X, v_x)

If the diameter is halved, the the drift velocity is would increase by a factor of 4. On the graph, we have the point (Y, 4v_x).

Since v ∝ 1 / d², the graph of drift velocity against position (diameter) would resemble the reflection of the graph of y = 1/x² about the x-axis (since here, from X to Y, the diameter is decreasing).}