$(a)$ State the commercial unit of electric energy and find its relation with its $SI$ unit.

$(b)$ The current through a resistor is made three times its initial value. Calculate how it will affect the heat produced in the resistor.

$(c)$ Find the increase in the amount of heat generated in a conductor if another conductor of double resistance is connected in the circuit keeping.all other factors unchanged.

$(a)$ The commercial unit of electric energy is kilowatt hour $(kWh).$

$SI$ unit of electrical energy is Joule $(J)$

$1\, kWh =1000$ watts $\times 60\, min$

$=1000 \times 60 \times 60 s$

$=3.6 \times 10^{6}$ watts seconds

$=3.6 \times 10^{6}$ Joules

$(b)$ Initial production of heat

$H = I ^{2} R t$

With the new resistance, heat produced

$=(3 I)^{2} \times R \times t$

$=9 I ^{2} R t$

So the heat produced will be nine times.

$(c)$ Initial heat, $H = I ^{2} \times R \times t$

New resistance $= R +2 R =3 R$

With the new resistance, heat produced

$= I ^{2} \times 3 R \times t$

$=3 I ^{2} R t$

$\therefore$ The heat produced will become thrice.