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Math Help - divergence

  1. #1
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    divergence

    Hi I have this problem that I cannot find the solution:

    2 students propose the following 2 different flux densities as solutions to an electrostatics problem:
    D1= ε* ρ^2 ρ (ρ, φ unit vector)

    D2= ε*ρ^2 *(sin(φ))^2 ρ+ 3*ε*ρ*z*(cos(φ))^2φ

    Show that the volume charge densities that correspond to these two fields are identical.


    I tried to apply the divergence to D1 and D2 but I don't find the same response.
    So I dont know what else I can do here.
    Please can I have some help?
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  2. #2
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    Quote Originally Posted by braddy View Post
    Hi I have this problem that I cannot find the solution:

    2 students propose the following 2 different flux densities as solutions to an electrostatics problem:
    D1= ε* ρ^2 ρ (ρ, φ unit vector)

    D2= ε*ρ^2 *(sin(φ))^2 ρ+ 3*ε*ρ*z*(cos(φ))^2φ

    Show that the volume charge densities that correspond to these two fields are identical.


    I tried to apply the divergence to D1 and D2 but I don't find the same response.
    So I dont know what else I can do here.
    Please can I have some help?
    These are both in cylindrical coordinates, right?

    I'll call epsilon e, rho p, phi f, and let d/dp stand for the partial derivative with respect to p, etc.

    Since D1 is entirely radial:
    (grad)(dot)D1 = 1/p * d/dp[p*D1]

    = e/p * d/dp[p^3] = 3pe

    (grad)(dot)D2 = 1/p * d/dp[p*D2(p)] + 1/p * dD2(f)/df
    where D2(p) is the p component of D2, etc.

    = e/p * sin^2(f) * d/dp[p^3] + 3epz/p * d/df[cos^2(f)]

    = 3ep*sin^2(f) + 3ez*sin(2f)

    I'm assuming D is the displacement vector, so
    (grad)(dot)D = (charge density)

    Thus I would agree with you. The two charge densities are not the same.

    -Dan
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