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Question

# Using Gauss's law, deduce the expression for the electric field due to a uniformly charged spherical conducting shell of radius R at a point (i) outside and (ii) inside the shell. Plot a graph showing variation of electric field as a function of r> R and r< R. (r being the distance from the centre of the shell)

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Solution

## Electric field due to a uniformly charged thin spherical shell : (i) When point P lies outside the spherical shell : Consider a spherical shell of radius R and centre O. Let q be the charge on the spherical shell. Suppose that we have to calculate electric field at the point P at a distance r (r > R) from its centre. Draw the Gaussian surface through point P so as to enclose the charged spherical shell. The Gaussian surface is a spherical shell of radius r and centre O. Let →E be the electric field at point P. Then, the electric flux through area element →ds is given by dϕ=→E.→ds Since →ds is also along normal to the surface, ∴ dϕ=Eds ∴ Total electric flux through the Gaussian surface is given by ϕ=∮sE.ds =E∮dsNow,∮ds=4πr2∴ ϕ=E×4πr2..........(i) Since the charge enclosed by the Gaussian surface is q, according to Gauss theorem, ϕ=qε0 ....... (ii) From equations (i) and (ii), we get E×4πr2=qε0 E=14πε0.qr2 (for r > R) (ii) When point P lies inside the spherical shell : In such a case the Gaussian surface encloses no charge, According to Gauss law, E×4πr2=0 i.e., E=0(r<R) Graph showing the variation of electric field as a function of r.

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