First slide

Electrical power

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

Moderate

A

1 : 2
B

2 : 1
C

1 : 8
D

8 : 1

Solution

$\begin{array}{l} R_{1} = ρ \frac{l_{1}}{A_{1}} and R_{2} = ρ \frac{l_{2}}{A_{2}} \\ \Rightarrow \frac{R_{1}}{R_{2}} = \frac{l_{1}}{l_{2}} . \frac{A_{2}}{A_{1}} = \frac{l_{1}}{l_{2}} {(\frac{r_{2}}{r_{1}})}^{2} \\ Given \frac{l_{1}}{l_{2}} = \frac{1}{2} and \frac{r_{1}}{r_{2}} = \frac{2}{1} or \frac{r_{2}}{r_{1}} = \frac{1}{2} \Rightarrow \frac{R_{1}}{R_{2}} = \frac{1}{8} \\ ∴ Ratio of heats, \frac{H_{1}}{H_{2}} = \frac{V^{2} / R_{1}}{V^{2} / R_{2}} = \frac{R_{2}}{R_{1}} = \frac{8}{1} \end{array}$

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