Ampere's Law physical significance

[member 392791]

Hello,

I am confused as to the physical significance of the dot product of B and ds. Why would we evaluate this scalar product. My textbook has it on here without any motivation for it.

Also, why is the analogous gauss' law used with flux, yet magnetic flux doesn't use amperes law?

 

[WannabeNewton]

Maxwell's equations for magnetostatics give us, assuming non-existence of magnetic charges, $\nabla\cdot B = 0, \nabla\times B = \mu_{0}J$ with the latter of the two being Ampere's law; the differential form of Ampere's law is, in my opinion, very visual so hopefully you can easily gleam some physical intuition just by looking at it. Given a portion of the current density within a region of compact support, we can use Stokes' theorem to write this as $\int (\nabla\times B)\cdot dA = \oint B\cdot dl = \mu_{0}\int J\cdot dA = \mu_{0}I_{\text{enclosed}}$ giving us the usual integral form of Ampere's law. As you can see, the dot product comes right out of Stokes' theorem applied to the differential form of Ampere's law. It gives us a relation between the magnetic field around some closed path to the enclosed current passing through the path.

Magnetic flux, under the assumption of no magnetic charges, will vanish identically as you can see by applying the divergence theorem to the other of the two magnetostatic equations: $\int (\nabla\cdot B )dV = 0 = \int B\cdot dA$. Why should it be related to Ampere's law in the manner of which you speak?

 

[member 392791]

I don't have enough math background as of right now to have a physical intuition for stoke's theorem or differential forms of ampere's law. I'm just taking the intro class for E&M. I was saying a physical significance, because I don't understand math as much, just something conceptually to soothe me.

I don't understand most of your post.

 

[micromass]

I don't have enough math background as of right now to have a physical intuition for stoke's theorem or differential forms of ampere's law. I'm just taking the intro class for E&M. I was saying a physical significance, because I don't understand math as much, just something conceptually to soothe me.

I don't understand most of your post.

Which part don't you understand? We can give you suitable references if you want.

 

[member 392791]

I don't understand what ∮B⋅dl means.

 

[sankalpmittal]

I don't understand what ∮B⋅dl means.

Well do you understand this sign : ∮ ?

I recommend you do a reading of this topic from a good textbook.

 

[physwizard]

I don't have enough math background as of right now to have a physical intuition for stoke's theorem or differential forms of ampere's law. I'm just taking the intro class for E&M. I was saying a physical significance, because I don't understand math as much, just something conceptually to soothe me.

I don't understand most of your post.

if you don't have enough math background, you will not be able to understand most of EM. you need to atleast know vector calculus. Isn't vector calculus a prerequisite for your EM course? Do you understand line, surface and volume integrals?

 

[member 392791]

The textbook uses the term line integral, but the way its treated, it seems to be the exact same thing as any other integral. So they'll say something like "do this line integral" then they integrate an expression just as normal. Yes I know what closed line integral means.

 

[e.chaniotakis]

First of all you need to understand that Ampere's law is Biot-Savart's law applied in problems where there is an underlying symmetry => thus the physics is the same , yet the math is more beautiful and elegant leading to well phrased physics results.

There is a heuristic way to present the topic :
the idea is simple:
Electric field, E has a source =>q
By analogy, magnetic field has a source => I

From the experiments conducted, we know that the magnetic field of an infinite wire cannot be radial (starting from the source and going outwards (as the electric field of a point charge).
We also know that it cannot be collinear with the current.
This leaves only the phi-component of the field => a.k.a the field is tangential.
(If you want a real derivation on this I suggest that you check R.Griffiths's Electrodynamics ,especially the part that explains magnetism from a relativistic point of view).

So far : B~I
This is not the only analogy between B and E.
the field is not uniform-> it decreases with distance (something like E)
Therefore we expect:
B~I/r

Now Ampere's law states the following.
If you go to a distance r from the wire, and draw a circle, the magnetic field will be the same in each point of the circle (since I,r are the same).
since B~I/r=> I~Br

To get the units right: μοΙ ~Βr

Now, I is a scalar, B is not, r is not either. If you take B X r you get a vector (non zero).=>Not what you want.
If you take B . r you get a scalar (zero).

Thus the best way to act is to take a unit vector dS tangential to the 'circle' where the field is tangential and do the dot product between B,dS.
Then you have:
B . dS =>scalar, non zero.

So far so good.

Now what remains is to think that the enclosed current , I, produces the magnetic field over the whole circle:
Thus you have to add the B.dS components to reproduce the current.
That's what the line integral does.

Therefore : μοΙ = ∮ Β.dS

Of course as I said this explanation is heuristic and has MANY bugs. At least I hope that I have explained the dot product properly.

