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humo90
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Homework Statement
How do I show that our equations for the E- and B-fields for the oscillating electric dipole do NOT satisfy Maxwell’s equations?
Homework 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.
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).
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