A wire of uniform resistance is bent into a circle of radius and another piece of wire with length is connected between points A and B (AOB) as shown in the figure. [2026]

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Solution

Q.
A wire of uniform resistance $λ Ω / m$ is bent into a circle of radius $r$ and another piece of wire with length $2 r$ is connected between points A and B (AOB) as shown in the figure. The equivalent resistance between points A and B is _______ $Ω$ . [2026]

1 $2 π λ r$

2 $\frac{3 π λ r}{8}$

3 $\frac{6 π λ r}{3 π + 16}$

4 $(π + 1) 2 r λ$

Ans.
(3)

$\frac{1}{R_{A B}} = \frac{2}{λ π r} + \frac{1}{λ \cdot 2 r} + \frac{2}{λ \cdot 3 π r}$

$= \frac{1}{λ r} [\frac{2}{π} + \frac{1}{2} + \frac{2}{3 π}]$

$= \frac{1}{λ r} (\frac{12 + 3 π + 4}{6 π}) = \frac{1}{λ r} (\frac{16 + 3 π}{6 π})$

$R_{A B} = λ r (\frac{6 π}{16 + 3 π})$

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