Question

Two heater coils $1$ and $2$ made of the same material are connected in parallel across the mains. The length and diameter of the wire of coil $2$ are double that of coil $1$. Find the ratio of heat generated between coil $1$and $2$.

• A. $1:2$
• B. $1:1$
• C. $2:1$
• D. $3:2$

Answer 1

The correct option is A $1:2$

Here, subscript $1$ denotes coil $1$ and $2$ denotes coil $2$

Given : diameter $d_2=2d_1$ , length $L_2=2L_1$

Since, both coils are of the same material${\rho }_1={\rho }_2=\rho$

Resistance of coil 1 $R_1=\rho \frac{L_1}{A_1}=\frac{4\rho L_1}{\pi d_1^2}$

Resistance of coil 2 $R_2=\rho \frac{L_2}{A_2}=\frac{4\rho L_2}{\pi d_2^2}=\frac{4\rho \left(2L_1\right)}{\pi (2d_1{)}^2}=\frac{R_1}{2}$

Since, Heat generated is given by $H=\frac{V^{2}t}{R}$

Since, the coils are connected in parallel , $V$ across both will be same,

Thus ratio of heat generated $\frac{H_1}{H_2}=\frac{V^{2}t}{R_1}.\frac{R_2}{V^{2}t}=\frac{R_2}{R_1}=\frac{1}{2}$