Three concentric spherical metallic shells $$X$$, $$Y$$ and $$Z$$ of radius $$a$$, $$b$$ and $$c$$ respectively $$a < b < c$$ have surface charge densities $$\sigma$$, $$-\sigma$$ and $$\sigma$$, respectively. The shells $$X$$ and $$Z$$ are at same potential. If the radii of $$X$$ & $$Y$$ are 2 cm and 3 cm, respectively. The radius of shell Z is _______ cm.

Given: Three concentric shells X (radius a=2cm), Y (radius b=3cm), Z (radius c) with surface charge densities $$\sigma, -\sigma, \sigma$$ respectively. X and Z are at the same potential.

Potential at shell X (radius a):

$$V_X = \frac{\sigma \cdot 4\pi a^2}{4\pi\varepsilon_0 a} + \frac{(-\sigma) \cdot 4\pi b^2}{4\pi\varepsilon_0 b} + \frac{\sigma \cdot 4\pi c^2}{4\pi\varepsilon_0 c} = \frac{\sigma}{\varepsilon_0}(a - b + c)$$

Potential at shell Z (radius c):

$$V_Z = \frac{\sigma \cdot 4\pi a^2}{4\pi\varepsilon_0 c} + \frac{(-\sigma) \cdot 4\pi b^2}{4\pi\varepsilon_0 c} + \frac{\sigma \cdot 4\pi c^2}{4\pi\varepsilon_0 c} = \frac{\sigma}{\varepsilon_0}\left(\frac{a^2 - b^2}{c} + c\right)$$

Setting $$V_X = V_Z$$:

$$a - b + c = \frac{a^2 - b^2}{c} + c$$

$$a - b = \frac{a^2 - b^2}{c} = \frac{(a-b)(a+b)}{c}$$

$$1 = \frac{a+b}{c}$$

$$c = a + b = 2 + 3 = 5 \text{ cm}$$

The radius of shell Z is 5 cm.