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Electric current - Current Electricity

Question

An electric current flows through a non-uniform metal rod AB of length L with end A at x = 0 and end B at x = L. The current density J in the rod varies with x as shown in Fig.
Which of the following statements is correct?

Moderate

A

The rod has a smaller radius at A than at B.
B

The rod has a higher radius at A than at B.
C

The current at A greater than that at B.
D

The potential gradient along the rod is constant.

Solution

The current I in a rod is independent of the area of cross-section. So the current at A = current at B and, therefore, choice (c) is wrong. The current density J is given by
$J = \frac{I}{a} = \frac{I}{{πr}^{2}}$
where a is the cross-sectional area. Since I is the same along the rod, J will be higher where a is smaller. From the given graph, it follow that J is greater at x = 0 (end A) than at x = L (end B). Hence, the radius r is smaller at A than at B. So choice (1) is correct. Since the cross-sectional area at end A is smaller that at B from Ohm's law, V=IR, the potential gradient $\frac{dV}{dx}$ is not constant (because R is not constant) along the rod.
So the only correct choice is (1).

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