Heat produced in a current carrying conducting wire depends on current $I$, resistance $R$ of the wire and time $t$ for which current is passed. Using these facts, obtain the formula for heat energy.
Heat produced in a current carrying conducting wire depends on current $I$, resistance $R$ of the wire and time $t$ for which current is passed. Using these facts, obtain the formula for heat energy.
Suppose, heat energy $\mathrm{H} \propto \mathrm{I}^{a} \mathrm{R}^{b} t^{c}$
$\therefore \mathrm{H}=k \mathrm{I}^{a} \mathrm{R}^{b} \mathrm{t}^{c} \ldots$ $(i)$ (where $k$ is a dimensionless constant.)
Now, writing the dimensional formula on both sides,
$\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}=(\mathrm{A})^{a}\left(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\right)^{b}(\mathrm{~T})^{c}$
$\therefore \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}=(\mathrm{A})^{a-2 b} \mathrm{M}^{b} \mathrm{~L}^{2 b} \mathrm{~T}^{c-3 b}$
${[\mathrm{H}]=\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}}$
${[\mathrm{I}]=\mathrm{A}[t]=\mathrm{T}}$
${[\mathrm{R}]=\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-2}}$
Equating the indices on both sides,
$a-2 b=0, b=1 \therefore a-2(1)=0 \therefore a=2$
$-3 b+c=-2$
$\therefore-3+c=-2$
$\therefore c=-2+3=1$
Thus, $a=2, b=1$ and $c=1$.
Substituting these values in equation $(i)$, we get
$\mathrm{H}=k \mathrm{I}^{2} \mathrm{R} t$
$\mathrm{H}=k \mathrm{I}^{2} \mathrm{R} t \text { But, } k=1 \quad \therefore \mathrm{H}=\mathrm{I}^{2} \mathrm{R} t$
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