An infinitely long thin wire, having a uniform charge density per unit length of \(5 nC/m,\) is passing through a spherical shell of radius \(1 m\), as shown in the figure. A \(10 nC\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points P and R, in Volt, is ______. [Given: In SI units \(\frac{1}{ 4πϵ0} = 9 × 10^9 , ln \ 2 = 0.7\). Ignore the area pierced by the wire.]
Correct Answer: 171
Solution and Explanation
Potential Difference Calculation
The potential difference is calculated by considering contributions from both a line charge and a sphere charge.
Step 1: Potential Difference Due to the Line Charge
The potential difference due to the line charge is given by: \[ (V_P - V_R)_{\text{line charge}} = 2k \lambda \ln \left( \frac{r_P}{r_R} \right) \] where:
- k: Coulomb’s constant
- \( r_P \): The distance of point P
- \( r_R \): The distance of point R
- \( \lambda \): The linear charge density
Given that: \[ r_P = 126 \, \text{V} \]
Step 2: Potential Difference Due to the Sphere Charge
The potential difference due to the sphere charge is: \[ (V_P - V_R)_{\text{sphere}} = kq \left( \frac{1}{r_R} - \frac{1}{r_R} \right) = kq \cdot \frac{2}{r_R} \] where:
- q: The charge on the sphere
- \( r_R \): The reference radius
Given: \[ (V_P - V_R)_{\text{sphere}} = 45 \, \text{V} \]
Step 3: Total Potential Difference
The total potential difference is: \[ V_P - V_R = 126 + 45 = 171 \, \text{V} \]
Final Answer:
The total potential difference between points P and R is 171 V.
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