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title: "A conducting material of uniform resistivity \\(\\rho\\) fills the region between two concentric, highly conducting spherical shells of inner radius \\(a\\) and outer radius \\(b\\), where \\(b > a\\). A steady total current \\(I\\) flows radially outward from the inner shell to the outer shell. Which of the following statements correctly describes the behavior of the total radial resistance \\(R\\) as the outer radius \\(b\\) approaches infinity (\\(b \\to \\infty\\))?"
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url: "https://nerd-notes.com/ubq/121551/"
date_modified: "2026-08-23T05:23:59+00:00"
---

# A conducting material of uniform resistivity \(\rho\) fills the region between two concentric, highly conducting spherical shells of inner radius \(a\) and outer radius \(b\), where \(b > a\). A steady total current \(I\) flows radially outward from the inner shell to the outer shell. Which of the following statements correctly describes the behavior of the total radial resistance \(R\) as the outer radius \(b\) approaches infinity (\(b \to \infty\))?

A conducting material of uniform resistivity \(\rho\) fills the region between two concentric, highly conducting spherical shells of inner radius \(a\) and outer radius \(b\), where \(b > a\). A steady total current \(I\) flows radially outward from the inner shell to the outer shell. Which of the following statements correctly describes the behavior of the total radial resistance \(R\) as the outer radius \(b\) approaches infinity (\(b \to \infty\))?

![A cross-sectional diagram showing two concentric circles centered at the origin. The inner circle has a solid boundary with radius labeled a, indicated by an arrow from the center to the inner boundary. The outer circle has a solid boundary with radius labeled b, indicated by a longer arrow from the center to the outer boundary. The annular region between the two circles has light gray hatching and is labeled with the symbol \rho. Exactly four outward-pointing radial arrows labeled I are spaced symmetrically at 0, 90, 180, and 270 degrees, each extending from the inner boundary across the shaded region toward the outer boundary. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462638-fuof5e.jpg)

- **A.** \(R\) approaches \(\infty\) linearly with \(b\), because the path length of the current through the resistive material increases without bound.
- **B.** \(R\) approaches \(0\), because the cross-sectional area available to the current becomes infinite as the outer boundary moves to infinity.
- **C.** \(R\) approaches the finite value \(\dfrac{\rho}{4\pi a}\), because the resistance contribution \(dR \propto \dfrac{dr}{r^2}\) of outer shells falls off rapidly enough for the total integral to converge.
- **D.** \(R\) approaches \(\infty\) logarithmically as \(\ln(b/a)\), because the radial current density decreases inversely with distance from the center.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121551/*