darkenlighten

The issue is that you are not solving the integral properly. If you are not sure and would like to know how to solve it properly look at http://www.scienceforums.net/forum/showpost.php?p=554213&postcount=47 This will make it clear as to why you cannot say dl is 1 meter, but it is the integration variable. It is simpler to solve for the magnetic field first (especially so you can see the correct way to solve the integral) and then substitute it in the force equation.

And to be clear, I originally solved the magnetic field for an infinite wire using Ampere's Law out of Maxwell's equations, but used the Biot-Savart Law to show you where that double integral came from.

Okay so this is what I was able to do:

[math] {\mathbf{r}} = r \hat{r} = r ( cos\theta\hat{\mathbf{x}} + sin\theta\hat{\mathbf{y}}) [/math]

[math] {\mathbf{v}} = v (-sin\theta\hat{\mathbf{x}} + cos\theta\hat{\mathbf{y}}) [/math]

[math] \mathbf{a} = \frac{d\mathbf{v}}{dt} = v (-\dot{\theta} cos\theta\hat{\mathbf{x}} - \dot{\theta}sin\theta\hat{\mathbf{y}}) [/math]

1) [math] \mathbf{a} = - v\dot{\theta}\hat{\mathbf{r}} [/math]

2) [math] \dot{\theta} = \frac{d( \omega t)}{dt} = \omega [/math]

[math] \dot{\mathbf{r}} = \mathbf{v} = r \dot{\theta}(-sin\theta\hat{\mathbf{x}} + cos\theta\hat{\mathbf{y}}) [/math]

3) [math] |\mathbf{v}| = v = \sqrt{ (r \dot{\theta})^2((-sin\theta\hat{\mathbf{x}} + cos\theta\hat{\mathbf{y}}))^2} = r \dot{\theta} [/math]

Then using 1), 2) and 3 you get [math] a = \frac{v^2}{r} [/math]

Is this what you were wanting?

It would still be geometry/trig. What is your concern with it?

You did not solve this equation given by the BIPM:

http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf

 

[math]

\mathbf{F}_{12} = \frac {\mu_0} {4 \pi} I_1 I_2 \oint_{C_1} \oint_{C_2} \frac {d \mathbf{s_2}\ \mathbf{ \times} \ (d \mathbf{s_1} \ \mathbf{ \times } \ \hat{\mathbf{r}}_{12} )} {r_{12}^2}

[/math]

 

Can you show your work?

I did, the equation above is a derivation from the Force equation:

[math] \mathbf{F}_{12} = I_2 \int d \mathbf{l} \times \mathbf{B}_1 ;[/math] where [math] d\mathbf{l} = d\mathbf{s}_2[/math]

and [math]\mathbf{B}_1 = \int\frac{\mu_0}{4\pi} \frac{I_1 d\mathbf{l} \times \mathbf{\hat r}}{|\mathbf{r}|^2};[/math] where [math] d\mathbf{l} = d\mathbf{s}_1[/math] and [math] \mathbf{r} = r_{12}\hat{\mathbf{r}}_{12} [/math]

[math] \Rightarrow \mathbf{F}_{12} = I_2 \int d \mathbf{s}_2 \times \int\frac{\mu_0}{4\pi} \frac{I_1 d\mathbf{s}_1 \times \mathbf{\hat r}_{12}}{|\mathbf{r}_{12}|^2}[/math]

cleaning it up and we get:

[math]\mathbf{F}_{12} = \frac {\mu_0} {4 \pi} I_1 I_2 \int \int \frac {d \mathbf{s_2}\ \mathbf{ \times} \ (d \mathbf{s_1} \ \mathbf{ \times } \ \hat{\mathbf{r}}_{12} )} {r_{12}^2}[/math]

As you can see I solved for B and then used that in the [math] \mathbf{F} = I \int d \mathbf{l} \times \mathbf{B} [/math] in my example in post # 59.

