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)

20

Answer 2

Let $\rho$ be resistivity of the material of the wire and r be radius of the wire.
Therefore, resistance of 1 m wire is
$R=\frac{\rho(1)}{?r^2}=\frac{\rho}{?r^2}\, \, \, \, \, \bigg(\because R =\frac{\rho l}{A}\bigg)$
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 $?T(=10^\circ C)$ in time t.
\therefore\, \, \, \frac{(10\varepsilon )^2}{R}t=ms?T \hspace40mm ...(i)
In second case,
Resistance of same wire of length 2 m is
$\, \, \, \, \, \, R'= \frac{\rho(2)}{?r^2}=\frac{2\rho}{?r^2}=2R$

Let n cells each of emf $\varepsilon$ are connected in series to heat the same wire of length 2 m, by the same temperature $?T(=10^\circ C)$ in the same time t.
\therefore\, \, \, \frac{(n\varepsilon )^2 t}{2R}=(2m)s?T \hspace30mm ...(ii)
Divide (ii) by (i), we get
$\, \, \, \, \frac{n^2 }{200}=2 \Rightarrow n^2 = 400 \, \, \therefore \, \, n = 20$