Now, about Gauss's law and Ampere's law you are talking about two different things.
Gauss's law is based on the fact that there is an 'electric monopole' (the electric charge), whereas there is not such thing for magnetism.
Gauss's law states that the electric flux coming out from a surface around the source is equal to the one produced. No source => no way to apply such a law in magnetism.

 

[physwizard]

how about surface and volume integrals? do i have to keep repeating my questions?

 

[BruceW]

The textbook uses the term line integral, but the way its treated, it seems to be the exact same thing as any other integral. So they'll say something like "do this line integral" then they integrate an expression just as normal. Yes I know what closed line integral means.

Yes. It is just the usual line integral, but also, with the specification that the start and end point are the same. So you could just write this in the limits of the integral, but writing the little loop is just a more convenient way.

I was not sure what the \oint meant for a while, haha, I think the lecturer had forgotten to explain what it meant. I connected the dots when he kept drawing loops, rather than just curves. Also, I guess 'closed line integral' the hint is in the name.

edit: and also, that three dots thing that physicists use to say 'therefore'. So in my first term of first year undergraduate, I was like 'what the **** is that thing he keeps drawing on the board?...' I guess I should have just put my hand up and asked, but you know what it is like when you are new, and you don't want to look like you are not as smart as everyone else, haha good times.

 

[e.chaniotakis]

Dear physwizard,
you questions were:
- I am confused as to the physical significance of the dot product of B and ds. Why would we evaluate this scalar product. My textbook has it on here without any motivation for it.

- was saying a physical significance, because I don't understand math as much, just something conceptually to soothe me.

I tried to explain in conceptual terms the problem you asked. What is it exactly that you did NOT comprehend?

 

[member 392791]

Hello e.chaniotakis. Thank you for taking the time to respond (physwizard is not the OP)

From what I gathered from your post, the physicists tweaked the math to fit what they found experimentally. Explaining it in such a way as "well they had to do this because doing it gave the right results" doesn't really give me that soothing feeling that I am still seeking, more of just them doing it because it works.

 

[vanhees71]

That's how (theoretical) physics works! Everything we know in physics (and all natural sciences by the way) comes from observations in nature. Then one tries to find a mathematical scheme, where you start with as general assumptions as possible to predict any phenomena related with the subject of these fundamental natural rules.

In the case of electromagnetics this set of rules are the (microscopic) Maxwell equations in differential form. They are, to the best of our knowledge, the general laws, from which you can derive all phenomena related with electromagnetism, except the cases where quantum theory is necessary for a refined description. So Maxwell's equations are the general laws of classical electromagnetism.

Usually, they are written in terms of three-dimensional Euclidean vector calculus, relating the electric and magnetic components \vec{E} and \vec{B} of the electromagnetic field among each other (homogeneous Maxwell equations) and to the electric charges and currents (inhomogeneous Maxwell equations). Later, you'll learn that behind this complicated looking set of equations there are quite simple laws, when rewritten in relativistically covariant form. However, first you have to learn about three-dimensional vector calculus, i.e., the differential operator \vec{\nabla} and the integral theorems by Stokes and Gauß. Then, you'll see that all the manipulations with line, surface and volume integrals make a lot of sense.

 

[physwizard]

Dear physwizard,
you questions were:
- I am confused as to the physical significance of the dot product of B and ds. Why would we evaluate this scalar product. My textbook has it on here without any motivation for it.

- was saying a physical significance, because I don't understand math as much, just something conceptually to soothe me.

I tried to explain in conceptual terms the problem you asked. What is it exactly that you did NOT comprehend?

hi. okay, probably there is some confusion here. my question was not directed to you. my question was directed to the original person who asked the question - woopydalan. i had asked him if he understood line, surface, and volume integrals. he replied saying that he understood line integrals. he did not make any mention about whether he understood surface and volume integrals or not. that's why i asked him that question again and whether i needed to keep repeating my questions.
@woopydalan :
please make an attempt to answer all parts of the question. unless you communicate how much math you actually know, it is going to be that much more difficult to answer your original question.

 

[e.chaniotakis]

physwizard, woopydalan sorry ,messed up the names:P

Woopydalan, the fundamental 'observation' is that the magnetic field appears when we have charges with nonzero velocity and that it 'circles' the charges' trajectory. The circles grow sparser as distance from the moving charges increase.
Ampere experimented with various geometries of current loops to figure out an underlying symmetry - and he did - . Finally his law was expressed.

Same goes with Gauss's law for electrostatics. The fundamental observation is that there is a source (charge) which excerts a force on other charges. The force is proportional to the charge and grows weaker as distance increases.
Gauss's law is a convenient way to express the physics content in an elegant way when the problem has an underlying symmetry.

So here we have a 'crossroad'. Is it the physics that you seek to comprehend better, or the math that we use?