Because it has already been done, refer to post #49 with the derivation (the math) that shows how to properly setup and integrate the Biot-Savart Law for a wire of length L and steady current I.

And so did I back on page 3, post #59 lol

Yes, and do you realize now we are actually solving the same problem from the very beginning with SINGLE ELECTRON. Yes, we are solving for INFINITESIMAL segments "dl", and that means we are solving for point charges again, only these magnetic fields in wires theoretically do NOT EXIST in front or behind their perpendicular plane, not in any of our equations anyway.

 

 

 

In two parallel wires, these "magnetic slices" do not interact with any other slices but only the ones of the other wire whose slices are in that same plane. There are no varying angles and no superposition here.

No this is not true. If you look at the derivation again.

[math] q \mathbf{v} = q \frac{d\mathbf{x}}{dt} = \frac{dq}{dt}\mathbf{x} = I \int{d\mathbf{l}} [/math]

This means the Current times the LENGTH of the wire, which is [math] x = \int_0^L d\mathbf{l} [/math] where L is the length of the wire. dl is not the length of the wire, you've stated that yourself, it is an infinitesimal piece of the whole length. You know vector calculus, so you should have no trouble understanding how a line integral works. And since dl IS a vector, it will have a magnitude and direction, which means it can change with respect to the distance away.

Therefore the contribution from the whole length is not confined to the perpendicular plane. I think the picture is confusing you. As swansont said, due to symmetry the field only points in the azimuthal direction, but is caused by the rest of the wire.

It's unfortunate you've misunderstood me for being arrogant, you might have stopped to think about what I've been posting and realized your mistake.

There is a fine line between arrogance and confidence. I have been studying this subject for almost 3 quarters now, it's not that I'm arrogant, it's that I know I am right. Just because you don't believe me does not make me arrogant, because you think that I think I'm right.

Also to make it clear:

[math] q \mathbf{v} = q \frac{d\mathbf{x}}{dt} = \frac{dq}{dt}\mathbf{x} = I \int{d\mathbf{l}} [/math]

I'm going to state this again...( I said this in a previous post)

Hopefully this will understand why you cannot look at just 1 element of dl, but the sum of all dl's (hence an integral)

At point "P" away from the wire there is a magnitude of the magnetic field that is caused by each segment of wire, not just by the segment below it, otherwise it would make it act no different than one moving charge, which is obviously not true.

For example: take multiple moving charges equally spaced apart traveling at the same speed (a steady current) and confine them in 1 dimension (a straight line). They all have their own magnetic field, but if you look at a point in space, the total magnetic field at that point will be due to the superposition of all of the moving charges magnetic fields. Since their magnetic fields depend on distance away, this correlates to the length of a wire.

No, I'm not talking about Biot-Savart law and imaginary scenarios, I'm talking about these four monsters that you supposedly DERIVED FROM MAXWELL'S EQUATIONS:

 

 

 

Have you derived these from Maxwell's equations and will you provide some reference finally?

Okay, though I've repeated myself. The first 2 equations, the E field I have there can be ignored, since we have already corrected that saying that the E field is zero, E = 0. The B field there is correct with the reference of the scanned pages of Griffith's text showing the full derivation using the Biot-Savart Law. I used Maxwell's equations to derive everything there, I don't need a reference using Maxwell's because my math is not wrong, so the reference I have is to show I have the correct answer.

As far as changing B field scenario, I do not have a reference since that was a scenario I made up, but since you do not believe the previous answer, you will not believe the last 2 I gave here. Which you apparently had a problem with a changing Magnetic field creating an Electric field...which is silly.

Considering where this argument brought us, that's a very good question. -- In relation to some constant distance from the wire it will be the same, that's clear from all these equations - magnitude of B field around wire depends only on current "I" and distance (1/r^2 at right angle), but in any case NOT on wire length - there is no such variable in Biot-Savart law equation.

