A solid non-conducting sphere of radius R carries a non-uniform charge distribution with charge density?

ρ=ρ1 (r/R) where ρ1 is a constant. Show that (a) the total charge on the sphere is Q=π(ρ1)(R^3), and (b) the electric field inside the sphere is given by E=(Qr^2)/(4πε_o R^4 )

ε_o is epsilon with supscript o. the permittivity of free space.

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  • 1 decade ago
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    a)

    The charge per unit volume is ρ=ρ1 (r/R) where r is the distance from the center.

    Divide the entire sphere into infinite concentric spherical shells of thickness dr.

    Consider one such shell with radius r.

    So

    volume of the spherical shell = 4πr^2 dr

    Amount of charge on this shell

    dq = volume*charge density = 4πr^2 ρ1 (r/R) dr

    Integrate from r = 0 to r = R:

    Q = π(ρ1)(R^3)

    b)

    Let the electric field at a distance r from the center be E.

    Consider a Gaussian Surface t o be a sphere of radius r.

    Charge enclosed by the surface can be calculated as in a. Integrate from r = 0 to r = r.

    q = π(ρ1)(r^4) / R

    Consider a small area element ds on the surface.

    E and ds are parallel everywhere.

    So

    E.ds = Eds

    d(phi) = Eds

    Integrate :

    Phi = EA = E (4πr^2)

    (Integral of ds is the surface area of the sphere.)

    From Gauss's Law :

    q = Phi * ε_o

    or

    E = ρ1 r^2 / (4 R ε_o)

    Substitute ρ1 = Q / (πR^3) from a.

    E = (Qr^2)/(4πε_o R^4 )

    Hope this helps.

    your_guide123@yahoo.com

  • 3 years ago

    Non Uniform Charge Density

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