Deriving Magnetic Field Through Solenoid Using Amperes Law

[Jjackson]

[SOLVED] Amperes Law

Homework Statement

Deriving the magnetic field through a solenoid.

Homework Equations

dB = mu0/4pi x Idlsintheta/ r^2
where I = current dl= a tiny change in length dB = a tiny change in flux density
sintheta is the angle to the normal r is the radius mu0 is the permeabiability of free space.

The Attempt at a Solution

Long integration that we have been give by our teacher, I understand the math. My issue however is not to do with the formula manipulation, but rather with the actual concept underlying this calculation. A look on wikipedia indicated that the problem should in fact be solved using amperes law, however I am unfamiliar with amperes law and would love a conceptual rather than formulaic explanation of the theorem. Unfortunately my high school teacher does not himself understand the concepts behind the equations he teaches, but rather simply copies notes from a textbook, he himself admitting so. In the event that I have used the template incorrectly, or am going about something wrong, I do strongly appoligise, and would appreciate any guidance in going about my query correctly.

 

[Raze2dust]

Have you studied Gauss Law?? If yes you can just go through Amperes law in wikipedia it shouldn't be difficult.

 

[Swatje]

Biot savart and ampères law. It is usually introduced by Biot-savard at first as you state.

However, ampères law is only usuable for trajectories where you know there is a certain symmetry or constant magnitude on those trajectories. For a solenoid, this might be any parallel line with the coils, but this is only the case when the solenoid has infinite length. But these are things you'll see later on.

So in this specific case, as you state, with angles theta1 to theta2, it is advisable to use biot-savard. Since you arent sure it is constant.

Now, Biot-Savart is a tricky law, because it often is just placed there as an introduction to magnetic fields, on which they then try to deduce the magnetic field for a single charge, which brings the complications and so on. The way to understand biot-savarts law best, is to imagine it in a row of charges instead of a conducting wire. If you would run along the charges, at the same speed of the charges? What would you get? Will there be a magnetic field? Will something replace this magnetic field? Compare your results to what you would find when you stand still. Think about why this is possible, and what the implications of conservation of energy are.

You will automatically find more understanding of what magnetism is... Hint: it involves special relativity.

Then, if you understand the formula, you'll understand that the formula is valid for any conducting wire. (Then when you start moving the positive charges will start moving in the opposite direction.) And hence, this solenoid is just a mathematical application of the formula through integration.

 

[Jjackson]

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.

 

[Swatje]

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...)

 

[Swatje]

Okay, quick addition:

Electric field of a uniformely charged, infinitely long rod:

E = L'/(2*pi*epsilon-0*R')

where L' is the amount of charges per meter of the rod.

If you would transform this to a fast observer, or the current moving fast, you would get several distortions:

a) the amount of charges per meter will change, because the "distances change". So L' will become L.
b) The distance towards the charges change, but the formula is only dependent on the "shortest distance" R. This distance will not change, because of it being perpendicular to the direction of movement.

So our new formula will be:

E=L/(2*pi*epsilon-0*R)

And L will be fewer than L', because space "stretched out". So the rod kinda stretched out now its moving fast. So you have the same rod, but with less density. But due to conservation of energy, we will have a new field, the magnetic field.

There are transformation formula's for this, but they're pretty complicated if you haven't seen special relativity, but you would get:

B= [(mu-0)*L'*v]/[2*pi*R]

Which is exactly the same what you would get if you would simple do biot-savart on a straight conductor. (Note that L'*v = I)
(The only difference between a straight line of charges and a conductor, is that when you start moving along with the charges in a conductor, the positive charges will move oppositely and create a magnetic field of there own, to sustain the conservation of energy. In a conductor there won't be any electric field standing still or moving, so the compensation has to come from somewhere.)

 

[Jjackson]

Swatje thankyou very much, you have helped me a lot, I really appreciate it.

Question Solved.