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Question
“A uniformly charged conducting spherical shell for the points outside the shell behaves as if the entire charge of the shell is concentrated at its centre”. Show this with the help of a proper diagram and verify this statement.
Solution
A shell of radius R is carrying uniformly distributed charge of charge density σ.
Electric field E at point P at a distance r from the centre of a uniformly charged spherical shell:
Consider a Gaussian surface to be a sphere of radius r >R and with centre O, passing through P.
Using the principles of spherical symmetry, E and area vector ΔS at every point are parallel, flux through each of the area element.
ΔΦ = E. ΔS
Summing over all ΔS,
Φ = E × 4 π R2
The charge enclosed is σ × 4 π R2
By Gauss’ law,
`"E" xx 4 pi "r"^2 = "q"/ε_o = (σ "X" 4pi "R"^2)/ε_o`
`"E" = (σ "R"^2)/(ε_o "r"^2) = "q"/(4 pi ε_o "r"^2)`
This formula is exactly similar to the field produced by a point charge q placed at the centre O, that is, as per Coulomb’s law.
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