Gauss' law for magnetism

[Conductivity]

We took today in a lecture gauss' law for magnetism which states that the net magnetic flux though a closed shape is always zero (Monopoles don't exist). The professor explained/proved it as following (Since it needs math theorems):
Draw any shape. From the fact that any magnetic field line that passes the shape must come out, This mean that the number of magnetic field lines going in and out is the same thus the law is proven.

And I have actually seen this proof in a couple of sites, But it doesn't makes sense to me. Magnetic field lines are just the tangent to the magnetic field at any point. So for example draw a shape when a magnetic field cross it for the first time there is a flux with angle theta and there is a B magnetic field, When you cross from the other way, There will be another angle and another value B (If the field changes inside the shape). So clearly for one line the net flux is not always zero.

What I did to fix this is to consider an infinitesimal cube where the magnitude and the direction of the magnetic field is constant then you can find that the net flux is zero and you can build any shape out of these cubes

Is there is something I said wrong? What is with the magnetic field lines proof?

 

[Charles Link]

Your second paragraph describes it fairly well, but flux lines are somewhat of a layman's description, and the density of flux lines represents the strength of the magnetic field. The magnetic lines of flux need to obey certain rules, and one of them is that their density doesn't spontaneously increase. The more mathematical description is $\nabla \cdot B=0$ , which in integral form becomes $\int B \cdot dA =0$ for any closed geometric shape. What this says is you will never have a case with magnetic lines of flux like you do for electric lines of flux from a charged particle, where the electric lines of flux radiate radially outward (from the charged particle). Hopefully this was somewhat helpful.

 

[Dale]

So for example draw a shape when a magnetic field cross it for the first time there is a flux with angle theta and there is a B magnetic field, When you cross from the other way, There will be another angle and another value B (If the field changes inside the shape). So clearly for one line the net flux is not always zero.

You are forgetting the area. It all works out. If the B decreases then the lines are further apart. So you have a lower flux density over a larger area. The net flux is then the same. If the field lines go in and out then the net flux is guaranteed to be 0.

 

[Conductivity]

You are forgetting the area. It all works out. If the B decreases then the lines are further apart. So you have a lower flux density over a larger area. The net flux is then the same. If the field lines go in and out then the net flux is guaranteed to be 0.

Let me just say what I know about magnetic field lines so we can be on the same page:
1) tangent to the direction.
2) represents the strength of magnetic field, further apart represents low magnetic field and the opposite.
and some other properties but not important here

I have always thought of them as merely for depiction, How do you tell the reader that there is a magnetic field stronger there? Draw them closer. So when you tell me they cover a bigger area, It is a good property of our choice to how represent them because it shows that the net flux will be zero or will work out but it is not a proof.

When you ask me to find the magnetic flux of a single line, I would get the first intersection with the surface. Get the B there and make a cross product, Add that to the flux at the 2nd point so we can have the net flux of a single line. Now there is no possible way that every line has zero flux it is not necessary. However overall if you take the contribution from other lines to it equals to zero.

That was my argument to the professor's proof, stating that merely the same number of magnetic fields enter and exit makes the net flux is zero is not enough proof.

 

[Dale]

but it is not a proof

I agree. It is not a proof, but it is a valid heuristic that quickly gets you to the right answer.

When you ask me to find the magnetic flux of a single line, I would get the first intersection with the surface. Get the B there and make a cross product, Add that to the flux at the 2nd point so we can have the net flux of a single line.

You are neglecting the area again. Insofar as it even makes sense to talk about a single line of flux you still need to specify the area covered by that line.

Also, it is a dot product, not a cross product.

$$\int \mathbf{B} \cdot d\mathbf{A}$$

That was my argument to the professor's proof, stating that merely the same number of magnetic fields enter and exit makes the net flux is zero is not enough proof.

Again, I agree that it isn’t a proof, but it is a valid explanation. You are misusing the concept of the field lines, your specific objections don’t hold.

 

[Conductivity]

You are neglecting the area again. Insofar as it even makes sense to talk about a single line of flux you still need to specify the area covered by that line.

Also, it is a dot product, not a cross product.

$$\int \mathbf{B} \cdot d\mathbf{A}$$

Thank you for replying
I meant dot product sorry.

Wouldn't the area be an infinitesimal area around the point of intersection? assuming I take it around every point which is the integral? So it is all the same.

What is wrong with this please?

 

[Dale]

Wouldn't the area be an infinitesimal area around the point of intersection?

If you have an infinite number of field lines, yes.

assuming I take it around every point which is the integral? So it is all the same.

What is wrong with this please?

Not all infinitesimals have the same size. That isn’t how calculus works. Infinitesimal is not a small fixed number, it is part of a limiting process where you look at ratios as the limit goes to zero.

Consider a simple vector field (not a magnetic field) in 2D where the field lines point radially out from the origin. Consider a Gaussian “surface” like a wedge with one side on the y-axis from 1 to 2, another side on the x-axis from 1 to 2, and with the inside being a 90 deg arc at a radius of 1 from the origin and the outside similarly but at a radius of 2 from the origin.