No its not clear to you, because this is incorrect. The length contributes and yes there is such a variable [math] \int d\mathbf{l} [/math]. You claim to understand vector calculus but you have been wrong about this the whole time. At point "P" away from the wire there is a magnitude of the magnetic field that is caused by each segment of wire, not just by the segment below it, otherwise it would make it act no different than one moving charge, which is obviously not true. For example: take multiple moving charges equally spaced apart traveling at the same speed (a steady current) and confine them in 1 dimension (a straight line). They all have their own magnetic field, but if you look at a point in space, the total magnetic field at that point will be due to the superposition of all of the moving charges magnetic fields. Since their magnetic fields depend on distance away, this correlates to the length of a wire.

The thing is, magnetic field around wire in these equations is ONE field, it is not combination of many fields, but it's one field just like single electron has one electric field, only this magnetic field instead of being a sphere is cylindrical, and instead of originating from a point, its "origin" is a line mathematically constructed with line integral. -- So the 2m wire will have overall more magnetic potential, i.e. more magnetic energy as a whole, but measuring this potential some constant distance from the wire will be the same regardless if the wire is 1, 2 or 100 meters long.

Technically it is more than one field, due to the example above (about multiple moving charges).

You only add up force vectors when they are acting on the same object. Imagine there is a spring between the two wires whose compression will determine the force between the wires. Both forces will add up.

That is not what it means when it says the forces between the wires. Is one force not left and the other right, hence one negative and one positive? If you sum them, you will get zero.

Say current I1= 0.5A and I2=1.5A, will their attraction be the same as when they were both 1 Amps? Yes, but you would not know that if you calculated the force from only one wire.

No you are misunderstanding what it means "the force between the wires". What it means is the force that one wire feels due to the external magnetic field of the other wire.

Is there any question that I ignored or did not answer directly? It is you who still needs to provide some reference for those equations supposedly derived from Maxwell's, or you could simply say that you were mistaken.

Once again I have already provided reference, but you refuse to accept it, why?

Two Questions ambros:

1) Do you think that a wire of length 1m with steady constant current I will have a different magnetic field than a wire of length 2m?

2) When you are solving for, what you call, the total magnetic force, why are you adding up the individual forces from 2 different objects and why is one of them not negative?

Answering these straight forward will put us back on the correct path. If you choose to ignore them and not answer directly, this discussion is going to continue to be circular.

Ay, caramba! There are two forces there and they act on TWO DIFFERENT objects. The "net force", of course, is the one that will correctly calculate the relative displacement, and both of these two forces play part there, as illustrated, of course.

That doesn't mean I can't consider this whole system, but that is hardly the point. There is only 1 Force acting on each object and that force is 2 x 10^-7 N/m derived as I have shown.

Do you not understand "distance ®" is the SHORTEST PATH?

 

 

Do you not see what is 'angle', what is 'dl' and what is 'r'? Now, compare that with your source - it's not the same, and if it was then "your" result would be wrong. Your source does not talk about any case scenario I have given here, what it talks about is simply insane.

Um yea it does. It solved the infinite wire with steady current I using the Biot-Savart Law...And once again do you understand why you look at dl and r? Think of it this way, say you start with a piece of wire that is 1L long (that somehow has a steady current), it will have a certain magnetic field. Then look at another piece of wire that is 2L, it will have another magnetic field different than that of the wire of length 1L. If you agree with that, then you should realize that dl x r is not 1 and should refer to the correct derivation 14 posts before this.

Why don't you trust my source?

 

http://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law

 

http://en.wikipedia.org/wiki/Magnetic_field

 

http://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law

 

What part do you think states you are correct? Maybe this: - "The physical origin of this force is that each wire generates a magnetic field (according to the Biot-Savart law), and the other wire experiences a Lorentz force as a consequence."? Ha-ha.

While these sources are correct, you do not solve them correctly, which is the half the point here.

