Question

Two wires ‘A’ and ‘B’ of the same material have their lengths in the ratio 1 : 2 and radii in the ratio 2 : 1. The two wires are connected in parallel across a battery. The ratio of the heat produced in ‘A’ to the heat produced in ‘B’ for the same time is

• A. $1:2$
• B. $2:1$
• C. $1:8$
• D. $8:1$

Answer 1

Answer: (D)

$8:1$

Answer 2

$R_{1} = \rho\frac{l_{1}}{A_{1}}$ and $R_{2} = \rho\frac{l_{2}}{A_{2}}$⇒ $\frac{R_{1}}{R_{2}} = \frac{l_{1}}{l_{2}}.\frac{A_{2}}{A_{1}} = \frac{l_{1}}{l_{2}}\left( \frac{r_{2}}{r_{1}} \right)^{2}$

Given $\frac{l_{1}}{l_{2}} = \frac{1}{2}$ and $\frac{r_{1}}{r_{2}} = \frac{2}{1}$ or $\frac{r_{2}}{r_{1}} = \frac{1}{2}$⇒ $\frac{R_{1}}{R_{2}} = \frac{1}{8}$

$\therefore$ Ratio of heats$\frac{H_{1}}{H_{2}} = \frac{V^{2}/R_{1}}{V^{2}/R_{2}} = \frac{R_{2}}{R_{1}} = \frac{8}{1}$