Sorry if this question is stupid, but I really need to understand this. I read that in a straight simple wire, the magnetic field lines are concentric about the wire, and the circles become larger as we move away from the wire. Now lets think about a circular wire. All I can think about is the wire making many of these circles around the curve, kind of like a spring. But if we look at the diagram of the field lines around a loop, they gradually become straight, and through the centre of the loop, the field line is perfectly straight. What is the reason behind this?
[Physics] the reason behind the shape of the magnetic field lines in a circular loop
electromagnetismmagnetic fields
Related Solutions
First doubt: Why do magnetic field lines form closed curves?
The premise is false!
Take the following image I generated as an example. The black circles here are two current loops arranged haphazardly. The blue line is a single magnetic field line, plotted for a really long length. It's still going, and it isn't ending any time soon.
The only statement of importance is that $\nabla \cdot \vec{B}=0$. This can be interpreted differently: the divergence of a vector field at a point can be approximated by the flux into a very small sphere of volume $V$ at that point:$$\nabla \cdot \vec{B}=\lim_{V\to 0}\frac{\oint_S\vec{B}\cdot d\vec{a}}{V} $$ ($S$ denotes the surface of the sphere volume $V$ centered at the point in question, and $d\vec{a}$ denotes a vector area element). Therefore, if a magnetic field line penetrates the tiny sphere and ends, and has some magnitude, then $\nabla \cdot \vec{B}\neq 0$ and you've violated a Maxwell law!
But a magnetic field line can actually end. For example, imagine two single loop solenoids on top of each other, pointing in opposite directions. As derived on this page, we might have:
$$B_z=-\frac{\mu_0 R^2 I}{2((z-a)^2+R^2)^{3/2}}+\frac{\mu_0 R^2 I}{2((z+a)^2+R^2)^{3/2}}$$
At $z=0$, the field is zero. At $z<0$, the field is positive and along the z axis. At $z>0$, the field is negative and along the z axis. So clearly the field line heads towards zero, but never reaches it.
Second doubt: Why do we say that the strength of the magnetic field is more where the lines are closer together?
The following page defines $B=\sqrt{\vec{B}\cdot \vec{B}}$ and $\hat{b}=\vec{B}/B$, and proves that as you walk along a field line:
$$\frac{dB}{d\ell}=\hat{b}\cdot \nabla B=-B \nabla \cdot \hat{b}$$
If the field lines are converging then $\nabla \cdot \hat{b}<0$ and so $B$ is increasing in magnitude, and if the field lines are diverging then $\nabla \cdot \hat{b}>0$ and so $B$ is decreasing in magnitude. So there's your vector calculus proof.
J.D Callen, Fundamentals of Plasma Physics, chapter 3
Third: Why do iron fillings acquire exactly the design of the magnetic field?
This is more complicated. Each iron filing forms a little magnet that attracts its neighbors, so the iron filings can't fill up space and instead join end to end in directions induced by the magnetic field. So they form lines. Which field lines are chosen depends on the whole, ugly dynamics of the situation.
Last doubt: The diagram of the magnetic field lines that we see (the 2D diagram with many curves), is that diagram 3D in reality?
Yep, Maxwell's equations in their vector calculus form work only in 3D, so the lines you get, in general, are three dimensional lines.
I have a simple argument in mind. for an infinitely long wire carrying current in the +X direction, the magnetic field at a given distance r from the wire, has 2 components(suppose) radial and tangential. Consider specifically the radial component and assume it points outwards(or inwards). Now, switch the current direction to -X. The radial component should also switch directions. But notice that this current is no different from what it was before, if you just turn the wire around(i.e) watch the wire from the other side, the situation is exactly similar to what it was before, yet the radial field direction has switched. This is an impossibility, proving that the radial field must be zero, leaving behind only circles, as the possible field shape.
EDIT: CASE 1: from side A you see the current moving to the right. and the magnetic field is (suppose) radially outward. Now you reflect yourself about the wire and reach side B. You see the current moving to the left now, but the situation is essentially same(just observing from 2 sides of a wire), so the radial field is still outward.
CASE 2:you return to side A, and now reverse the current(it now travels left), so the radial component must switch(inward). But havent you already seen this current? A current flowing left was encountered in CASE 1 from side B!, where you observed the radial field to be essentially outward. But now you observe it to be inward. CONTRADICTION. thus no radial field can exist.
Best Answer
(source: gsu.edu)
Do you know a straight line is a circle with infinite radius.
Similarly at centre you get a circle with infinite radius.
When wire is bent in form of circle, magnetic field is no more concentric circles. Electric field at centre of ring is much stronger than outside. This causes field to form ovals.
The oval in centre can bend in neither direction, so it forms a circle with infinite radius, i.e. a straight line..