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10 July, 14:59

A solid uniformly charged insulating sphere has uniform volume charge density p and radius R. Apply Gauss's law to determine an expression for the magnitude of the electric field at an arbitrary distance r from the center of the sphere, such that r < R, in terms of rho and r

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  1. 10 July, 15:23
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    electric field E = (1 / 3 e₀) ρ r

    Explanation:

    For the application of the law of Gauss we must build a surface with a simple symmetry, in this case we build a spherical surface within the charged sphere and analyze the amount of charge by this surface.

    The charge within our surface is

    ρ = Q / V

    Q ' = ρ V '

    The volume of the sphere is V = 4/3 π r³

    Q ' = ρ 4/3 π r³

    The symmetry of the sphere gives us which field is perpendicular to the surface, so the integral is reduced to the value of the electric field by the area

    I E da = Q ' / ε₀

    E A = E 4 πi r² = Q ' / ε₀

    E = (1/4 π ε₀) Q ' / r²

    Now you relate the fraction of load Q 'with the total load, for this we use that the density is constant

    R = Q ' / V' = Q / V

    How you want the solution depending on the density (ρ) and the inner radius (r)

    Q ' = R V'

    Q ' = ρ 4/3 π r³

    E = (1 / 4π ε₀) (1 / r²) ρ 4/3 π r³

    E = (1 / 3 e₀) ρ r
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