Q. 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
Explanation
(d) $R_{1} = \rho\frac{l_{1}}{A_{1}}$ and $R_{2} = \rho\frac{l_{2}}{A_{2}} \Rightarrow \frac{R_{1}}{R_{2}} = \frac{l_{1}}{l_{2}}.\frac{A_{2}}{A_{1}} = \frac{l_{1}}{l_{2}}\left( \frac{r_{2}}{r_{1}} \right)^{2}$
Given$\ \frac{l_{1}}{l_{2}} = \frac{1}{2}$ and $\frac{r_{1}}{r_{2}} = \frac{2}{1}\ $or $\frac{r_{2}}{r_{1}} = \frac{1}{2} \Rightarrow \frac{R_{1}}{R_{2}} = \frac{1}{8}$
$\therefore$ Ratio of heats $\frac{H_{1}}{H_{2}} = \frac{V^{2}/R_{1}}{V^{2}/R_{2}} = \frac{R_{2}}{R_{1}} = \frac{8}{1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.