Total power radiated from two dipoles (electrodynamics)

[CAF123]

Homework Statement

The retarded vector potential $\mathbf{A}(\mathbf{r}, t)$ in Lorenz gauge due to a current density $\mathbf{J}(\mathbf{r}, t)$ contained entirely within a bounded region of size d is $$\mathbf{A}(\mathbf{r},t) = \frac{1}{4\pi c}\int_V' \frac{\mathbf{J}(\mathbf r', t')}{|r-r'|} dV',$$ where $t' = t-|r-r'|/c$ If this current density is due to a charge density oscillating harmonically with frequency $\omega \ll c/d,$ then at distances $r \gg d,$ $\mathbf{A}(\mathbf{r}, t)$ the current density is proportional to the real part of $i\omega \mathbf{p}e^{i\omega(t−r/c)}$ , where $\mathbf p \cos \omega t$ is the dipole moment of the charge distribution.

Now consider two oscillating electric dipoles of equal moment $\mathbf p$ that are placed next to each other on the z axis, a small distance d apart. The two dipoles have the same angular frequency $\omega$, but are out of phase with each other by $\pi$. Show that the total power radiated by this system is smaller than that of the single dipole by a factor $\omega^2 d^2/5c^2$

Homework Equations

E and B field formulae and the Poynting vector. Total power radiated given by $\langle P \rangle = \int \langle S \rangle \cdot d \mathbf A$

The Attempt at a Solution

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I have solved the time averaged power for the single dipole by computing the far field E and B from the given A and then computing the poynting vector before the total time averaged power. Since this second dipole is out of phase by pi relative to the other, then can write $\mathbf p_1(t) = \mathbf p \cos \omega t$ for the first one and $\mathbf p_2(t) = -\mathbf p \cos \omega t$ for the second. I am just a bit unsure of what to do next. I guess I am to superimpose the E and B fields from the two dipoles and then compute P but how does the superposition work? Thanks!

 

[mark726]

I would first clarify the problem by stating the equations and assumptions being used. The given equations seem to be using the Lorenz gauge and are based on the retarded vector potential. The problem also assumes that the current density is due to oscillating electric dipoles with a small frequency compared to the size of the bounded region.

Next, I would suggest using the superposition principle to solve the problem. This principle states that the total field at any point is the sum of the individual fields from each source. In this case, we have two dipoles placed next to each other, so we can calculate the total field at any point by adding the fields from each dipole.

To calculate the total power radiated by the system, we can use the Poynting vector, which gives the energy flux per unit area. We can integrate this over a closed surface to get the total power radiated. In this case, we can use a spherical surface centered at the origin to capture the fields from both dipoles.

Finally, we can use the given equation for the single dipole to calculate the power radiated by a single dipole and compare it to the total power radiated by the two dipoles. We should see a reduction in the power radiated due to the interference between the two dipoles, resulting in the factor $\omega^2 d^2/5c^2$.