And from the source http://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law it shows what I am saying read it:

[math]F_m = 2 k_A \frac {I_1 I_2 } {r} [/math]

...and your equations, that's just hideous. How did that happen? Because you did not PROVIDE ANY REFERENCE, and I'm talking about these four monsters that you supposedly DERIVED FROM MAXWELL'S EQUATIONS:

 

 

Not quite.

On the contrary: You would still be wrong if that meant the net force, but the net force here is zero, and that is not what it means when saying the force between the wires. And you are off by a half on each and since you are working with a unit of 1m it being r^2 or r will not make a difference for this example.

This is the correct answer:

[math] \mathbf{B}_2 = \frac{\mu_0 I_2}{2\pi s} \hat{\phi} ;[/math] [math] \mathbf{F}_1 = I_1 \int{d\mathbf{l} \times \mathbf{B}_2} [/math] [math]\Rightarrow F_1 = \frac{\mu_0 I_1 I_2}{2\pi s} \mathbf{\hat{r}} [/math] per unit length

[math] \mu_0 = 4\pi\times 10^{-7};[/math] [math] I_1 = I_2 = 1 A;[/math] [math] s = 1m [/math]

[math]\Rightarrow F_1 = 2\times 10^{-7}N[/math] per unit length

One force is negative one force is positive, look at your own diagram.

So [math] F_{net} = F_1 + F_2 = 2\times 10^{-7} - 2\times 10^{-7} = 0[/math]

Please answer me again when I ask: Why do you not trust my source?

Also, where you got the picture from even states I am correct: http://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law

--- EXAMPLE Q1 ------------------------------------------------------

Wire along x-axis has steady current of 1 ampere, solve for E® and B®.

 

 

darkenlighten:

[ATTACH]2452[/ATTACH] [ATTACH]2453[/ATTACH]

 

Do you mean to say this is correct answer? Are these equations valid at all, is that correct derivation of Maxwell's equations? - Do you also mean to agree with him about Biot-Savart law and deny that actually this is the correct expression, where both the full and simplified version can be applied to this problem:

So as stated before, we already determined that [math] \mathbf{E} = 0 [/math] and the B field I gave there is correct.

This is not correct, reason being is that you are dealing with a wire and all of the pieces contribute towards the total magnetic field, therefore there has to be an integral to sum up all of the pieces, which also means that [math] d\mathbf{l} \times \mathbf{r} [/math] is not 1 but what is shown in my previous post with the reference from Griffith's text. So why do you refuse to believe a highly reputable source?

Can you solve this example? Can you CLEARLY write down the two equations derived from Maxwell's that actually can solve E® and B® for this scenario, and can you please use CORRECT NOTATION or point some reference?

 

--- CHANGING CURRENT ---------------------------------------------------

 

darkenlighten:

 

Do you mean to say these equations are correct? Do you mean to agree this is true: - "for a changing current there will be an electric field around the wire... E field exists due to a changing B Field"?

I've already gave enough information for why these are correct. I am using correct notation, as cylindrical coordinates are best, if you are not familiar with them, read about them and it will make things easier. Also I think the underlying issue is that you do not believe that a changing B field creates an electric field??? Which I don't understand considering that's electric motors work. A changing magnetic field induces an emf (electromotive force) which is a vector potential (a voltage) that corresponds to an electric field.

I'm sorry for being rash. This is just kind of frustrating, so I will probably stop here.

No. I already stated that was an example for a changing B field. Stop being a fool and read back through. I am done.

The reference in my previous post.

The former is known as 'ignorance', it qualifies the later as 'arrogance'. Even if you derived those equations properly they would still give DIFFERENT results than Biot-Savart and Coulomb's law, just because of *different* constants and distance relation.

Call it what you will, doesn't bother me any.

Wire positioned along x-axis has steady current of 1 ampere, solve for E® and B®.

