For constant current, the magnetic field seems to be easy to calculate because of Biot-Savart Law. For a loop having varying magnitude of current (simplifying an electromagnet), however, it seems that you need to figure out retarded potentials from the current density vector and then the fields from the retarded potentials. Am I wrong in saying this? If incorrect, what should I do instead?
If I'm correct, I would like to know how to define parts of the integral $$\mathbf{A}(\mathbf{r},t) = \frac{\mu}{4\pi}\int{\frac{\mathbf{J}(\mathbf{r},t)}{\mathbf{r'}}}d\tau,$$ i.e. vector potential formula used for non-static sources, for a loop carrying a sine dependent current (simplifying the electromagnet, and assuming that each loop is infinitely thin). These are the problems I'm getting:
- The current density vector is: $$\mathbf{\vec{J}} = I_o \sin{(2\pi ft)}(R \cos{(2\pi f_2t)} \hat i +R\sin{(2\pi f_2t)}\hat j +f_2t \hat k)$$ having magnitude of $I_o\sin{(2\pi ft)}$ where f is dependent on input current. f2 here depends on the drift velocity of electrons in the conductor. Since I'm defining a magnitude and a direction for the vector, it makes sense right? Should I not have a time dependent position vector for J? Component of z axis represents completion of one loop at time period of flow.
- How do I define the τ that is supposed to represent the length of the loop? My first response was a simple repetition of $R \cos{\theta}\hat i + R \sin{\theta}\hat j$. But something does not seem right. I will have squared R in my answer, and the complete expression looks a little strange to me.
- I still have not touched upon the effect on expression due to retarded time. That is because I do not know what kind of effect it has in this case, since in each loop, all loop elements are at the same distance from the point of calculation. Some clarification would be helpful.
As I am writing this down, I think this looks like a complete mess. I am essentially alone in learning and solving problems such as these, due to various circumstances. Hence, any knowledgeable help to solve the problems I mentioned will be greatly appreciated.
Note: As a reminder, my goal is to find an expression of vector potential that will be used to figure magnetic (and induced electric) field of an electromagnet due to a sine current flowing through its conductors. I am simplifying it to a group of loops for ease of calculation.