Two wires A and B of equal lengths are connected in left and right gap of a metre bridge, null point is…

Two wires A and B of equal lengths are connected in left and right gap of a metre bridge, null point is obtained at $40 \mathrm{~cm}$ from left end. Diameters of the wires $\mathrm{A}$ and $\mathrm{B}$ are in the ratio $3: 1$. The ratio of specific resistance of $A$ to that of $B$ is
  1. 8:1
  2. 6:1
  3. 4:1
  4. 3:1

Solution

$\begin{aligned} \ell_{\mathrm{A}} &=\ell_{\mathrm{B}} \quad \frac{r_{\mathrm{A}}}{r_{\mathrm{B}}}=\frac{3}{1} \\ \frac{R_{\mathrm{A}}}{R_{\mathrm{B}}} &=\frac{4}{6}=\frac{2}{3}=\frac{\rho_{\mathrm{A}} \ell_{\mathrm{B}}}{A_{\mathrm{A}}} \times \frac{A_{\mathrm{B}}}{\rho_{\mathrm{B}} \ell_{\mathrm{B}}} \\ &=\frac{\rho_{\mathrm{A}}}{\rho_{\mathrm{B}}} \times \frac{\ell_{\mathrm{A}}}{\ell_{\mathrm{B}}} \times \frac{r_{\mathrm{B}}^{2}}{r_{\mathrm{A}}^{2}} \\ \therefore \quad \frac{\rho_{\mathrm{A}}}{\rho_{\mathrm{B}}} &=\frac{2}{3} \times 9=\frac{6}{1} \end{aligned}$

Asked in: MHT CET 2020 (16 Oct Shift 1)

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