Undertanding Maxwell's 3rd. Did Maxwell get it wrong, or did I...

[CasualKilla]

I am considering a DC current I flowing through a conductor. From Ampere's Law, we know this creates a Constant magnetic field circling the conductor.

 

I am trying to reconcile Faraday's Law with the phenomenon described above. I know that Ampere's Law describes the phenomenon, but my reasoning is that it should also not violate Faraday's Law.

 

Using Faraday's Law, to analyse the situation, I came to the conclusion that the magnetic field must change with an for a DC current through the conductor.

 

Could someone please try follow my steps (below) and try show me where I am going wrong.

 

 

Note there is a mistake in step 3: J = oE not E = oJ, but it does not make difference to my reasoning.

Edited March 18, 2015 by CasualKilla

[swansont]

Why do you think the curl of E is nonzero?

[CasualKilla]

Why do you think the curl of E is nonzero?

Is there not a net swirl of the curl in the clock-wise direction?

 

Unless I am misunderstanding the curl operation, (step 4)

Edited March 18, 2015 by CasualKilla

[swansont]

Is there not a net swirl of the curl in the clock-wise direction?

 

Unless I am misunderstanding the curl operation, (step 4)

No. Why do you think there is an E like that, that would give you a nonzero curl?

[studiot]

Where have you accounted for continuity in step3 to step 4?

 

[math]\nabla {\bf{.J}} = \frac{{\partial \rho }}{{\partial t}}[/math]

 

Note continuity is not one of Maxwell's equations.

 

In free space you also have the conduction current J = 0

 

 

Edited March 18, 2015 by studiot

[CasualKilla]

No. Why do you think there is an E like that, that would give you a nonzero curl?

Oh ok, well since there is current in the conductor, we have a drift current density J.

 

 

And Ohm's law says that if there is a drift current density, we will have a magnetic field E.

 

 

Why I drew the E field as little vectors like that I have no idea. But I am partially certain there will be an E-field in the same direction and position as the current flow in the wire. I am new to vector functions, curl, divergence etc. I taught it to myself over the last 2 days, so I apologize if my understanding is a bit basic.

[swansont]

Oh ok, well since there is current in the conductor, we have a drift current density J.

 

current-relationship.gif

 

And Ohm's law says that if there is a drift current density, we will have a magnetic field E.

 

ohms-law.gif

 

Why I drew the E field as little vectors like that I have no idea. But I am partially certain there will be an E-field in the same direction and position as the current flow in the wire. I am new to vector functions, curl, divergence etc. I taught it to myself over the last 2 days, so I apologize if my understanding is a bit basic.

 

E is the electric field, and the current is going in a straight line. There's no loop.

[CasualKilla]

Where have you accounted for continuity in step3 to step 4?

 

[math]\nabla {\bf{.J}} = \frac{{\partial \rho }}{{\partial t}}[/math]

 

Note continuity is not one of Maxwell's equations.

 

In free space you also have the conduction current J = 0

 

 

So the drift current coming from a point/volume is equal to the change of charge at the point /( in that volume). Ok makes sense, so we can't have current flow if there are no electrical charges. ie. vacumm. That makes sense. But I am considering the current flowing in that a conductor.

Ok I see that I have gone full retard, apparently the E field is zero inside a conductor, but I still don't get that. Is that only true if we assume that the conductivity is infinite (ie Rcond = 0 ohms)?. (thinking about ohms law here)

Edited March 18, 2015 by CasualKilla

[swansont]

The E is what drives the current. It's inside the conductor.

[CasualKilla]

The E is what drives the current. It's inside the conductor.

Ok, i thought so initially, but then misread something about Faraday cages talking about conductors, which is why I said the E field was zero. Ok that makes more sense then, so there is a finite E field in the wire.

 

So the issue now is, why is the E field not creating a net curl vector at H?

[studiot]

Edit sorry I missed a minus sign

 

[math]\nabla {\bf{.J}} = - \frac{{\partial \rho }}{{\partial t}}[/math]

[CasualKilla]

Edit sorry I missed a minus sign

 

[math]\nabla {\bf{.J}} = - \frac{{\partial \rho }}{{\partial t}}[/math]

 

Current coming from a source corresponds to a decrease of charge in the source. Makes sense. But doesn't really solve my problem XD

Edited March 18, 2015 by CasualKilla

[swansont]

Ok, i thought so initially, but then misread something about Faraday cages talking about conductors, which is why I said the E field was zero. Ok that makes more sense then, so there is a finite E field in the wire.

