electric-currentelectromagnetismmagnetic fields
At which point inside a circular current carrying loop the strongest? Some sources say it is at the centre and some say it is at the circumference.
And how can we mathematically prove this?
And also, is the magnetic field line through the centre of the loop exactly a straight line? Because this diagram shows so:
However if the field lines are concentric circles centered at the circumference then they can never become a straight line.
(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..
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
As a rough estimate for the behaviour, I have plotted a graph.
Taking a slice of the loop, the field from the left and right current elements fall off like 1/r^2, here I have modelled the graph such that r is radius of the loop, and x is the displacement from the origin.
See how inside the loop, the magnetic field is actually weakest in the center, and approaches infinity after we reach the radius of the loop
Outside the loop, the magnetic field falls off rapidly, which is the expected behaviour.
The behaviour of this graph is for infinitely thin currents, however I believe the generalisation to volume currents is similar, with this function having a domain restriction once we reach the wires radius to avoid infinite magnetic fields.
I could be wrong, but I am fairly confident that this is a good approximation (obviously ignoring there is a full circle, not 2 current elements). Focusing on the rough shape, and not the actual numbers atleast.