The scalar potential induced by the current line (Скалярный потенциал, индуцируемый линией тока)

We need a distribution of U_C along the circle (рис. выше). Let's remember some properties of the magnetic field which is induced by the current. In principle we can consider a scalar potential, if we will exclude an area which is filled with a current from the problem domain – the first condition for existing of the scalar potential. The second condition: we should make a proper cut in the space. We can propose for example such problem domain. The border of the problem domain starts from right side. We can come to our system along this line, then will come around the circle, so that to exclude the wire from the problem domain and then continue moving to the left side. So the upper half space without this wire will be a space where we can introduce a scalar magnetic potential.

How to calculate it? If we will introduce here a radial coordinate system than we can say: there is a point at the end of the vector . Let’s decide: the potential on this horizontal line is equal to zero. In such case we can calculate potential in the point which located at the end of the vector .

Let us choose a point (line) of zero potential here

Arc length = r · The field intensity along the arc = const =

The potential in the point :

The dependence U_c is anti-symmetric with respect to the angular coordinate:

U_c = -U_c-

The current potential in the cylindrical coordinate system (Потенциал от линии с током в цилиндрической системе координат)

We need to find . To do this we need to do several transformations with angles  and φ trying link them together

The current potential in the complex plane (Потенциал от линии с током в комплексной плоскости)

Let’s look at the same system, but in the complex plane. There are two axes: y – imaginary and x – real.

We can write an expression for point (we get this point by parallelogram rule): . Also, we can write an expression for point : . Then we will take

On the other hand: ⇒

Let us introduce a parameter: (расстояние от центра цилиндра до точки b всегда будет больше, чем его радиус R ⇒ )

Finally:

Expansion of the current potential in the cylindrical coordinate system (Разложение потенциала от линии с током в цилиндрической системе координат)

We should consider (-b) as a complex number. Vector which corresponds to this number starts at zero and then it simply corresponds to the point that mirrored from the point b left. So (-b) is the point with the angle = π. ⇒

- Taylor series (Ряд Тейлора, x<1)

Applying this transformation: ,

(сюда подставляем все, что получили выше)

Finally:

Also we can found:

The potential which is induced by the currents is just proportional to the θ angle. So we have found an expression for this potential: