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Electric Field Due to a Sphere and Thin Shell

The electric ...

Question

The electric field at a distance r from the centre in the space between two concentric metallic spherical shells of radii $r_{1}$ and $r_{2}$ carrying charge $Q_{1}$ and $Q_{2}$ is $(r_{1} < r < r_{2})$ :

\frac{Q_{1} + Q_{2}}{4 π ε_{0} {(r_{1} + r_{2})}^{2}}

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\frac{Q_{1} + Q_{2}}{4 π ε_{0} r^{2}}

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\frac{Q_{1}}{4 π ε_{0} r^{2}}

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\frac{Q_{2}}{4 π ε_{0} r^{2}}

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Solution

The correct option is C $\frac{Q_{1}}{4 π ε_{0} r^{2}}$
Consider a spherical surface with radius r where $r_{1} < r < r_{2}$
Thus the charge inside the surface is only $Q_{1}$
By Gauss's law $E . d S = \frac{Q_{i n}}{ϵ_{0}}$
or $E . 4 π r^{2} = \frac{Q_{1}}{ϵ_{0}}$
or $E = \frac{Q_{1}}{4 π ϵ_{0} r^{2}}$

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