 

 

COULOMB vs. "MAXWELL" by darkenlighten:

 

[math] E = {1 \over 4\pi\varepsilon_0}{Q_{net}\over r^2}[/math] -VS- [math]E = \frac{\mu_0 I}{4 \pi}\hat{r} [/math]

Lets start with this. First that is not what I gave for this specific example. If you would read back through, which I feel you fail to do. This was the example I gave for a changing magnetic field, where I was NOT constant. And we already determined for this E = 0 for a wire, due to its net charge, but this is not true if the B field is changing, which was my example.

BIOT-SAVART vs. "MAXWELL" by darkenlighten:

 

[math]B = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2}[/math] -VS- [math]B = \frac{\mu_0}{2\pi} \frac{I}{r}\hat{r}[/math]

 

E field at point 'r' can change with varying current, change in voltage, or moving magnets around the wire? -- If you think I do not represent Biot-Savart and Coulomb's law correctly for THIS PARTICULAR example and/or if you think I misinterpreted "your" equations, then go ahead and finally write the "correct" solutions already yourself, get rid of the 'Phi' and spherical coordinates so we can COMPARE the equations properly.

Yes it can, but not the E field that is due to an electrostatic charge, but due to changing B field, hence a changing current or moving magnet.

Stop wasting everyone's time, that's embarrassing. You are making it only worse for Maxwell because you do not have that term in there at all, and I just got it out because it equals to -ONE- for this particular case scenario. - Logically, and obviously from the illustration, in the case with wires the distance vector is always at 90 degrees to wires, so: dl x r = sin(90) = 1

 

[math]B®= \frac{\mu_0 I d\mathbf{l} \times \mathbf{r}}{4\pi r^2} = \frac{\mu_0 I sin(angle)}{4\pi r^2} = \frac{\mu_0 I sin(90)}{4\pi r^2} = \frac{\mu_0 I * 1}{4\pi r^2} = \frac{\mu_0 I}{4\pi r^2}[/math]

You are wrong.

So lets take a look at the derivation and as to why you are wrong:

Taken from "Introduction to Electrodynamics" by David J. Griffiths Page 216-217

Magnitude of E filed at distance 'r' can change with frequency?

Looking at Faraday's equation, you can see that an Electric Field can exist if there is a changing B Field. Therefore, in MY example for a changing current [math] I = I_0 cos(\omega t )[/math], there will be an electric field around the wire due to its changing magnetic field.

Stop hallucinating equations and dreaming up nonsense. We are COMPARING two different sets of equation by trying to apply them on the same scenario and get them in THE SAME FORMAT. -- You have lost all your credibility, I will not consider any more empty arguments from you without some reference provided and equations written in proper and comparable format.

Hate to break it to you but [math] \hat{r} [/math] is a polar coordinate, hence a cylindrical coordinate, so we are in the same format, you just don't see it.

Magnitude of E filed at distance 'r' can change with changing B field?

 

What "E field" are you talking about? Where and how do we measure it?

Once again due to the changing B in MY example.

I have gave another example where an E field exists due to a changing B Field. And once again in YOUR example with constant I, E = 0 .

What equation is wrong, "Biot-Savart" or "Maxwell"?

 

http://en.wikipedia.org/wiki/Biot-savart

 

If you mean to disagree be specific and write down EXACTLY what you think is the correct solution for this example given by BOTH formulas so we can compare, and yes provide some reference, online preferably. - Meanwhile, I will repeat that this IS the solution (simplified) given by the Biot-Savart law for the B field in relation to electric current and distance from a straight wire, so to be applied to the given example: B®= m0/4Pi * I/r^2

I am saying what you have, [math] \mathbf{B}®= \frac{\mu_0 I}{4\pi r^2} [/math], is wrong. You cannot ignore the [math] d\mathbf{l} \times \mathbf{r} [/math]. So I will repeat, that is not the simplified solution. Please refer back to my earlier example showing what the actual solution is. I don't feel like scanning my book again, but trust me it is a legitimate source. Heck it's even what wikipedia has sourced.

