A wire of length $L$ and $3$ identical cells of negligible internal resistances are connected in series. Due to current, the temperature of the wire is raised by $\Delta T$ in a time $t$. A number $N$ of similar cells is now connected in series with a wire of the same material and cross-section but of length $2\, L$. The temperature of the wire is raised by the same amount $\Delta T$ in the same time $t$. the value of $N$ is
IIT 2001, Medium
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Let $R$ and m be the resistance and mass of the first wire, then the second wire has resistance $2R$ and mass $2\,m$.
Let $E =$ $emf$ of each cell, $S =$ specific heat capacity of the material of the wire. For the first wire, current
${i_1} = \frac{{3E}}{R}$ and $i_1^2Rt = mS\Delta T$
For the second wire, ${i_2} = \frac{{NE}}{{2R}}$ and $i_2^2(2R)t = 2\,mS\Delta T$. Thus, ${i_1} = {i_2}$ or $N = 6$.
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