A solid sphere of radius R has a charge Q distributed in its volume with a charge density
ρ = κra, where κ and a are constants and r is the distance from its centre. If the electric
field at r =R/2 is 1/8 times that at r = R, find the value of a.

Dear Anantika Shakya,

Ans:- The total charge is Q. Hence if we integrate the charge we shall get Q

Let us consider a thin spherical shell of thickness dr it'vol is 4∏r² dr and charge will be kra times of this hence From Gauss's law we get the value of the field at r=R/2 is

E 4∏(R/2)²=∫4∏kar³ dr=∏ka(R/2)^4

E=kaR²/16ε...............(1)

again the field at r=R is =Q/4∏R²ε...............(2)

The ratio is 1:8 hence we get

a=Q/(2∏kR^4)

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