Total power radiated from two dipoles (electrodynamics)

In summary, the problem involves calculating the total power radiated by two oscillating electric dipoles placed next to each other on the z axis, using the superposition principle and the Poynting vector. The resulting factor for the total power radiated is ##\omega^2 d^2/5c^2##, showing a reduction in power due to interference between the two dipoles.
  • #1
CAF123
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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


[/B]
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!
 
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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##.
 

Related to Total power radiated from two dipoles (electrodynamics)

1. What is meant by "total power radiated" from two dipoles?

The total power radiated from two dipoles refers to the combined electromagnetic energy emitted by two dipole antennas in all directions.

2. How is the total power radiated from two dipoles calculated?

The total power radiated from two dipoles can be calculated by summing the individual power radiated by each dipole, taking into account their relative positions and orientation.

3. What factors affect the total power radiated from two dipoles?

The total power radiated from two dipoles is affected by the distance between the dipoles, their orientation, the frequency of the electromagnetic waves, and the material properties of the surrounding medium.

4. Can the total power radiated from two dipoles be increased?

Yes, the total power radiated from two dipoles can be increased by optimizing the distance and orientation between the dipoles, using higher frequency waves, and utilizing materials with higher conductivity.

5. What is the significance of understanding the total power radiated from two dipoles?

Understanding the total power radiated from two dipoles is important in designing and optimizing antenna systems for various applications, such as communication, radar, and broadcasting. It also helps in predicting the coverage and efficiency of the system.

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