Let $\rho$ be resistivity of the material of the wire and r be radius of the wire.

Therefore, resistance of 1 m wire is

$R = {{\rho \left( 1 \right)} \over {\pi {r^2}}} = {\rho \over {\pi {r^2}}}$
$\left(\because {R = {{\rho l} \over A}} \right)$

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(= 10°C) in time t.

$\therefore {{\left( {10\varepsilon } \right)} \over R}r = ms\Delta T$    ...(i)

In second case,
Resistance of same wire of length 2 m is

$R' = {{\rho \left( 2 \right)} \over {\pi {r^2}}} = {{2\rho } \over {\pi {r^2}}} = 2R$


Let n cells each of emf $\varepsilon$ are connected in series to heat the same wire of length 2 m, by the same temperature $\Delta$T (= 10°C) in the same time t.

$\therefore {{{{\left( {n\varepsilon } \right)}^2}t} \over {2R}} = \left( {2m} \right)s\Delta T$    ...(ii)

Divide (ii) by (i), we get

${{{n^2}} \over {200}} = 2 \Rightarrow {n^2} = 400$

$\therefore$ n = 20