Question

# A solid non-conducting sphere of radius R has a nonuniform charge    distribution of volume charge...

A solid non-conducting sphere of radius R has a nonuniform charge    distribution of volume charge density ρ = rρs/R, where ρs is a constant and    r is the distance from the centre of the sphere.

Show that:

(a) the total charge on the sphere is Q = π ρsR3 and

(b) the magnitude of the electric field inside the sphere is given by the    equation

E = (Q r2 / 4π ε0R4)

here radius of sphere = R variable charge density p = r ps / R

as we know that the volume harge density

p = dQ / dV

so the total charge

Q = integration of [ p . dV ]

= integration of [ p . A . dR ]

= integration of [ ( r.ps / R ) ( 4 x pie x R2 ) dR ]

= ps x pie x 4 x R integration of [ r . dR ]

= ps x pie x 4 x R x ( R2 / 2 )

= 2 x pie x ps x R3 Cb Ans

(b) as we know that

first let a gauss surface of radius r

then flux fie = EA

and the flux is also given as fie = Q / eo

so EA = Q / eo

E( 4 x pie x r2 ) = pV / eo

so the electric field in the gauss surface

E = pV / ( 4 x pie x eo x r2 )

= pr / 3eo  r^

here p = Q / Vtotal

so E = Qr / ( 4/3 x pie x R3 ) x 3 x e0

E = Qr / ( 4 x pie x eo x R3 ) r^

here vector r^ = r / R

so the magnitude of electric field

E = Qr2 / ( 4 x pie x eo x R4 )