Take the example above, the one we are already talking about, then take some magnet and move it around that wire any way you want. There will be dB/dt, but what E field do you think would change - where and how would you measure it?

I really don't feel like it lol

I really want to know what is this "E field" you are talking about in terms of measurement, instruments and in what units is that "E field" of yours, especially since we just previously concluded there will be no electric field around the wire due to superposition of positively charged nucleus in the wire, regardless of any change in current or application of voltage, didn't we?

 

We are already talking about that scenario, with straight wire along x-axis and steady current of 1 ampere, we said there will be no electric field around it due to superposition, so what in the world is "cos(wt)" and what any angles have to do with E field?

So the new example I gave had current as [math] I = I_o cos(\omega t) [/math], so therefore this is not a constant current, but it is a steady current. This is what type of current comes out of the wall with a frequency of 60hz, aka alternating current.

a.) "Fi" is "flux" symbol in these equations, you need to use "dl" to denote the direction of electric current, is your "Fi" a unit vector or variable magnitude?

Okay so it might not be easy to tell, but that is not correct. Flux is actually denoted by capital phi: [math] \Phi [/math] and the lower case phi: [math] \hat{\phi} [/math] is the azimuthal direction in cylindrical coordinates. So in my case it is a unit vector describing a direction. And as I stated before, [math] d\mathbf{l} [/math] is not just a direction, but a vector, just dropping it out and assigning its direction is not the whole story. which is why the equation you gave is wrong for our earlier example with constant current I.

b.) Are you saying it's "m0/2Pi" and not "m0/4Pi"? Distance is not squared?

Once again it has to do with the [math] d\mathbf{l} [/math]

c.) Do you not see that your dB/dt equals to zero, actually, and what does it take for you to realize there will be no E field around the wire regardless of current, voltage or any change in any magnetic fields?

Not at all...how do you figure this, I don't know how many times I need to stress this. Do you know what the derivative of [math] cos(\omega t) [/math] is? It is definitely not zero (spoiler it is [math] - \omega sin(\omega t) [/math]). And yes there will be an E field in this scenario since the B field is changing!!! That is the whole point.

Ay, caramba! Is that magnetic or electric constant? What field are you talking about, "around the wire" or something else? Wires are made of neutrally charged atoms - superposition of positive and negative charges is what keeps it all electrically neutral regardless of any B or dB/dt and amperes and voltage, only position, i.e DISTRIBUTION of E fields/charges is what matters: (-q +q), and no B fields go into this vector addition calculus, ok?

 

E= Ke* (-q+q)/r*2

 

What you did is hideous, you completely removed the distance!!?! You have no idea what E filed you are talking about, without the distance what would you measure, where? - Your conclusions are completely devoid of any practical and experimental linkage, you are still yet do differentiate between the basic terms and their physical meaning, like charge, potential, voltage, current, and what "field(s)" have to do with any of that, in real world.

If there is changing B, there will be an Electric Field and since the Magnetic Field depends on current, the Electric Field will also. If the current is constant means no electric field (no change), if the current is changing, there will bean electric field.

Put it this way ambros, I don't feel that you will really understand what is going on here until you further your studies in vectors, spherical and cylindrical coordinates and integrals. Because I have more than enough shown you when and why Maxwell's equations are extremely useful and that they are also not always useful, and sometimes you need to use Coulomb's or Biot-Savart Laws. But you are lacking if you don't know when to use what and why. I'm sorry I cannot help you further if my explanations do not do it justice. Thank you though. I'm not sure how much more effort I am going to put into this. We will see how much time I have, Spring Quarter is starting tomorrow and I need some sleep

Sounds like you should have went to Ohio State and gone for Engineering Physics with a concentration in Electrical and Computer Engineering, which is what I am doing.

