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As shown in the figure below, two concentric conducting spherical shells, centered at r = 0 and having radii r = c and r = d are maintained at potentials such that the potential V(r) at r = c is V1 and V(r) at r = d is V2. Assume that V(r) depends only on r, where r is the radial distance. The expression for V(r) in the region between r = c and r = d is

1. V(r) = cd(V2 − V1)/((d − c)r) − (V1 c + V2 d − 2V1 d)/(d − c)
2. V(r) = cd(V1 − V2)/((d − c)r) + (V2 d − V1 c)/(d − c)
3. V(r) = cd(V1 − V2)/((d − c)r) + (V1 c − V2 c)/(d − c)
4. V(r) = cd(V2 − V1)/((d − c)r) + (V2 c − V1 c)/(d − c)

Correct answer: V(r) = cd(V1 − V2)/((d − c)r) + (V2 d − V1 c)/(d − c)

Solution

The correct option is derived from the principles of electrostatics, where the potential difference between two points in a radial electric field is expressed in terms of the distances and potentials at the boundaries. This option correctly accounts for the linear variation of potential in the region between the two spherical shells, ensuring that it satisfies the boundary conditions set by the potentials at r = c and r = d.

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