Similarity between magnetic field created by a straight wire and wire loop 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?
 A: 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!!