I wanted the same approach. I was initially thinking Electrical Engineering, but really liked physics and wanted to understand why things work and not just how things worked. Its been really good, but as stated before, if you are good at what you do, you should get far. I would still see what other colleges offer you, and what it would take to get into one of your liking.

Ke=1/4Pi*e0

Coulomb's law: E= Ke* (-q+q)/r^2

 

Km=m0/4Pi

Biot-Savart law:B= Km* I/r^2

-simplification without "dlxR", as distance is measured at right angle

==================================================

 

Ok, let's now look at those constants - what in the world is "4Pi" doing there? It seem arbitrary in these equations as they already describe spherical geometry without it, it's an unnecessary scalar in this vector equation, how did it get there? In any case, we should note the whole term "4Pi*r^2" is the description of 'surface area of a sphere', i.e. A= 4Pi*r^2.

To start, the reason I (and if you work with these equations) do not use the constant k, is because it does not always appears because some things cancel out the 4 or the pi. This has to do with the geometry of the electric and magnetic fields. And yes what you get is necessary for vector form because it describes the whole picture.

1.) ELECTRIC FIELDS

 

1st: divE= p/e0

 

3rd: rotE= -dB/dt

------------------

 

First time you gave us something like Coulomb's law, so please decide and write down what is the final equation for E field you are offering as the solution for this example, I'm not sure if your distance there is inverse squared or not.

The reason it looks different (and the field is different), is because it is due to the geometry of the object. All of the examples I have given have been different scenarios with different geometry.

In any case there is something strange about the 3rd one, where do we see E field depend on the change of B field, and how can this 3rd equation do anything without either magnetic or electric constant? Are there not any examples where we can use this 3rd (and 2nd) equation?

Here a couple of scenarios: A magnet in motion near a stationary object, causing a change in the B field Or a time-varying current causing a changing B field. The reason you don't "see" any constants is because you can derive the changing B field or you might already know it. Which would be done from Ampere's Law. Take a look:

Say you have wire with current [math] I = I_0 cos(\omega t)[/math] what E field is it producing?

So [math]\triangledown \cdot \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}[/math]

[math] \Rightarrow \oint \mathbf{E} \cdot d\mathbf{l} = - \int \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{a}[/math]

and using Ampere's Law you get [math] \mathbf{B} = \frac{\mu_0 I_0 cos(\omega t)}{2 \pi s} \hat{\phi} [/math]

[math] \Rightarrow \mathbf{E} = - \frac{\partial}{\partial t}[\frac{\mu_0 I_0 cos(\omega t)}{4\pi}] \hat{s}[/math]

[math] \Rightarrow \mathbf{E} = \frac{\mu_0 I_0 \omega sin(\omega t)}{4\pi} \hat{s} [/math]

which makes sense for an infinite wire that E would not depend on distance away

2.) Magnetic fields

 

2nd: divB= 0

 

4th: rotB= J + dE/dt

---------------------

 

Where, why and how do we ever use 2nd equation and how can it work without magnetic constant? Anyhow, here is this 4th equation and we finally have some real answers, so let's see if those make any sense...

 

a.) "Are we talking about the same fields?"

 

MAXWELL: B= m0/2Pi * I/r

BIOT-SAVART: B= m0/4Pi * I/r^2

 

"2Pi" is not the same as "4pi" and "r" is not the same as "r^2". What we have here is 'circumference of a circle': C= 2Pi*r, versus 'surface area of a sphere': A= 4Pi*r^2. This again points to "two-dimensionality" of these equations, but most importantly the two formulas are not IDENTICAL, so which one is wrong?

The equation you have is wrong, I can show you the derivation from "Introduction to Electrodynamics" if you would like.

And as far as the second equation goes. I am unfortunately not experienced with that, from what I've seen it is more of a reason why these equations work the way they do rather than using it to actually solve for the B field. So I do not currently have an example for the 2nd equation that can solve directly the B field, but don't worry we have Ampere's Law mainly for that.

b.) "That same old equation for loops, again."

