Due to cold weather a 1 m water pipe of cross-sectional area 1 cm² is filled with ice at $-$10$^\circ$C. Resistive heating is used to melt the ice. Current of 0.5A is passed through 4 k$\Omega$ resistance. Assuming that all the heat produced is used for melting, what is the minimum time required? (Given latent heat of fusion for water/ice = 3.33 $\times$ 10⁵ J kg^$-$1, specific heat of ice = 2 $\times$ 10³ J kg^$-$1 and density of ice = 10³ kg/m³

Given, the length of the water pipe, L = 1 m<br/><br/>The cross-sectional area of the water pipe, A = 1 cm² = 10^$-$4 m²<br/><br/>The temperature of the ice = $-$ 10$^\circ$C<br/><br/>Current passing in the conductor, I = 0.5 A<br/><br/>Resistance of the conductor, R = 4 k$\Omega$<br/><br/>The latent heat of fusion for ice, L_f = 3.33 $\times$ 10⁵ J/kg<br/><br/>The density of the ice, d = 1000 kg/m³<br/><br/>The specific heat of the ice, c_{p, ice} = 2 $\times$ 10³ J/kg<br/><br/>Heat required to melt the ice at 10$^\circ$C to 0$^\circ$C<br/><br/>Q = mc_p$\Delta$T + mL_f $\Rightarrow$ Q = dVc_p$\Delta$T + dVL_f<br/><br/>= 1000 $\times$ 10^$-$4 $\times$ 2 $\times$ 10³ $\times$ (10) + 1000 $\times$ 10^$-$4 $\times$ 3.33 $\times$ 10⁵ ($\because$ V = A $\times$ L)<br/><br/>= 35300 J<br/><br/>According to the Joule's law of heating,<br/><br/>H = I²Rt<br/><br/>$\Rightarrow$ 35300 = (0.5)²(4000) (t)<br/><br/>$\Rightarrow$ t = 35.3 s<br/><br/>Thus, the minimum time required to melt the ice is 35.3 s.