# Radiation from stationary changing charge

I wish to calculate the radiation from a surface charge density excited by some incident light. i am aware that the larmor formula assumes a constant quantity of charge multiplied by its acceleration squared as the source of radiation. my question is, would it be equivalent focus on a single point and calculate $$\ddot{q}$$? another version of the larmor formula is in terms of $$\ddot{p}$$. could I not then keep the dipole the same size and simply oscillate the charge quantity to get $$\ddot{p}$$ = $$\omega^{2}$$$$\ddot{q}$$ d ~ $$\ddot{\sigma}$$ (where sigma is 1D charge density)? it seems the two situations should be somehow equivalent. looking forward to some insight

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Staff Emeritus
2019 Award
It's best to solve these kinds of problems in terms of multipole moments.

the problem is being solved numerically due to the complexity of the geometry involved (I am modelling light scattering off a metallized AFM tip above a dielectric substrate). I have the real and imaginary part of the displacement field, D, as a result of a finite element simulation. I wish to know how my system will radiate (it is basically an antenna) as a function of the distance between tip and sample and the sample's complex permittivity.

My approach was to calculate the 2nd time derivative of the surface charge at both tip and sample interfaces. Is this wrong?

Thanks

Claude Bile
Good question,

My first thought is that an oscillating charge would produce EM waves over a range of directions (or k-space if you want to get technical). I think it is feasible to calculate the source function in the way you have proposed, but to calculate the emitted field, you would need to integrate the characteristic emission of an accelerating point charge over the area of your conducting surfaces; I'm not sure whether that is the most efficient approach from a computational point of view, though I must admit, my intuition at this level is not fantastic :).

Have you thought about trying to calculate the Poynting vector and going from there? That seems the most rigorous approach.

Claude.

Staff Emeritus
2019 Award
The thing about the Poynting vector is that it requires knowing H as well. I wonder if that's in the modeling.

The other issue with numerical modeling is the behavior at the "edge." Discontinuities (if there are any) can do funny things to the rest of the calculation.

Well the simulation can calculate power flow. But this is obscured by the incident wave being much bigger than the scattered one.

To get around that, I can calculate the scattered electric field and the scattered magnetic field (where the incident is subtracted out). From those I could reconstruct the Poynting vector.

However I seem to be getting decent results by calculating the polarization charge density at each interface so maybe the poynting vector approach is not necessary. So my more or less philosophical question remains: does $$\ddot{q}$$ radiate in an equivalent way as a point dipole at that location $$e^{2}a^{2}$$? If so where can I find a derivation of that?

Claude Bile