Resistance of a sphere of resistivity rho and radius S

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SUMMARY

The discussion focuses on calculating the resistance between two points on a sphere with resistivity ρ and radius S. The user employs the formula dR = ρ * dl/A, where dl = -S Sin(θ) d(θ) and A(θ) = π {S Sin(θ)}². The challenge arises when evaluating the definite integral from θ = 0 to π, as the strip element reduces to a point at the endpoints, leading to difficulties in obtaining a finite resistance value. The user seeks assistance in resolving this integral issue.

PREREQUISITES
  • Understanding of electrical resistance and resistivity concepts
  • Familiarity with calculus, particularly integration techniques
  • Knowledge of spherical coordinates and their applications
  • Experience with mathematical modeling of physical phenomena
NEXT STEPS
  • Study the evaluation of definite integrals in spherical coordinates
  • Research the application of resistivity in three-dimensional geometries
  • Explore advanced calculus techniques for handling limits and singularities
  • Investigate numerical methods for approximating integrals in complex geometries
USEFUL FOR

Physics students, electrical engineers, and mathematicians interested in advanced resistance calculations and integral evaluation in spherical geometries.

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I calculated the resistance between two points (P and Q) on a sphere that are located on the two ends of a diameter by dividing the sphere into thin strips of thickness dz that are in series perpendicular to the line PQ (say the z-axis). I can get an expression for the indefinite integral in theta (the polar angle theta, z = S Cos (theta), and S= radius), however I run into problem when evaluating this definite integral for theta = 0 to pi. Apparently the fact that the strip element becomes a points at the ends is causing this problem. I use dR=rho * dl/A with dl = -S Sin(theta) d(theta) and A(theta) = pi {S Sin(theta)}^2. What is going wrong here? Because I know that there must be a finite resistance between two such points.
 
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