JEE Main 2023 — Current Electricity Question with Solution

Let's analyze the initial situation where the potential $$V_0$$ is applied across the entire length of the wire with resistance $$R$$.

The power dissipation, $$P_1$$, can be calculated using the formula:

$$P_1=\frac{V_0^2}{R}$$

Now, let's consider the case where the wire is cut into two equal halves. Each half will have half the original resistance, $$\frac{R}{2}$$. The potential $$V_0$$ is applied across the length of each half.

For each half of the wire, the power dissipation, $$P^{\prime }$$, can be calculated using the formula:

$$P^{\prime }=\frac{V_0^2}{\frac{R}{2}}=\frac{2V_0^2}{R}$$

Since there are two halves of the wire, the total power dissipation across the two wires, $$P_2$$, is:

$$P_2=2P^{\prime }=2\left(\frac{2V_0^2}{R}\right)=\frac{4V_0^2}{R}$$

Now, let's find the ratio $$P_2:P_1$$:

$$\frac{P_2}{P_1}=\frac{\frac{4V_0^2}{R}}{\frac{V_0^2}{R}}=4$$

Comparing this to the given ratio $$\sqrt{x}:1$$, we have:

$$\frac{P_2}{P_1}=\sqrt{x}$$

So, $$\sqrt{x}=4$$.

Squaring both sides, we get:

$$x=16$$

The value of $$x$$ is 16.