Solution:

Given, three concentric spherical shells have radii a, b and c (a < b="" />< c)="" and="" have="" surface="" charge="" densities="" sigma_{z}="" -="" sigma="" and="" a="" />

We need to find the relation between the potentials of the three shells when c = a, b.

Let us assume that the potential at infinity is zero.

Using the formula for potential due to a charged spherical shell, we get:

- Potential at any point inside a charged spherical shell of radius R and surface charge density sigma is given by: V = (sigma * R^2)/(3 * epsilon_0).

- Potential at any point outside a charged spherical shell of radius R and surface charge density sigma is given by: V = (sigma * R)/(3 * epsilon_0).

Using the above formulas, we can find the potentials of the three shells as follows:

- Potential of the outermost shell (c): V_{c} = (sigma * c)/(3 * epsilon_0).

- Potential of the middle shell (b): V_{b} = (sigma * b)/(3 * epsilon_0).

- Potential of the innermost shell (a): V_{a} = (sigma * a)/(3 * epsilon_0).

Let us now consider the three cases:

Case 1: c = a

- Potential of the outermost shell (c): V_{c} = (sigma * c)/(3 * epsilon_0).

- Potential of the innermost shell (a): V_{a} = (sigma * a)/(3 * epsilon_0).

- Potential of the middle shell (b): V_{b} = (sigma * b)/(3 * epsilon_0).

We can see that the potentials of the three shells are not equal. Hence, option (A) is incorrect.

Case 2: c = b

- Potential of the outermost shell (c): V_{c} = (sigma * c)/(3 * epsilon_0).

- Potential of the middle shell (b): V_{b} = (sigma * b)/(3 * epsilon_0).

- Potential of the innermost shell (a): V_{a} = (sigma * a)/(3 * epsilon_0).

We can see that the potentials of the three shells are not equal. Hence, option (B) is incorrect.

Case 3: b = a

- Potential of the middle shell (b): V_{b} = (sigma * b)/(3 * epsilon_0).

- Potential of the innermost shell (a): V_{a} = (sigma * a)/(3 * epsilon_0).

- Potential of the outermost shell (c): V_{c} = (sigma * c)/(3 * epsilon_0).

We can see that the potentials of the three shells are equal. Hence, option (D) is correct.

Option (C) is incorrect as it is not possible for the potentials of three concentric shells to be equal at different radii.

Therefore, the correct answer is option (D) V_{c} = V_s = V_n.