First slide

Cell

Question

Ten identical cells connected in series are needed to heat a wire of length one meter and radius ‘r’ by $10^{o} C$ 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’ ?

Moderate

A

20
B

30
C

40
D

10

Solution

Let $ρ$ be resistivity of the material of the wire and r be radius of the wire. Therefore, resistance of 1 m wire is
$R = \frac{ρ (1)}{π r^{2}} = \frac{ρ}{π r^{2}} (∵ R = \frac{ρ l}{A})$
Let $ε$ be emf of each cell.
In first case,
10 cells each of emf $ε$ are connected in series to heat the wire of length 1 m by $Δ T (= 10^{o} C)$ in time t.
$∴ \frac{(10 ε)^{2}}{R} t = m s Δ T . . . . . . . . (i)$
In second case, Resistance of same wire of length 2 m is
$R^{'} = \frac{ρ (2)}{π r^{2}} = \frac{2 ρ}{π r^{2}} = 2 R$
Let n cells each of emt $ε$ are connected in series to heat the same wire of length 2 m, in the same temperature $Δ T (= 10^{o} C)$ in the same time t
$∴ \frac{(n ε)^{2} t}{2 R} = (2 m)_{s Δ T} . . . . . (i i)$
Divide (ii) by (i), we get
$\frac{n^{2}}{200} = 2 \Rightarrow n^{2} = 400 ∴ n = 20$

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