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.

<p>Given three concentric spherical shells X, Y, and Z with radii a, b, and c respectively, and with surface charge densities ( $\sigma$ ), ( $-\sigma$ ), and ( $\sigma$ ) respectively, we know that the potential at the surface of a sphere due to a uniform surface charge is given by:</p> <p>$ V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} $</p> <p>where ( $\epsilon_0$ ) is the permittivity of free space, ( Q ) is the total charge on the sphere, and ( r ) is the radius of the sphere.</p> <p>However, in this case, the total charge on each sphere is given by its surface charge density ( $\sigma$ ) times its surface area ( $4\pi r^2$ ). Substituting this into the formula for ( Q ) gives:</p> <p>$ Q = \sigma 4\pi r^2 $</p> <p>So the potential at the surface of each sphere is given by:</p> <p>$ V = \frac{1}{4\pi\epsilon_0} \frac{\sigma 4\pi r^2}{r} = \frac{\sigma r}{\epsilon_0} $</p> <p>We are given that the potential at X and Z are the same. Thus:</p> <p>$ V_X = V_Z $</p> <p>Substituting the formula for the potential into this equation gives:</p> <p>$ \frac{\sigma a}{\epsilon_0} = \frac{\sigma c}{\epsilon_0} $</p> <p>This simplifies to:</p> <p>$ a = c $</p> <p>However, we also need to take into account the effect of the charge on shell Y on the potentials at X and Z. The potential at any point due to a charged shell is the same everywhere outside the shell, so we can add the potential due to shell Y at X to both sides of the equation. This gives:</p> <p>$ \frac{\sigma a}{\epsilon_0} - \frac{\sigma b}{\epsilon_0} + \frac{\sigma c}{\epsilon_0} = \frac{\sigma a}{\epsilon_0} + \frac{\sigma c}{\epsilon_0} $</p> <p>This simplifies to:</p> <p>$ c(a - b + c) = a^2 - b^2 + c^2 $</p> <p>Further simplification gives:</p> <p>$ c(a - b) = a^2 - b^2 $</p> <p>So:</p> <p>$ c = a + b $</p> <p>Given that the radii of X &amp; Y are 2 cm and 3 cm, respectively, we have:</p> <p>$ c = 2\, \text{cm} + 3\, \text{cm} = 5\, \text{cm} $</p> <p>Therefore, the radius of shell Z is 5 cm.</p>