If you have 360 field lines then 90 will go through the wedge. The area for each will be $\pi/2/90$ at the inner surface and $\pi/90$ at the outer. The ratio is 2. If you have 900 field lines going through the wedge then the area for each will be $\pi/2/900$ at the inner surface and $\pi/900$ at the outer. The ratio is 2. In the limit as the number of lines goes to infinity then the area is infinitesimal at both the inside and the outside, but still the ratio is 2.

 

[Conductivity]

If you have an infinite number of field lines, yes.

Not all infinitesimals have the same size. That isn’t how calculus works. Infinitesimal is not a small fixed number, it is part of a limiting process where you look at ratios as the limit goes to zero.

Consider a simple vector field (not a magnetic field) in 2D where the field lines point radially out from the origin. Consider a Gaussian “surface” like a wedge with one side on the y-axis from 1 to 2, another side on the x-axis from 1 to 2, and with the inside being a 90 deg arc at a radius of 1 from the origin and the outside similarly but at a radius of 2 from the origin.

If you have 360 field lines then 90 will go through the wedge. The area for each will be $\pi/2/90$ at the inner surface and $\pi/90$ at the outer. The ratio is 2. If you have 900 field lines going through the wedge then the area for each will be $\pi/2/900$ at the inner surface and $\pi/900$ at the outer. The ratio is 2. In the limit as the number of lines goes to infinity then the area is infinitesimal at both the inside and the outside, but still the ratio is 2.

Ah, I don't think infinitesimal as small finite number but as something that neither is zero nor finite something that can be expressed with a symbol like dx which has properties we want.

I didn't understand some bits, But I will re-read all your posts again and try to understand where I went wrong again.

Thank you dale for the help.

 

[Conductivity]

I re-read everything a couple of times, understood each point and the infinitesimal argument.

Just this last point because it wasn't discussed, If we define a surface element to be as $\lim_{n \rightarrow +\infty} {\frac {A_t} n}$ ( I defined it as that so we can have the same infinitesimal area, Hopefully nothing wrong with that) and the flux on each surface element as $B . dA$, Does it make sense to talk about the flux of a single magnetic field line (only care about the vectors on each end) ? if it does, Then it is not necessary equal to zero. However, the overall contribution from all the field lines must equal to zero.

If this doesn't make sense, Scrap it because I couldn't get past this point.

 

[Charles Link]

I re-read everything a couple of times, understood each point and the infinitesimal argument.

Just this last point because it wasn't discussed, If we define a surface element to be as $\lim_{n \rightarrow +\infty} {\frac {A_t} n}$ ( I defined it as that so we can have the same infinitesimal area, Hopefully nothing wrong with that) and the flux on each surface element as $B . dA$, Does it make sense to talk about the flux of a single magnetic field line (only care about the vectors on each end) ? if it does, Then it is not necessary equal to zero. However, the overall contribution from all the field lines must equal to zero.

If this doesn't make sense, Scrap it because I couldn't get past this point.

The flux that you are calculating with this $\int B \cdot dA=0$ equation is over a volume. Flux going into any box that you draw picks up a minus sign on the $B \cdot dA$ because of the dot product. What the equation says is the flux going in is equal to the flux coming out for any box that you draw. $\\$You don't have any magnetic sources that flux can always come out of, so that if you were to encompass such a source, you could have all the flux going out, and nothing coming in. Such magnetic sources don't exist. $\\$ Magnetic "poles", (e.g. the magnetic poles on a permanent magnet), always have the magnetic field ($B$) flux lines pass right through them. (The reason why the $B$ field lines go from minus to plus inside a permanent magnet, (i.e. the flux lines go into the positive pole inside the magnet, and emerge from the positive pole outside the magnet), really is not apparent, but its origins can be explained if you assume the magnetic field arises not from the magnetic poles, but rather from surface currents on the outer cylinder of the magnet). $\\$ For electric fields we have $\int E \cdot dA=\frac{Q}{\epsilon_o}$, (electric field flux over a box), and $Q$ is the electrical charge inside the box. Here the electric charges are "sources" of electric field, so that only in the absence of charge inside the box do we have $\int E \cdot dA=0$ for electric fields. $\\$ Hopefully this is helpful.

 

[Dale]

Just this last point because it wasn't discussed, If we define a surface element to be as limn→+∞Atnlimn→+∞Atn\lim_{n \rightarrow +\infty} {\frac {A_t} n} ( I defined it as that so we can have the same infinitesimal area, Hopefully nothing wrong with that) and the flux on each surface element as B.dAB.dA B . dA ,

This is fine, nothing wrong with it.

Does it make sense to talk about the flux of a single magnetic field line (only care about the vectors on each end) ?

At this point, no, you have pretty much abandoned the flux line concept. If you would want to recover it then you would have to determine how many flux lines go through each area. Since the areas are fixed size the number of flux lines would have to vary.

You can talk about a fixed number of flux lines and determine the area, or you can talk about an fixed area and determine the number of flux lines. Either way works.

 

[Conductivity]

Thank you both, Thank you truly.