On a solid sphere of radius R, or resistivity σ, there are two points a, b on the equatorial line at a distance x from each other... What is the equivalent resistance between the two points...?

To find the equivalent resistance between two points on the surface of a solid sphere, we need to consider the geometry of the sphere and the properties of the material. Let's break this down step by step.

Understanding the Setup

We have a solid sphere with radius R and resistivity σ. The two points, a and b, are located on the equatorial line of the sphere, separated by a distance x. The goal is to determine the resistance between these two points.

Geometry of the Sphere

First, we need to recognize that the distance x on the surface of the sphere corresponds to a certain angle θ in spherical coordinates. The relationship between the arc length x and the angle θ can be expressed as:

• x = R * θ

From this, we can derive the angle:

• θ = x / R

Current Distribution

When a potential difference is applied between points a and b, the current will spread out over the surface of the sphere. Due to the symmetry of the sphere, we can assume that the current will flow uniformly across the surface. This means we can treat the sphere as a series of infinitesimally thin spherical shells.

Calculating Resistance of a Shell

Consider a thin shell of thickness dr at a radius r from the center of the sphere. The resistance dR of this shell can be expressed as:

• dR = (1 / (σ * A)) * dr

where A is the surface area of the shell, given by:

• A = 4πr²

Thus, the resistance of the shell becomes:

• dR = (1 / (σ * 4πr²)) * dr

Integrating for Total Resistance

To find the total resistance between points a and b, we need to integrate this expression over the appropriate limits. However, since we are interested in the resistance only along the surface between the two points, we can simplify our approach by considering the effective path of current flow.

Using Symmetry

Due to the symmetry of the sphere, the equivalent resistance R_eq can be approximated using the formula for resistance between two points on a spherical surface:

• R_eq = (σ * x) / (2πR)

This formula arises from considering the uniform distribution of current and the geometry of the sphere. The factor of 2πR accounts for the circular path the current takes around the sphere.

Final Expression

Thus, the equivalent resistance between the two points a and b on the surface of the sphere is given by:

• R_eq = (σ * x) / (2πR)

This result highlights how the resistivity of the material and the geometry of the sphere influence the resistance between two points on its surface. The larger the distance x or the lower the resistivity σ, the lower the equivalent resistance will be.