Three concentric spherical metallic shells X, Y and Z of radius a, b and c respectively [a < b < c] have surface charge densities σ,-σ and σ 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.

Solution

Physics sir · 4 steps

Write the potential at X and at Z as a sum of three shell potentials, set them equal, and c drops out neatly.

1. One shell's potential. A shell of radius R with density σ has charge q = 4π R² σ. At a point inside or on it, its potential is q/(4πε₀ R) = (σ R)/ε₀. At a point outside, at distance r, it is (σ R²)/(ε₀ r).
2. Potential of X. Shell X sits inside Y and Z, so all three add their surface values: V_X = σ/ε₀(a - b + c).
3. Potential of Z. Shell Z is outside X and Y, so those two act like point charges at distance c: V_Z = σ/ε₀(a²/c - b²/c + c).

   a - b = (a² - b²)/c = (a-b)(a+b)/c ⇒ c = a + b

4. So. Setting V_X = V_Z and cancelling a - b (which is not zero) gives c = a + b = 2 + 3 = 5 cm.

The idea to keep. For concentric shells, the potential at a point is the sum of each shell's contribution: use its own radius for shells outside the point, and the point's distance for shells inside it.

Answer from the NTA final answer key. Solution by PrepWiser.