Question

The ratio of heat dissipated per second through the resistance 5 ohm and 10 ohm in the circuit given below is :

• A. 1:2
• B. 2:1
• C. 4:1
• D. 1:1

Answer 1

The Correct Option is B

Given circuit:
The $5 \, \Omega$ and $10 \, \Omega$ resistors are connected in parallel.

Step 1: Calculating the Equivalent Resistance
The equivalent resistance $R_{\text{eq}}$ of the parallel combination is given by:

$$\frac{1}{R_{\text{eq}}} = \frac{1}{5} + \frac{1}{10}.$$

Calculating:

$$\frac{1}{R_{\text{eq}}} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10} \implies R_{\text{eq}} = \frac{10}{3} \, \Omega.$$

Step 2: Current Division in Parallel Resistors
Let $i_1$ be the current through the $5 \, \Omega$ resistor and $i_2$ be the current through the $10 \, \Omega$ resistor. By the current division rule:

$$\frac{i_1}{i_2} = \frac{R_2}{R_1} = \frac{10}{5} = 2.$$

Thus, $i_1 = 2i_2$.

Step 3: Calculating the Power Dissipated
The power dissipated $P$ in a resistor is given by:

$$P = i^2R.$$

The ratio of the power dissipated in the $5 \, \Omega$ resistor to the $10 \, \Omega$ resistor is:

$$\frac{P_1}{P_2} = \frac{i_1^2R_1}{i_2^2R_2} = \left( \frac{i_1}{i_2} \right)^2 \times \frac{R_1}{R_2}.$$

Substituting the values:

$$\frac{P_1}{P_2} = (2)^2 \times \frac{5}{10} = 4 \times \frac{1}{2} = 2.$$

Therefore, the ratio of heat dissipated per second through the $5 \, \Omega$ and $10 \, \Omega$ resistors is $2:1$.