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Maxwell’s equations for oscillating electric dipole

  1. Apr 17, 2012 #1
    1. The problem statement, all variables and given/known data

    How do I show that our equations for the E- and B-fields for the oscillating electric dipole do NOT satisfy Maxwell’s equations?




    2. Relevant equations

    After approximations in retarded potentials, we have our E- and B-field as following:

    E = -ω2[itex]μ_{0}[/itex][itex]p_{0}[/itex](4∏r)-1sin(θ)cos[ω(t-[itex]\frac{r}{c}[/itex])][itex]\hat{θ}[/itex] (Griffiths 11.18)

    and

    B = -ω2[itex]μ_{0}[/itex][itex]p_{0}[/itex](4∏cr)-1sin(θ)cos[ω(t-[itex]\frac{r}{c}[/itex])][itex]\hat{\phi}[/itex] (Griffiths 11.19)

    Where ω is angular frequency for the oscillating charge moving back and forth, c is the speed of light, r is the distance where E and B are to be calculated, θ is the angle between dipole axis and the distance r, [itex]p_{0}[/itex] is the maximum value of dipole moment, [itex]μ_{0}[/itex] is permeability of free space, t is time, [itex]\hat{\phi}[/itex] is direction in azimuthal angle, and [itex]\hat{θ}[/itex] is direction in polar angle.





    3. The attempt at a solution

    I got divergence of B is satisfied (2nd eq. of Maxwell's), also, I got faradays law satisfied (3rd eq. with curl of E).

    I am stuck in the other two equations:

    For Gauss's law (1st eq.) I got div. of E does not equal zero, but maybe that because of the charge density. So, I am not sure whether this equation is satisfied or not, and I do not know how to show that.

    Also, the same argument For Curl of B. I got the same result for time derivative of E in addition to an extra component in [itex]\hat{r}[/itex] direction which may be the volume current density term in 4th Maxwell's equation (Ampere's and Maxwell's law).
     
    Last edited: Apr 17, 2012
  2. jcsd
  3. Apr 18, 2012 #2

    rude man

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    Umm -they don't? Whose equations?
     
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