A positive spherical charge core extends from the origin to a distance r0 and ha

Question

A positive spherical charge core extends from the origin to a distance r0 and has a uniform charge density p0 c/m3 negative charge lies in a shell between 2r0 and 4r0 and has a charge density py = p0r20/r2. Find the static electric flux density for all r > 0. What is the chargc unbalance? Positive charge is placed within a spherical volume of radius a about the origin, with volume density py = Ar C/m3 (A = constant). In addition, negative charge is distributed over a spherical shell of radius 2a, with a surface. density ps = -Aa2 12C/m2. Derive the expressions for the electric field strength and for the electric potential at all points.

Explanation / Answer

a)

for r< Ro

flux density, phi = Qin/eps0 = rho0*4/3 pi r^3 / eps0

for Ro<r<2Ro

phi = rho0*4/3 pi Ro^3 / eps0

for 4Ro>r> 2Ro

phi = rho0*4/3 pi Ro^3 / eps0 - (rho0*Ro^2/r^2 )* 4/3 pi (r^3-(2*Ro)^3) / eps0

for r> 4Ro

phi = rho0*4/3 pi Ro^3 / eps0 - (rho0*Ro^2/(4*Ro)^2 )* 4/3 pi ((4*Ro)^3-(2*Ro)^3) / eps0

b)

by Gauss theorem

integral (E.dS) = Qin/eps0

for a spherical element of radius r

for r<a

E1 * 4pi*r^2= Ar*(4/3) pi r^3/eps0

E1 = (A/3) r^2/eps0   

for a<=r<2a

E2 = (A/3)*a^4/(r^2*eps0 )

for r>= 2a

E3*4pi * r^2 = ((4/3)*a^3*A*a - (Aa^2/12)*4*pi*(2a)^2 )/eps0

= 0

E3 = 0 for r>=2a

assume V=0 at infinity

V3 = integral (E.dr) =0 for r>=2a

delta V2= -integral(E.dr) from r=2a to r=r

V2(r) - 0 = -(A/3)*a^4/eps0 *(-1/r + 1/2a) for a<r<2a

V2(r=a) = (A/3)*a^3/2*eps0

V1(r) - V(r=a) = -integral( E.dr) = -(A/3) r^3/(3*eps0)

V1(r) = (A/3)*a^3/(2*eps0) - (A/3)(r^3/(3*eps0)

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