Question

A wire of length L and three identical cells of negligible internal resistance are connected in series. Due to the current, the temperature of the wire is raised by $\Delta T$ in time t. A number N of similar cells is now connected in series with a wire of same material and cross section but of length $2L$. The temperature of the wire is raised by the same amount $\Delta T$ in the same time. The value of N is

• A. 4
• B. 6
• C. 8
• D. 9

Answer 1

The correct option is B 6

Let emf of each cell is E and resistance wire is R.
In first case: Total emf, $E_t=3E$
Heat produced, $H=\left(V^2/R\right)t=ms\Delta Tor\frac{9E^2}{R}t=ms\Delta T...\left(1\right)$ where $s=$ specific heat.
We know that: $R=\frac{\rho L}{A}\Rightarrow R\propto L$ and mass $m=d\left(\pi r^{2}L\right)\Rightarrow m\propto L$ where $\rho =$ resistivity and $d=$ mass density.
Thus, when the length (L) of the wire is doubled, resistance and mass both are doubled.
Therefore, in the second case, $\frac{(NE{)}^2}{2R}=\left(2m\right)s\Delta T...\left(2\right)$
$\left(2\right)/\left(1\right),\frac{N^2}{18}=2\Rightarrow N^2=36$

$\Rightarrow N=6$