# Maxwell’s equations for oscillating electric dipole

1. Apr 17, 2012

### humo90

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

and

B = -ω2$μ_{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.

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 $\hat{r}$ 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. Apr 18, 2012

### rude man

Umm -they don't? Whose equations?