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Due to cold weather a $1\, {m}$ water pipe of cross-sectional area $1\, {cm}^{2}$ is filled with ice at $-10^{\circ} {C}$. Resistive heating is used to melt the ice. Current of $0.5\, {A}$ is passed through $4\, {k} \Omega$ resistance. Assuming that all the heat produced is used for melting, what is the minimum time required ? (In ${s}$)
(Given latent heat of fusion for water/ice $=3.33 \times 10^{5}\, {J} {kg}^{-1}$, specific heat of ice $=2 \times 10^{3}\, {J}$ ${kg}^{-1}$ and density of ice $=10^{3}\, {kg} / {m}^{3}$
$0.353$
$35.3$
$3.53$
$70.6$
Solution
mass of ice ${m}=\rho {A} \ell=10^{3} \times 10^{-4} \times 1=10^{-1}\, {kg}$
Energy required to melt the ice
${Q}={ms} \Delta {T}+{mL}$
$=10^{-1}\left(2 \times 10^{3} \times 10+3.33 \times 10^{5}\right)=3.53 \times 10^{4} {J}$
${Q}={i}^{2} {RT} \Rightarrow 3.53 \times 10^{4}=\left(\frac{1}{2}\right)^{2}\left(4 \times 10^{3}\right)({t})$
$\text { Time }=35.3 \,{sec}$
