Question

Ten identical cells connected in series are needed to heat a wire of length one meter and radius '$r$' by $10^{\circ} C$ in time '$t$'. How many cells will be required to heat the wire of length two meter of the same radius by the same temperature in time '$t$' ?

• A. 20
• B. 30
• C. 40
• D. 10

Answer 1

Answer: (A)

Let $\rho$ be resistivity of the material of the wire and $r$ be radius of the wire.
Therefore, resistance of $1\, m$ wire is
$R=\frac{\rho(1)}{\pi r^{2}}=\frac{\rho}{\pi r^{2}} $
$\left(\because R=\frac{\rho l}{A}\right)$
Let $\varepsilon$ be emf of each cell.
In first case,

$10$ cells each of emf $\varepsilon$ are connected in series to heat the wire of length $1 m$ by $\Delta T\left(=10^{\circ} C\right)$ in time $t$.
$\therefore \frac{(10 \varepsilon)^{2}}{R} t=m s \Delta T \ldots \text { (i) }$
In second case,
Resistance of same wire of length $2 m$ is
$R^{\prime}=\frac{\rho(2)}{\pi r^{2}}=\frac{2 \rho}{\pi r^{2}}=2 R$

Let $n$ cells each of emf $\varepsilon$ are connected in series to heat the same wire of length $2 m$, by the same temperature $\Delta T\left(=10^{\circ} C\right)$ in the same time $t$.
$\therefore \frac{(n \varepsilon)^{2} t}{2 R}=(2 m) s \Delta T \ldots \text { (ii) }$
Divide (ii) by (i), we get
$\frac{n^{2}}{200}=2 $
$\Rightarrow n^{2}=400 $
$\therefore n=20$