A charge Q is distributed uniformly within the material of a hollow sphere of inner and outer radii r1 and r2 figure. Find the electric field at a point P a distance x away from the centre for r1 < x < r2. Draw a rough graph showing the electric field as a function of x for 0 < x < 2r2 figure.



Given:


Amount of charge distributed within the hollow sphere=Q


Inner and outer radius of hollow sphere=r1 and r2


Volume charge density of hollow sphere is given by



….(i)


To find the electric field at a point P at a distance x away from the centre consider a spherical gaussian surface of radius x()



By symmetry all points on this surface are equivalent and electric field at all points have same magnitude and is in radial direction


Therefore flux through this surface is given by


..(ii)


Charge enclosed by this surface is given by




…(iii)


We know that,


By Gauss’s law, flux of net electric field (E⃗ ) through a closed surface S equals the net charge enclosed (qin) by the surface divided by ϵ0



Using gauss’s law and eqn.(ii) and (iii)




This electric field is directly proportional to x for r1<x<r2


For


Considering a similar gaussian spherical surface of radius x such that


Charge enclosed by this surface =Q


Total electric flux through this surface =


Using gauss law we can write



The graph showing electric field as a function of x is shown as follows



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