# A paradox regarding the Larmor formula

According to the Larmor formula, the power radiated by an accelerated electron is $$P_0=\frac{e^2 a^2}{6\pi \varepsilon_0 c^3}$$.

Radio waves can be radiated from an antenna by accelerating electrons in the antenna. Suppose the number of accelerating electrons is $$n$$. The total power radiated can be calculated by two methods:

1. In the formula above for $$P_0$$, we can replace $$e$$ by $$ne$$. The total power is $$n^2$$ times $$P_0$$, giving $$P_2=n^2 P_0$$

2. For a scalar quantity like power, we can simply add (superpose) the power radiated by each electron to calculate the total radiation power. Therefore, $$P_2=nP_0$$

How can we solve this paradox?

You can not add the power of two waves. You can only add the electric and magnetic fields. Imagine that you have two charges producing e.m. waves with power $$P$$. If both waves have an electric field and magnetic field that destruct each other, you couldn't add the powers. Indeed, you would have that the powers have been counteracted. However, if e.m. waves have constructive interference, then you will have that the power will have been multiplied by $$4$$ as:

$$P \propto E \times B = 2E_0 \times 2B_0 = 4 (E_0 \times B_0)$$

To add to Marc's excellent answer. The electrons in the antenna are not all independently radiating energy. The radio waves generated by one electron will "push against" all of the other electrons and will change the amount of radio they themselves emit. They can either enhance or suppress one another's emission. This is the exact mirror image of the constructive/destructive interference of the emission Marc raises and a calculation will give the same result.

The same is true of electrons in different radio antennas. If two antennas are set up so that their emission destructively interferes in all directions (requiring some weird shapes most likely) then their combined power will be a lot less than either by itself. (Energy is conserved, the current pumped into the antennas in this case just feels less radiation resistance). Conversely, they could be set up to instead emit more than the sum of either by itself.

The effect is very important in quantum optics, where it is known by the names "subradiance" and "superradiance". In these cases people consider fluorescent atoms emitting visible light instead of classical radio antennas, but the emission process is very similar.

• The only way to get two antennas to destructively interfere everywhere is to put them in the same place. But you might consider a Yagi antenna, where all the passive dipoles combine to destructively interfere with the driven element in most directions. The radiation resistance of the driven element is indeed suppressed relative to a bare dipole, so you must feed it more current at less voltage to deliver the same power. But energy is conserved, so the destructive interference in most directions is matched by constructive interference in a particular direction, which is the intention. Apr 5 at 12:32

Radiated power/area is described by the poynting vector ( from which the larmor formula is derived).

Consider 2 independant sources of EM radiation.

$$\vec{S_{1}} = \vec{E_{1}} × \vec{B_{1}}$$

$$\vec{S_{2}} = \vec{E_{2}} × \vec{B_{2}}$$

The total poynting vector is therefore

$$\vec{S_{1,2}} = (\vec{E_{1}}+\vec{E_{2}}) × (\vec{B_{1}} + \vec{B_{2}})$$

(aka the total poynting vector of the total field)

Adding up the poynting vector from the two sources seperately, we can see that:

$$S_{1,2} ≠ S_{1} + S_{2}$$

Meaning we cannot just "add up" the power radiated from 2 independant sources.

Method #2 is not correct, because the Larmor formula is applicable only when some conditions are satisfied:

• the accelerated charge body is extended in space, i.e. not point particle;
• we define power as integral of Poynting vector over a closed surface far from the region of space where the radiation was created by the accelerating charged body;
• the only electromagnetically interacting body inside the surface is our charged body;
• EM field is given by retarded solution of Maxwell's equations with our charged body as the source (no other sources of radiation outside the surface or "free field" present).

Then the Larmor formula gives correctly the energy leaving the region per unit time.

In your antenna example, at least one of these is violated: there are multiple charged particles in the region, they interact with each other, and this interaction modifies the EM field. The resulting EM field is similar to what one big particle with charge $$ne$$ would produce. So method #1 is much better (although not completely correct either, because particles are not at the same point of space).