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title: "Two concentric, thin conducting spherical shells have inner radius \\(a\\) and outer radius \\(b\\), respectively. The region between the shells (\\(a \\le r \\le b\\)) is filled with a material whose conductivity varies radially according to \\(\\sigma(r) = \\sigma_0 \\left(\\dfrac{a}{r}\\right)\\), where \\(\\sigma_0\\) is a positive constant and \\(r\\) is the radial distance from the center. A constant potential difference \\(V_0\\) is maintained between the inner and outer shells. Which of the following expressions represents the total leakage current \\(I\\) flowing radially between the two shells?"
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url: "https://nerd-notes.com/ubq/118405/"
date_modified: "2026-08-04T08:09:56+00:00"
---

# Two concentric, thin conducting spherical shells have inner radius \(a\) and outer radius \(b\), respectively. The region between the shells (\(a \le r \le b\)) is filled with a material whose conductivity varies radially according to \(\sigma(r) = \sigma_0 \left(\dfrac{a}{r}\right)\), where \(\sigma_0\) is a positive constant and \(r\) is the radial distance from the center. A constant potential difference \(V_0\) is maintained between the inner and outer shells. Which of the following expressions represents the total leakage current \(I\) flowing radially between the two shells?

Two concentric, thin conducting spherical shells have inner radius \(a\) and outer radius \(b\), respectively. The region between the shells (\(a \le r \le b\)) is filled with a material whose conductivity varies radially according to \(\sigma(r) = \sigma_0 \left(\dfrac{a}{r}\right)\), where \(\sigma_0\) is a positive constant and \(r\) is the radial distance from the center. A constant potential difference \(V_0\) is maintained between the inner and outer shells. Which of the following expressions represents the total leakage current \(I\) flowing radially between the two shells?

![Cross-sectional view of two concentric circles centered at the origin. The inner circle has radius a and is labeled a. The outer circle has radius b and is labeled b. The annular region between radii a and b is shaded. Four radially outward arrows point from the inner circle to the outer circle. A potential difference V_0 is indicated between the inner and outer shells. No other labels or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830995-5l0Kjg.jpg)

- **A.** \(\dfrac{4\pi \sigma_0 a V_0}{\ln\left(\dfrac{b}{a}\right)}\)
- **B.** \(\dfrac{4\pi \sigma_0 a b V_0}{b - a}\)
- **C.** \(\dfrac{2\pi \sigma_0 a V_0}{\ln\left(\dfrac{b}{a}\right)}\)
- **D.** \(\dfrac{2\pi \sigma_0 a b V_0}{b - a}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118405/*