 

So the issue now is, why is the E field not creating a net curl vector at H?

 

 

What do you mean? Look at your drawing. The H has a curl to it (same as B in free space).

[CasualKilla]

 

 

What do you mean? Look at your drawing. The H has a curl to it (same as B in free space).

Sorry, you asked me why I think there is a curl at at B, so I was assumed you were saying that there was not a net curl, ie. my steps 3 to 5 were incorrect. So I was going based on that assumption.. Also by curl I mean the mathematical curl created by the E-field at the test point with magnetic flux density B, not the the fact that the B field is actually 'curling' around the wire. Just making sure we on the same page.

 

Note. Probably should not have named that point B, it is rather confusing, but there is a B vector at the point, but I am trying to see why B is not increasing.

Edited March 19, 2015 by CasualKilla

[studiot]

Although I realise you are interested in testing out Maxwell, you don't need Maxwell and vector calculus to solve this one.

Indeed having a physics feel for what is going on is important.

 

With a steady direct current, the electric field situation is the same as a line of charges, viz the charge/current density never changes and the field has radial symmetry.

The E (and D) field lines must be at right angles to the B (and H) field lines.

As you rightly observe the B field lines form concentric circles around the line of the conductor.

 

How would another set of concentric circles be orthogonal to these?

 

You asked where the flaw in your reasoning lies and I pointed to the step from 3 to 4.

 

In step 3 you correctly say that if the E field is circular then curl E is non zero..

 

The error is in chosing the circular diagram, not the star in the top left of your attachment.

 

The electric field cannot form closed loops.

It starts or ends on a charge or goes to infinity.

 

The magnetic field, however cannot have ends. It can only form closed loops.

 

You can also apply Maxwell to show the consistency of this.

[swansont]

Sorry, you asked me why I think there is a curl at at B, so I was assumed you were saying that there was not a net curl, ie. my steps 3 to 5 were incorrect. So I was going based on that assumption.. Also by curl I mean the mathematical curl created by the E-field at the test point with magnetic flux density B, not the the fact that the B field is actually 'curling' around the wire. Just making sure we on the same page.

 

Note. Probably should not have named that point B, it is rather confusing, but there is a B vector at the point, but I am trying to see why B is not increasing.

 

 

I assumed B was the field, so I guess we weren't.

[studiot]

 

I assumed B was the field, so I guess we weren't.

 

 

 

So did I, isn't it?

[CasualKilla]

 

 

So did I, isn't it?

Well the B is a vector representing the field at a certain point in space, but since we are trying to determine what this value of B will be at that point, I am saying it might have been more accurate to to assign an arbitrary name to the point, say "P" and then calculate the value of B at that point using Maxwell equations. It still makes sense to assume there is a B field vector at that point and solve for it, so I guess i shouldn't have said anything, sorry for the confusion.

 

My reasoning was that the value of B at point P should be changing with time according to Faraday's Law.

 

New Post:

So the curl of the E-field is zero in step 3, so my error lies in the understanding of the curl in step 4. I have tried to understand this by making a vector function describing the field, then applying the curl operator, but can't seem to derive said vector function.

 

So the way curl at a point was described to me intuitively is like summing up the torque of all the points on the vector function around the point in question. This is how I get a non-zero curl, since there is a clockwise 'torque' around B.

 

However if I understand Studiot correctly, vectors need to create a continuous loop around the point to create a curl, and the infinite straight-line E field shown in step 3 does not do this, so the curl is zero.

 

I am trying to dive into the maths of the curl operator at the moment, since clearly my intuitive understanding does not suffice or is plain incorrect.

 

If someone could help with the derivation of a vector function of the E field in step 3, that would help loads.

