The region between two concentric conducting spheres with...

The region between two concentric conducting spheres with radii $a$ and $b$ is filled with a conducting material with resistivity $\rho$. (a) Show that the resistance between the spheres is given by $$ R=\frac{\rho}{4 \pi}\left(\frac{1}{a}-\frac{1}{b}\right) $$ (b) Derive an expression for the current density as a function of radius, in terms of the potential difference $V_{a b}$ between the spheres. (c) Show that the result in part (a) reduces to Eq. (25.10) when the separation $L=b-a$ between the spheres is small.

The region between two concentric conducting spheres with radii $a$ and $b$ is filled with a conducting material with resistivity $\rho$.
(a) Show that the resistance between the spheres is given by
$$
R=\frac{\rho}{4 \pi}\left(\frac{1}{a}-\frac{1}{b}\right)
$$
(b) Derive an expression for the current density as a function of radius, in terms of the potential difference $V_{a b}$ between the spheres. (c) Show that the result in part (a) reduces to Eq. (25.10) when the separation $L=b-a$ between the spheres is small.

Key Concepts

Small Separation Approximation

The small separation approximation involves simplifying the expression for resistance by assuming that the difference between the radii is small compared to the absolute values of the radii. This approximation allows the curved geometry of the spherical configuration to be treated as nearly planar over the small distance, leading to simpler forms of the resistance equation that are often easier to compare with or reduce to known results in planar geometries.

Current Density Distribution

The current density is defined as the current per unit area and can vary with position in non-uniform geometries. In spherical configurations, the current density at a given radius is determined by dividing the total current by the surface area of a sphere at that radius. When an external potential difference is applied, the spatial variation of the current density is intrinsically linked to the radial dependence of the electric field, which in turn is governed by the potential gradient.

Spherical Symmetry

Spherical symmetry implies that all points at a given radial distance from the center of the configuration are equivalent. In problems involving concentric spheres, this symmetry greatly simplifies the analysis because the physical quantities, such as current density and electric field, depend only on the radial coordinate. This allows for the assumption that the current flows radially and is uniformly distributed over spherical shells.

Differential Resistance and Integration

When dealing with geometries where the cross-sectional area for conduction varies with position, the concept of differential resistance becomes important. The method involves considering an infinitesimally thin shell in which the resistance can be expressed as the resistivity multiplied by an incremental length divided by the local cross-sectional area. Integrating this differential resistance over the entire region between conductors yields the total resistance, an approach that is particularly useful in spherically symmetric conditions.

Resistivity

Resistivity is a fundamental property of materials that quantifies how strongly a material opposes the flow of electric current. In the context of conduction, it relates the electric field to the current density through Ohm’s law in its differential form, where a higher resistivity indicates a material in which current is more impeded. This concept is central for calculating the resistance across any conductor given its geometry and material properties.

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