Question

If a wire of resistance R is connected across Vo, then power is Po. The wire is cut into two equal parts and connected with Vo individually, then the sum of power dissipated is P1, then \(\frac{Po}{Pi}\) is \(\frac{1}{x}\). Find the value of x.

Answer 1

Step 1: Power Dissipation Formula
The power dissipation across a resistor is given by: \[ P = \frac{V^2}{R}. \]

 Step 2: Resistance of Each Half
When the wire is cut into two equal halves, the resistance of each half becomes: \[ R_{\text{half}} = \frac{R}{2}. \] 

Step 3: Power Dissipation Across Each Half
When \( V_0 \) is applied across each half: \[ P_{\text{half}} = \frac{V_0^2}{R_{\text{half}}} = \frac{V_0^2}{\frac{R}{2}} = \frac{2V_0^2}{R}. \] 

Step 4: Total Power Dissipation Across Two Halves
The total power dissipation across the two halves is: \[ P_2 = 2 \times P_{\text{half}} = 2 \times \frac{2V_0^2}{R} = \frac{4V_0^2}{R}. \] 

Step 5: Ratio of Power Dissipation
The original power dissipation \( P_1 \) is: \[ P_1 = \frac{V_0^2}{R}. \] The ratio \( P_2 : P_1 \) is: \[ \frac{P_2}{P_1} = \frac{\frac{4V_0^2}{R}}{\frac{V_0^2}{R}} = 4. \] 

Step 6: Relation to \( \sqrt{x} \)
The ratio \( P_2 : P_1 \) is given as \( \sqrt{x} : 1 \), so: \[ \sqrt{x} = 4 \quad \Rightarrow \quad x = 16. \]