The magnetic field created by a straight wire is $B = \dfrac{\mu_0 I}{2 \pi d}$ where $I$ is the current flowing through that wire and $d$ the distance from the wire.
The magnetic field created by a current loop is $B = \dfrac{\mu_0 I}{2 R}$ where $I$ is the current flowing through the loop and $R$ its radius.
They look so similar, don't they? Does anyone know of an intuitive explanation behind that similarity?
We will start by looking at the solution to the problem of finding the magnetic field of a finite wire at a point $P$ which is $r$ distance away from the wire.The answer is :
$$B = \frac{\mu_0 i}{4\pi r}(cos\alpha+cos\beta)$$
where $\beta=the\ angle\ formed\ by\ the\ wire\ and \ the\ line\ joining\ an\ endpoint\ with\ P\\ \alpha=the\ angle\ formed\ by\ the\ wire\ and \ the\ line\ joining\ the\ other\ endpoint\ with\ P$
For a wire of $\infty$ length, $\alpha=\beta=0$, and putting this in gives
$$B = \frac{\mu_0 i}{2\pi r}$$
For a circular loop, using the Biot-Savart Law gives :
$$B = \frac{\mu_0 i}{2r}$$
We know that a finite wire can always be looped to a circle. Then, let us now loop the finite wire and find the magnetic field using the very first equation!
First we write $cos \alpha = \frac{h}{\sqrt{l^2+r^2}}$ where $l$ is the length of the wire and $h=d\ cot\alpha$
($h$ is actually the length of wire from the foot of perpendicular of P on the wire and the endpoint with the $\alpha$ angle )
Also, $cos\beta = \frac{l-h}{\sqrt{(l-h)^2+r^2}}$
Putting everything together,
$$B = \frac{\mu_0 i}{4\pi r}(\frac{h}{\sqrt{l^2+r^2}}+\frac{l-h}{\sqrt{(l-h)^2+r^2}})$$
When we loop this wire, the point $P$ becomes the centre of the circle, the length $l$ of the wire becomes $2\pi r$ and since all points on a circle are equivalent foots of perpendiculars dropped from P, so $h$ is just an arbitrary arc length, and for sake of calculation we set it equal to 0 (it can be anything, the answer doesn't change)
[You can see the last fact from the simple observation that all points on a circle can have a tangent perpendicular to the radius, and these tangents are the instantaneous direction of current at that point].
Rewriting our equation in light of these new facts :
$$B = \frac{\mu_0 i}{4\pi r}(\frac{2\pi r}{\sqrt{((2 \pi r)^2+r^2)}})$$
Cancelling terms and taking an $r$ out from the square root gives us :
$$B = \frac{\mu_0 i}{2 r}(\frac{1}{\sqrt{(4\pi^2+r^2)}})$$
So we have the magnetic field at the centre of a circular loop (equation 2 from top) with a correction term. The correction term is because the wire was finite. Working with an infinite wire results in no correction terms ($\infty$s are powerful cancelling factors sometimes).
(The actual reason why the correction term comes is because the situation of looping a wire can never be ideal, there has to be some input and output current wires for the loop, which does not allow the loop to be complete, but is not a problem for a straight wire... You can work it out)
So, its all in the mathematics. The form is similar because of the Biot Savart Law, and the underlying symmetries of nature !
Keep on thinking!
Cheers!!
