Question

Two spherical conductive shells of radii a and b(b> a) are separated by a material with conductivity a. Find an expression for the resistance between the two spheres.(hint assuming +Q at r = a and -Q at r = b)

Answer 1

The expression for the resistance between two spherical conductive shells of radii a and b with conductivity σ is R = 1 / (σ • 4π) (1/a - 1/b), obtained by integrating over the spherical shell thickness.

Step-by-step explanation:

To solve the mathematical problem completely and find an expression for the resistance between the two spherical conductive shells, we can use the concept of conduction resistance in a spherical shell. Since the material between the spheres is conductive with conductivity σ, we assume a current flowing radially between them due to charges +Q and -Q placed on them.

For a spherical shell of inner radius r and infinitesimally small thickness δr, the resistance δR is given by δR = 1 / (σ • 4πr²). To find the total resistance R between the radii a and b, we integrate this expression from a to b.

The total resistance R between the spheres is given by:

1. R = ∫ a to b δR
2. R = ∫ a to b 1 / (σ • 4πr²) dr
3. R = 1 / (σ • 4π) ∫ a to b 1 / r² dr
4. R = 1 / (σ • 4π) [-1/r] from a to b
5. R = 1 / (σ • 4π) (1/a - 1/b)

Thus, the expression for the resistance between two concentric spherical conductive shells separated by a material with conductivity σ is R = 1 / (σ • 4π) (1/a - 1/b).