Maxwell’s equations for oscillating electric dipole

[humo90]

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 = -ω²μ_{0}p_{0}(4∏r)^-1sin(θ)cos[ω(t-\frac{r}{c})]\hat{θ} (Griffiths 11.18)

and

B = -ω²μ_{0}p_{0}(4∏cr)^-1sin(θ)cos[ω(t-\frac{r}{c})]\hat{\phi} (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, p_{0} is the maximum value of dipole moment, μ_{0} is permeability of free space, t is time, \hat{\phi} is direction in azimuthal angle, and \hat{θ} 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 \hat{r} direction which may be the volume current density term in 4th Maxwell's equation (Ampere's and Maxwell's law).

 

[rude man]

Umm -they don't? Whose equations?