Question 729: Electrostats & Capacitance

Question

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.

Accepted Answer

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:
$ V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} $
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.
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:
$ Q = \sigma 4\pi r^2 $
So the potential at the surface of each sphere is given by:
$ V = \frac{1}{4\pi\epsilon_0} \frac{\sigma 4\pi r^2}{r} = \frac{\sigma r}{\epsilon_0} $
We are given that the potential at X and Z are the same. Thus:
$ V_X = V_Z $
Substituting the formula for the potential into this equation gives:
$ \frac{\sigma a}{\epsilon_0} = \frac{\sigma c}{\epsilon_0} $
This simplifies to:
$ a = c $
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:
$ \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} $
This simplifies to:
$ c(a - b + c) = a^2 - b^2 + c^2 $
Further simplification gives:
$ c(a - b) = a^2 - b^2 $
So:
$ c = a + b $
Given that the radii of X & Y are 2 cm and 3 cm, respectively, we have:
$ c = 2\, 	ext{cm} + 3\, 	ext{cm} = 5\, 	ext{cm} $
Therefore, the radius of shell Z is 5 cm.

Electrostats & Capacitance

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