Maxwell's equations: E, B and distance

[Dunnis]

                P                   
                |
                | r
                |               
----------------A------------------wire W--->

Q: Wire positioned along x-axis has steady current of 1 ampere, solve for E(r) and B(r).


What Maxwell's equations are supposed to apply here, and how to make derivation so to get them in this format where the magnitude is evaluated as a function of distance?

 

[Born2bwire]

Well, you can get the electric field simply from Gauss' law . The magnetic field can be derived using Ampere's Law. You would have to use their integral forms.

 

[Dunnis]

Well, you can get the electric field simply from Gauss' law . The magnetic field can be derived using Ampere's Law. You would have to use their integral forms.

What version, 'free charge' or 'total charge'?

$$\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A = Q_{f}(V)$$ -OR- $$\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf E\;\cdot\mathrm{d}\mathbf A = \frac{Q(V)}{\varepsilon_0}$$


$$\oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t}$$ -OR- $$\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \varepsilon_0 \frac {\partial \Phi_{E,S}}{\partial t}$$


I actually need to use differential form as I need to find the solution for what is called infinitesimal segment of wire "dl" in a single instant in time, and I can integrate those "pieces of a segment" later depending on geometry, that was the plan anyway - so is there any particular reason why not to use differential form?

In any case I could not find anything on the internet about any of this, so can you just print down the final solution for E(r) and B(r), and if possible point some place where the derivation is explained?

 

[Born2bwire]

Free charge, total charge, doesn't matter you get the same answer. The key here is to use the symmetry of the problem to make assumptions on the form of the electric field.

You should use the integral equations. Just make use of delta functions to effectively grab the infinitesimal contribution. Still doesn't matter because of the symmetry of the problem because the problem is invariant in your d\ell direction.

I'm not going to give you the answers. This is actually a very simple problem and you should be able to work this out or at the very least find the appropriate equations yourself.

 

[jtbell]

Hyperphysics is usually a good place to look for stuff like this on the Web.

http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon

Note the sections on Gauss's Law and Ampere's Law. They discuss the specific applications that you're interested in.

 

[jtbell]

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html#c1

 

[Dunnis]

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html#c1

Thanks, but I do not see how did they get that, that's impossible. Biot-Savart law equation is supposed to stay in integral form - ITS FULL FORM, so we can actually apply it to different scenarios with different lengths and different angles. What they did is to derive and specialize this general formula which was supposed to stay general, i.e. to be applicable to any problem.

 \                 e-------f
  \               /        |  h-----> wire 2
   a-------b     /    m    | / 
            c---d    /\    g/
                    /  \
     k-------------l    n--------------> wire 1
    /
 --j

$$\mathbf{B} = \int\frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat r}}{|r|^2}$$

That equation is not supposed to be derived symbolically but integrated numerically. Note that you can not enter any lengths and angles in this "other" equation:

$$\mathbf{B} =\frac{\mu_0I}{2\pi*r}$$

This "quasi-biot-savart" equation can not be a product of integration, integrals are supposed to be evaluated numerically with any new given setup, and for that we need the full equation in its full form so we can input all the parameters and apply it generally to any situation and geometry. I do not think I can use that equation to solve my problem (diagram above), I simply have no way to input any segment lengths and define their geometrical orientation.


Now, the reason I'm asking for these equations is that I have seen a lot of different ones that claim to do the same thing, so all of them are simply wrong, but one. Which one? I can use Biot-Savart law in its full form, but I have no experimental data to verify my results and I am baffled that there are so many different equations for this and they all predict different results while some of them do not seem to even apply at all. I want see where do these equations come from and who is making them up. -- Actually, the best way to help me, would be to point some experimental data so I can then see for myself what equations make correct predictions and which ones came from the Wonderland.