Calculating Force on Particle Using Maxwell Stress Tensor

[the keck]

Consider a particle with charge q in an static, homogeneous electric field. Using the fact that the net force on the particle in the surface integral of the Maxwell Stress Tensor, and assuming the surface is a sphere around this particle:

a) Find the net force on the particle (This part I can do)

b) Show that this approach can be applied even if the external magnetic field is not homogeneous and time dependent.

F = Int (T[ij] S)

I am not really how to go about doing the second part. Do I break up the electric field into its vector components?

Thanks

Regards,
The Keck

 

[Jhero]

Science Team

Dear Keck Science Team,

Thank you for this interesting question. Yes, you are correct in thinking that you will need to break up the electric field into its vector components. To show that this approach can be applied even if the external magnetic field is not homogeneous and time dependent, you will need to use the full Maxwell Stress Tensor, which includes the contributions from both the electric and magnetic fields.

To do this, you will first need to write out the full Maxwell Stress Tensor, which has 9 components. Then, you can use the surface integral formula given in the forum post to calculate the net force on the particle. This integral will involve summing the products of the Maxwell Stress Tensor components and the surface normal components over the surface of the sphere.

Since the Maxwell Stress Tensor takes into account both the electric and magnetic fields, this approach can be applied even if the external magnetic field is not homogeneous and time dependent. The key is to make sure that you include all the necessary components in your calculations.

I hope this helps. Please let me know if you have any further questions.