Poisson's Equation, using Fourier Transforms!!

Hi! Today we are going to solve Poisson's Equation using Fourier Transforms. This equation gives us the electric potential when we have some charge density in space. I know the video is long, but don't worry, the problem is not hard or exremelly long, the reason is that I made an introduction about the Fourier Transform, Maxwell's Equations, Poisson's Equation, and that, as always, I wanted to explain everything with a lot of detail so nobody gets lost. So, if you are just interested in the answer or on some certain part, you can just go to the time stamps of the video, or even you can put it in x2 speed.

The problem is really interesting and the answer (spoiler alert) is actually a really well known result, which we learn when we start taking electromagnetism courses.

Here are the time stamps for the video so you can navigate it faster:
0:00 Introduction: Fourier Transform and its Properties
8:30 Multivariable Fourier Transform and its Inverse
13:55 Statement of the Problem we want to solve
15:22 Poisson's Equation: Derivation from Maxwell's Equations
26:10 We can start solving the problem!
26:50 1. Boundary Conditions
29:20 2. We take the Fourier Transform of the equation
33:20 3. Convolution Theorem
39:06 4. Inverse Fourier Transform of h(r): We go back to the (x,y,z) world
40:40 5. Choosing a new Coordinate Axes
45:05 6. Dot product between r and k
46:45 7. Passing to Spherical Coordinates
51:00 8. Solving the Integral
1:02:05 9. Expression for h(r)
1:02:50 10. Expression for the Potential
1:08:30 11. Result: Final Expression for the Potential

I hope you enjoyed the problem and see you in the next video!

Видео Poisson's Equation, using Fourier Transforms!! канала Sandro’s Space