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Current Electricity

JEE Main level, single correct, medium

97 of 380

Question 97 of 380

A wire of uniform resistance $\lambda \Omega / \mathrm{m}$ is bent into a circle of radius $r$ and another piece of wire with length $2 r$ is connected between points $A$ and $B(A O B)$ as shown in figure. The equivalent resistance between points $A$ and $B$ is $\_\_\_\_$ $\Omega$ .

Show answer and solution

Answer: (B) $\frac{6 \pi \lambda r}{3 \pi+16}$

The wire bent into a circle of radius

r

has resistance

\lambda \cdot \pi r

for the semicircular arcs, and the straight wire of length

2r

has resistance

\lambda \cdot 2r

. The three paths between A and B (two semicircular arcs of the circle and one straight wire through center) are in parallel.

\frac{1}{R_{AB}} = \frac{2}{\lambda \pi r} + \frac{1}{\lambda \cdot 2r} + \frac{2}{\lambda \cdot 3\pi r}

= \frac{1}{\lambda r} \left[\frac{2}{\pi} + \frac{1}{2} + \frac{2}{3\pi}\right]

= \frac{1}{\lambda r} \left(\frac{12 + 3\pi + 4}{6\pi}\right) = \frac{1}{\lambda r} \cdot \frac{16 + 3\pi}{6\pi}

R_{AB} = \lambda r \left(\frac{6\pi}{16 + 3\pi}\right) = \frac{6\pi \lambda r}{3\pi + 16}

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JEE Main 2025 Apr 2 Shift 1 question paper

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