Edited March 19, 2015 by CasualKilla

[imatfaal]

if your vector is

[latex]

 

\mathbf{V}=V_x\textbf{i}+V_y\textbf{j}+V_z\textbf{k}

[/latex]

 

then your curl is

[latex]

\nabla \times \textbf{V}=\left [ \frac{\partial V_z}{\partial y} -\frac{\partial V_y}{\partial z}\right ]\mathbf{i}+\left [ \frac{\partial V_x}{\partial z} -\frac{\partial V_z}{\partial x}\right ]\mathbf{j}+\left [ \frac{\partial V_y}{\partial x} -\frac{\partial V_x}{\partial y}\right ]\mathbf{k}

[/latex]

[CasualKilla]

if your vector is

[latex]

 

\mathbf{V}=V_x\textbf{i}+V_y\textbf{j}+V_z\textbf{k}

[/latex]

 

then your curl is

[latex]

\nabla \times \textbf{V}=\left [ \frac{\partial V_z}{\partial y} -\frac{\partial V_y}{\partial z}\right ]\mathbf{i}+\left [ \frac{\partial V_x}{\partial z} -\frac{\partial V_z}{\partial x}\right ]\mathbf{j}+\left [ \frac{\partial V_y}{\partial x} -\frac{\partial V_x}{\partial y}\right ]\mathbf{k}

[/latex]

The vector function must be evaluated at point B I assume? In that case the E field is 0i + 0j + 0z and it is not changing in the x,y or z directions, so are partials are also 0. Therefore: Curl(E) at B = 0i + 0j + 0k

 

Somehow I feel this is not right though..

Edited March 19, 2015 by CasualKilla

[imatfaal]

What is point B? I see a Magnetic Field labelled with a barred B - but no point B. Have a reread of Studiot's post above - it is a better explanation than I could give. And get step 4 drawn out properly with labels of fields on at the right places - this is where I think you are coming a cropper and getting it straight on paper will get it straight in your head

[studiot]

For static charges or steady current, the statements:-

 

The E field is conservative,

has a scalar potential, V (as introduced by imatfaal),

has a zero curl,

has a line integral around any closed curve equal to zero,

 

are all equivalent statements that are proved by simpler means before Maxwell is introduced.

 

Since these are all mathematically equivalent only one is chosen as the basic and the others follow mathematically.

 

One of the difficulties in replying to this is avoiding a circular argument.

 

So where does your course start?

Also where do you stand on vector algebra and vector calculus in 3 dimensions?

Edited March 20, 2015 by studiot

[studiot]

The book by H M Schey is loved by some and hated by others.

It is called

 

Div Grad Curl and all that

 

I am noting it here because it is an introduction to vector calculus by an unusual route.

It is only a little book, 163 pages all in and easy reading.

 

Although a mathematical book, it takes the electromagnetic field as its working material and develops the subject in terms of EM theory as far as Maxwell's equations.

 

It provides a pretty chatty approach to the subject matter of the relationship between the EM fields, the divergence and the curl and so on, starting from first principles.

 

It might suit you.

Edited March 20, 2015 by studiot

[Bignose]

+1 for div, grad, curl and all that. I recommend it to many beginners to vector calculus. It certainly is not super-rigorous, but I find it to normally be a gentle introduction to the subject and good at the beginnings of building intuition on the subject.

[CasualKilla]

For static charges or steady current, the statements:-

 

The E field is conservative,

has a scalar potential, V (as introduced by imatfaal),

has a zero curl,

has a line integral around any closed curve equal to zero,

 

are all equivalent statements that are proved by simpler means before Maxwell is introduced.

 

Since these are all mathematically equivalent only one is chosen as the basic and the others follow mathematically.

 

One of the difficulties in replying to this is avoiding a circular argument.

 

So where does your course start?

Also where do you stand on vector algebra and vector calculus in 3 dimensions?

So you cant have an E-field like I have drawn since you need to consider that that E-field will need to be created by some voltage source, which will make the curl 0?

 

So my understanding of the curl was correct, but my understanding of the E-field was fundamentally flawed.

 

Also that E field as I drew it only exists at a single point in the z and y plane, how would you compute the partial derivatives for that, it is like a piece wise vector function.

 

My vector algebra is ok since we calculate vector current and voltage for circuits, though this is 2 dimensional. All I know about vector calculus I have thought myself, but I have a good understanding of partial derivatives.