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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 $\Delta T$ 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 $2 L$ . The temperature of the wire is raised by the same amount $\Delta T$ in the same time $t$ . the value of $N$ is
Option: 1
4

Option: 2
6

Option: 3
8

Option: 4
9

Answers (1)

best_answer

In the first case, $i=\frac{3 E}{R}$

where $E$ is the emf of each cell and $R(\propto L)$ is the resistance of the wire.

Also $i^2 R t=m . s . \Delta T$ $(1)$

where $m$ is the mass of $L$ length of wire and $S$ is the specific heat of the material of the wire.
In the second case, $i^{\prime}=\frac{N E}{R^{\prime}} \quad$ where $R^{\prime} \propto 2 L$

and $\quad i^{\prime 2} R^{\prime} t=m^{\prime} s \Delta T$ $(2)$

Dividing equation (2) by equation (1)

$\left(\frac{i^{\prime}}{i}\right)^2 \cdot\left(\frac{R^{\prime}}{R}\right)=\frac{m^{\prime}}{m} \quad \Rightarrow \quad \frac{N^2}{9} \cdot \frac{R}{R^{\prime}}=\frac{m^{\prime}}{m} \quad \Rightarrow \quad \frac{N^2}{9} \cdot \frac{1}{2}=2$

$\begin{array}{ll} \Rightarrow & N=6 \\ \therefore & \text { (b) } \end{array}$

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