 

- Your equation in its original form actually has some "time-varying" terms in it, so how and why did you pick that one to start with, since we have a 'steady current' in our example?

 

- Did you start with "Formulation in terms of FREE charge and current" or "Formulation in terms of TOTAL charge and current" and how did you make the decision which one suits this example better - are we dealing here with "displacement current: dD/dt", or with "time-varying electric field" dE/dt"? How to obtain the value for "dD/dt" and/or "dE/dt", what is their physical meaning and what do these terms represent in our example?

Once again please, please look and understand why we are using loops when solving for B, it shows how the Magnetic Field is behaving.

As for why I had a t in there for the electric field for the most recent example was due to that fact the electric field is caused by charges (or a changing B), so if you have a current I that is equal to [math] I = \lambda v[/math], where [math]\lambda[/math] is a line charge. So that was my attempt to get it into terms of charge, since I is Coulomb's per second, the charge in your example for 1 ampere would have been 1 Coulomb. So my equation was like a snapshot or an observation in a certain time interval, since the charge was changing at a rate proportional to current.

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BY THE WAY, why is this important? This is exactly what defines the unit of Ampere, that's why, it is actually this magnetic field that goes all the way around and back to bait its tail and define both electricity and magnetism. All this got even more complicated as the units got redefined ("Rationalized") over time and many things became self-referenced and circularly defined. So, electric aka *Coulomb force constant* and magnetic constant lost their physical meaning as experimentally determined values and now it is almost as if they jumped out of Maxwell's equations...

What are you asking about when say "why is this important"?

Ah crap we are doomed...

YOUR EQUATION: E®= Ke* I(t)/r ??

 

What is that red "t", how do I "plug" that in, what is that? Please get out of cylindrical coordinates and tell us if your distance is inverse squared or not? -- There will be no electric field around wire because conductors are macroscopically electrically neutral due to superposition, as there is about equal amount of negative and positive charges in close proximity, so wires have net zero external electric charge, regrades of any current and voltage applied. That's how crazy formulas can make you blind even to the most obvious everyday experience.

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Yea so the equation I derived is a segment of the wire that would have a net charge if you only considered the electrons charge. So yes you are right there should, then, be no electric field around the wire in that case. So that is done.

 

YOUR EQUATION: B®= Km* I*Fi/r ??

 

Is that "Fi" for flux, how do I "plug" in that one? I believe this below is what you meant to say, but that is still not Biot-Savart law.

 

Well no the [math]\hat{\phi}[/math] is the direction since it is a vector, so that is the correct B field. And believe it (or not), yes that is what you get for an infinite wire, or an approximation for a really long wire or one "looking" very close to the wire.

 

Using Biot-Savart you can get:

 

[math] \frac{\mu_o I}{4\pi s}(sin\theta_2 - sin\theta_1)[/math]

 

and thanks Cap'n Refsmmat

http://en.wikipedia.org/wiki/Biot-savart_law

-"The vector field B depends on the magnitude, direction, length, and proximity of the electric current, and also on a fundamental constant called the magnetic constant. The law is valid in the magnetostatic approximation, and results in a B field consistent with both Ampère's circuital law and Gauss's law for magnetism."

 

Your demonstration so far suggests you have not used any of those equations, ever. You need to realize these two sets of equations must be strictly IDENTICAL as they are describing the same fields and same interaction. So, for example, if 2nd Maxwell's equation does not equal to Biot-Savart law, then one of them is wrong or not complete, and you will never realize which one if you do not start USING them.

 

Q1: Wire W1 positioned along x-axis has steady current of 1 ampere, what is the magnitude of E and B field at some arbitrary distance, solve for E® and B®?

Okay lets do this I will solve this for now and get to the other ones later when I have more time: