Question

If 50 cm of wire were used in heating the element of a 6 V, 30 W heater, what length of wire of the same material having double the cross - sectional area would be required for a 12 V, 30 W heating element , assuming that the final temperature is the same?

Answer 1

The power of a heating element

$$P=\frac{V^2}{R}$$

We get

$$\frac{V_1^2}{R_1}=\frac{V_2^2}{R_2}\\ R_2=R_1\frac{V_2^2}{V_1^2}=4R_1$$

The resistance of wire

$$R_1=\rho\frac{L_1}{A_1}\\ R_2=\rho\frac{L_2}{2A_1}\\ 4\rho\frac{L_1}{A_1}=\rho\frac{L_2}{2A_1}$$

Finally

$$L_2=8L_1=8\times 0.5\:\rm m=4\;\rm m.$$