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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 =EdsNow,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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