Question

The resistance per unit length of a wire is ${10}^{-6}\Omega /m$. It is bent in the form of a circle of diameter 2 m. A wire of same material and cross sectional area is connected across its diameter. The total resistance across its diameter $AB$ will be

• A. $\frac{4}{3}\pi \times {10}^{-6}\Omega$
• B. $\frac{2}{3}\pi \times {10}^{-6}\Omega$
• C. $0.88\times {10}^{-6}\Omega$
• D. $14\times {10}^{-6}\Omega$

Answer 1

The correct option is C $0.88\times {10}^{-6}\Omega$

Resistance per unit length,
$a={10}^{-6}\text{For upper arc}{R}_1=a\pi r\text{For lower arc}{R}_2=a\pi r$
For wire $AB=2ra$

$$\begin{array}{cc}\frac{1}{R_{eff}}& =\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\\ & =\frac{1}{a\pi r}+\frac{1}{a\pi r}+\frac{1}{2ar}\end{array}$$
$R_{eff}=0.88\times {10}^{-6}\Omega$