A wire of length L and 3 identical cells of negligible internal resistances are connected in series. Due to the current, the temperature of the wire is raised by in a time . A number N of similar cells is now connected in series with a wire of the same m

Question

A wire of length L and 3 identical cells of negligible internal resistances are connected in series. Due to the current, the temperature of the wire is raised by $Δ T$ in a time $t$ . A number N of similar cells is now connected in series with a wire of the same material and cross-section but of length 2L. The temperature of the wire is raised by the same amount $Δ T$ in the same time $t$ . The value of N is [2001]

Smart Achievers · Accepted Answer

(2)

Let $R$ be the resistance of the wire.

Energy released in $t$ second, $H = \frac{{(3 V)}^{2}}{R} \times t$ $(∵ H = \frac{V^{2}}{R} t)$

Also, $H = m c Δ T = \frac{9 V^{2}}{R} \times t$ $(∵ H = m c Δ T)$ $\dots (i)$

Let $R'$ be the resistance of the wire of double length $' 2 l'$ .

$∴$ Now energy released in $t$ -seconds

$H' = \frac{{(N V)}^{2}}{2 R} \times t$ $(∵ R \propto l)$

Also $H' = m' c Δ T = (2 m) c Δ T$

$2 m c Δ T = \frac{N^{2} V^{2}}{2 R} \times t$ $\dots (i i)$

Dividing eq. (i) by (ii)

$\frac{m c Δ T}{2 m c Δ T} = \frac{\frac{9 V^{2}}{R} \times t}{\frac{N^{2} V^{2}}{2 R} \times t}$ $or,$ $\frac{1}{2} = \frac{9 \times 2}{N^{2}}$

$or,$ $N^{2} = 18 \times 2$ $∴$ $N = 6$

Solution

1 4

2 6

3 8

4 9

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