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Using Gauss’s law in electrostatics, obtain the expression for electric field due to a uniformly charged thin spherical shell at a point 

i. outside the shell and 

ii. inside the shell

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Let σ be the uniform surface charge density of a thin spherical shell of radius R Field outside the shell. 

Consider a point P outside the shell at a distance r from the centre of the shell. Imagine a gaussian sphere of radius ‘r’ The electric flux at P due to surface ∆S is 

∆φ= \(\vec E. \vec{\Delta S}\) = E∆S COSθ = E∆S {COS θ = 1} 

Total electric flux due to the sphere is 

φ = E4πr2 ……….(1) 

From Gauss law the electric flux 

\(\varphi = \frac{1}{\varepsilon_0}\) total charge = \(\varphi=\frac{1}{\varepsilon_0}q\)

where q = total charge enclosed by the surface From (1) & (2) 

E4\(\pi\)r2\(\frac{1}{\varepsilon_0}q\) \(\vec E\) = \(\frac{1}{4\pi \varepsilon_0}\frac{q}{r^2}\hat r\)

b) electric field inside shell E = 0

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