Ncert Exercises & Solutions

A metallic spherical shell has an inner radius \(R_1\) and an outer radius \(R_2.\) A point charge \(Q\) is placed at the center of the spherical cavity. What are the surface charge densities \(\sigma_{in}\) and \(\sigma_{out}\) on the inner and outer surfaces of the shell, respectively?
1. \(\sigma_{in} = -\dfrac{Q}{4\pi R_1^2},~ \sigma_{out}=\dfrac{Q}{4\pi R_2^2}\)
2. \(\sigma_{in} = \dfrac{Q}{4\pi R_1^2},~ \sigma_{out}=0\)
3. \(\sigma_{in} = 0,~ \sigma_{out}=\dfrac{Q}{4\pi R_2^2}\)
4. \(\sigma_{in} = \dfrac{Q}{4\pi R_1^2},~ \sigma_{out}=-\dfrac{Q}{4\pi R_2^2}\)

Hint: Use the concept of induction of charge.

Step 1: Find the charge induced on the inner and outer surfaces.

Here, the charge placed at the centre of the spherical cavity is positively charged. So, the charge created at the inner surface of the sphere, due to induction will be -Q and due to this charge created at the outer surface of the sphere is +Q.

Now, the surface charge density on the inner surface

= \frac{- Q}{4 π R_{1}^{2}}

, and,

Surface charge density on the outer surface

= \frac{+ Q}{4 π R_{2}^{2}}

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