Magnetic Vector Potential

Its denoted by $\mathbf{A}$, is a fundamental concept in classical electromagnetism. It is defined such that the curl of $\mathbf{A}$ yields the magnetic field $\mathbf{B}$:

$\mathbf{B}=\nabla \times \mathbf{A}$

For a current loop, let’s consider a circular loop of radius $R$ carrying a steady current $I$ flowing in the counterclockwise direction when viewed from above. We can find the magnetic vector potential $\mathbf{A}$ at a point $P$ located at a distance $r$ from the center of the loop, in the plane of the loop.

The magnetic vector potential $\mathbf{A}$ at point $P$ due to a current element $d\mathbf{l}$ at a point $Q$ on the loop is given by:

dA=μ04πI dl∣r−r′∣dA=4πμ0​​∣r−r′∣Idl​

Where:

• μ0μ0​ is the permeability of free space,
• $I$ is the current flowing in the loop,
• $d\mathbf{l}$ is an infinitesimal element of current in the loop, and
• $\mathbf{r}$ and r′r′ are position vectors from the current element to the point $P$ and the element $d\mathbf{l}$ respectively.

Now, if the current loop lies in the $xy$-plane with its center at the origin, we can express $d\mathbf{l}$ in terms of $d\theta$ (an infinitesimal angle) as R dθ z^Rdθz^ where z^z^ is the unit vector along the $z$-axis.

Then, using the expression for $d\mathbf{A}$, the total magnetic vector potential $\mathbf{A}$ at point $P$ due to the entire loop can be found by integrating over the entire loop:

A=μ0I4π∮CR dθ z^∣r−Rcos⁡(θ)x^−Rsin⁡(θ)y^∣A=4πμ0​I​∮C​∣r−Rcos(θ)x^−Rsin(θ)y^​∣Rdθz^​

Where $C$ denotes the path of integration around