Q81 / 120 · VITEEE 2019 · Physics

The electric resistance of a certain wire of iron is $R$. If its length and radius are both doubled, then

  • A
    the resistance and the specific resistance, will both remain unchanged
  • B
    the resistance will be doubled and the specific resistance will be halved
  • Correct answer
    C
    the resistance will be halved and the specific resistance will remain unchanged
  • D
    the resistance will be halved and the specific resistance will be doubled

Solution

$\mathrm{R}_{1}=\frac{\rho \ell_{1}}{\mathrm{~A}_{1}}$, now $\ell_{2}=2 \ell_{1}$ and $\mathrm{r}_{2}=2 \mathrm{r}_{1}$ $$ \begin{array}{l} \mathrm{A}_{1}=\pi \mathrm{r}_{1}^{2} \\ \mathrm{~A}_{2}=\pi\left(\mathrm{r}_{2}\right)^{2}=\pi\left(2 \mathrm{r}_{1}\right)^{2}=4 \pi \mathrm{r}_{1}^{2}=4 \mathrm{~A}_{1} \\ \mathrm{R}_{2}=\frac{\rho \ell_{2}}{\mathrm{~A}_{2}} \\ \therefore \quad R_{2}=\frac{\rho\left(2 \ell_{1}\right)}{4 A_{1}}=\frac{\rho \ell_{1}}{2 A_{1}}=\frac{R_{1}}{2} \end{array} $$ $\therefore$ Resistance is halved, but specific resistance remains the same.

Hence, the correct answer is option: 'C'.

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