Jjackson said:
I had a long think about the Biot Savert law, and what I can understand from it is the following: It pertains to a length of wire that is so small it can be approximated to be a point
the 1/4pir^2 factor is because at a distance r, all of the flux is being spread across the surface of a sphere of that surface area, hence the flux density should be divided by that factor. the Sin theta is to do with direction of the field with respect to the angle to the point - relevant for solenoids. The I is proportional to the flux produced by the field, and the permeability is a factor to scale it to our defined units. the part where I am sort of lost is the dl part... I am not really sure what it means or what it is talking about.
As a sidenote I have not studied Gauss Law, although if you have a good link to an explanation or if you would like to explain it yourself I would be very appreciative.
I don't really know anything about special relativity or what it is, I have only just started studying magnetism.
Well I'll help you let me type.
Okay.
First of all its a law for currents and currents only. These must not be finited currents otherwise they would be observed as a row of sinlge moving charges. You'll see later on that you can't use biot-savart on a single charge. I'm going to explain biot-savart, as if it were for a straight line of charges, but you could generalize it too any wire with moving charges, there is just a geometrical difference that is taken up in the integral. (Explanation: It works for a straight line, you could chop up the straight line into miniature pieces and reattach them in a curvy way so you don't have a straight line. But if you look at every small chopped up piece as a inifintely small part of a straight line, it will work out.)
Now, the real source of biot-savart is special relativity. When you look at a row of charges, that are moving very fast, you see a magnetic field éh. When you move along the charges, you see a standing still uniformly charged row of charges. This will create an electric field. You probably already saw how to calculate the electric field at a distance R from a uniformely charged and infinitely long row right?
So:
Standing still: Electric field + magnetic field.
Moving along: Electric field.
Now that's funny isn't it? Conservation of energy states this is impossible. We're both looking at same thing, we just move at different speed. And the energy of the observer can hardly influence the energy of the object being observed right?
Energy of electric field standing still + magnetic field = energy of electric field moving along.
This is correct. Now why is that the electric field increases when moving along? Quite simple. But if you imagine, a case where you had a infinite row of charges, like said before, but they're all standing still, and then you start moving really fast alongside of them. This is exactly the same case as before, its actually indistinguishable, and then it is:
Electric field standing still = electric field moving + magnetic field.
This is the same thing, just looked upon from a different perspective.
Now how is this possible? Well it's quite simple. When two observers move really fast from each other, they're experience of reality space and time changes. These are the famous things einstein introduced, like that time goes slower when you go really fast compared to the speed of light (and we can postively say that electrons in a conducting wire can go really fast, not extreme, but pretty fast to catch up with.). And also space gets distorted. Now a magnetic field, is something that exists when charges move fast, and the faster the stronger. If there is no electric field/charge when the charge is standing still, there will be no magnetic field either. So they are essentially linked. A magnetic field is some sort of pseudo-electricfield. Now how does this come to exist? Well, through some complicated formula's which are called lorentztransformations. You could essentially see them as sort of converters from fast observer to slow observer and the other way around.
Now that you understand what magnetism is, ill go more into specific on biot-savart. When you have a uniformly lined up amount of charges, with zero speed, but infinitely long, you can easily resolve the electric field through integrating using coulomb's law. You will find a result, which only depends from the distance to the pillar of charges. Which is quite logically, due to the symmetery. It would be strange if it would vary if you moved up and down, because its infinitely long, and the top will always be the same as the bottom.
Since it only dependent from the distance to the line of charges, this gives a very great advantage for the "transformation formula's" to fast observer i talked about earlier. You could easily transform them into biot savart's law, to find what the compensation of the magnetic field due to the space/time distortions really are. You could then generalize this to replace the line of currents to something wider, like a wire, with an amount of amps. That's just terminology, but it's the same thing basically. (You will see later on, that if you have a cilindric conductor, like a wire, you can pretend its running through the heart of the wire, if all current is evenly distributed, just like you can look at a uniformly charged ball as a point charge.)
Now, you might be thinking, why are we doing all this for charged rows, and why can't we do it for just one charge? Well, you could use biot-savart for one charge, but you'll get an error. This error is due to the fact that a single charge works radially, and if it moves very fast, the distortions will become more complicated. That's what the great advantage of the infinitely long row of charges gives, it gives you symmetry, and only dependent on the distance to the object, and not from what angle your lookint at the object... When you look at a single charge, angles will start becoming important, because you can look at a ball from many different angles, however looking at a row is much more limited. So the distortions of the angles are non-existent.
I hope this explains magnetism a bit. (PS. This is my interpretation. I mightve made errors...)
