A $${5^ \circ }C$$  rise in temperature is observed in a conductor by passing a current. When the current is doubled the rise in temperature will be approximately

Solution :
Energy loss in conductor, $$Q = {i^2}Rt$$
where, $$i =$$  current flowing through it
$$R =$$  resistance of conductor
$$t =$$  time for which current is passed
Heat developed $$ = ms\Delta \theta $$
$$\therefore ms\Delta \theta = {i^2}Rt$$
Since $$m,s,R,t$$   remains constant.
So, $$\Delta \theta \propto {i^2}$$
So, for two different cases
$$\eqalign{ & \frac{{\Delta {\theta _2}}}{{\Delta {\theta _1}}} = \frac{{i_2^2}}{{i_1^2}} \cr & {\text{or}}\,\,\Delta {\theta _2} = {\left( {\frac{{{i_2}}}{{{i_1}}}} \right)^2}\Delta {\theta _1}\,......\left( {\text{i}} \right) \cr} $$
Given, $${i_2} = 2{i_1},\Delta {\theta _1} = {5^ \circ }C$$
So, from Eq. (i),
$$\eqalign{ & \therefore \Delta {\theta _2} = {\left( {\frac{{2{i_1}}}{{{i_1}}}} \right)^2} \times 5 \cr & = 4 \times 5 \cr & = {20^ \circ }C \cr} $$