So in the last lecture, I have discussed that we have now a differential equation for our Victa potential
that looks just like the song equation and that we can carry over all of the results to easily determine
the vector potential A and A magnetic field B and the equation we will get is called the bias of our
law.
So we have here our differential equation that looks just like the song equation and the solution from
Electrostatic who looked like this.
So if you remember, we did it in the following way.
We considered a point charge or charged sphere, a single one, and then we calculated the electrostatic
potential fine and the electric field e and then we generalize this result by considering many charged
spheres, which gave us some of these individual potentials here.
And if we made our spheres smaller and smaller so that we ended up with a charged density instead of
individual charges, we transform the sum to the integral.
And this was the result.
The corresponding electric field looked like this because it is just.
Yeah, the little gradient of this file.
Now we take these results and carry it over for the differential equation of a and then we can also
determine, B, why this relation here?
So you can see the solution for the vector potentially looks very similar to the solution for the electrostatic
potential fire.
The difference is that instead of this constant one over Epsilon zero, we have now here Ammu zero,
which of course comes from this factor and this factor here instead of the scalar charge danceteria,
we have now the vector, which is the current density Jaafar, which is here now, the magnetic field
B is just the rotation of A instead of the minus gradient here.
So the solution is here that the magnetic field is Muzio over for PI and then we have the integral of
the current density time and then the crust product of this vector here and divide it by the distance
in the position vectors to the power of three.
And we integrate over the whole space in all cases.
And so this law is called the Bill Savala.
And it's really helpful because if we have some current some current loop, for example, we can easily
calculate the magnetic field by calculating this integral here.