Magnetic Flux through a Closed Surface and Maxwell's laws

[leo9999]

Hi everyone, I have a question about Maxwell's laws. According to Maxwell the magnetic flux of a magnetic field through a close surface is 0.
But his third law says the circuitation of an electric field depends from magnetic flux variation. I can't understand how this can be possible since magnetic flux is always 0.
Sorry for my English, but it isn't my mother tongue.

 

[etotheipi]

Careful! Maxwell's second equation is$$\nabla \cdot \boldsymbol{B} = 0$$You can apply the divergence theorem over a closed surface $\partial \Omega$ that is the boundary of some region $\Omega$, i.e.$$\int_{\Omega} (\nabla \cdot \boldsymbol{B}) d\tau = \oint_{\partial \Omega} \boldsymbol{B} \cdot d\boldsymbol{S} = 0$$That is the statement that the magnetic flux through any closed surface is zero. Next, you have Maxwell's third equation$$\nabla \times \boldsymbol{E} = - \frac{\partial \boldsymbol{B}}{\partial t}$$You can apply Stokes' theorem over an open surface $S$ which is bounded by a curve $\partial S$,$$\oint_S (\nabla \times \boldsymbol{E}) \cdot d\boldsymbol{S} = \oint_{\partial S} \boldsymbol{E} \cdot d\boldsymbol{x} = - \frac{d}{dt} \oint_S \boldsymbol{B} \cdot d\boldsymbol{S}$$Note that, a closed surface has no boundary, so the integral of $\partial_t \boldsymbol{B}$ over a closed surface is guaranteed to be zero, by Stokes' theorem. All this just amounts to saying that the magnetic flux through a closed surface is zero, at all times.

 

[Ssnow]

Hi, the magnetic flux variation is different from the value of the magnetic flux ... in other terms if the flux is constant there is no variation ...
Ssnow

 

[leo9999]

Careful! Maxwell's second equation is$$\nabla \cdot \boldsymbol{B} = 0$$You can apply the divergence theorem over a closed surface $\partial \Omega$ that is the boundary of some region $\Omega$, i.e.$$\int_{\Omega} (\nabla \cdot \boldsymbol{B}) d\tau = \oint_{\partial \Omega} \boldsymbol{B} \cdot d\boldsymbol{S} = 0$$That is the statement that the magnetic flux through any closed surface is zero. Next, you have Maxwell's third equation$$\nabla \times \boldsymbol{E} = - \frac{\partial \boldsymbol{B}}{\partial t}$$You can apply Stokes' theorem over an open surface $S$ which is bounded by a curve $\partial S$,$$\oint_S (\nabla \times \boldsymbol{E}) \cdot d\boldsymbol{S} = \oint_{\partial S} \boldsymbol{E} \cdot d\boldsymbol{x} = - \frac{d}{dt} \oint_S \boldsymbol{B} \cdot d\boldsymbol{S}$$Note that, a closed surface has no boundary, so the integral of $\partial_t \boldsymbol{B}$ over a closed surface is guaranteed to be zero, by Stokes' theorem. All this just amounts to saying that the magnetic flux through a closed surface is zero, at all times.

Thanks, I really appreciate your answer. I'm a med student tho so all these integral equations are like Chinese to me, I've never studied those integral theorems. I've tried to to google the divergence theorem and Stoke's theorem but I've no fundation to understand them.
Could you try to explain it qualitatively? My lecturer only gave us the equations without any explanation, so I'm trying to understand how it works

 

[leo9999]

Hi, the magnetic flux variation is different from the value of the magnetic flux ... in other terms if the flux is constant there is no variation ...
Ssnow

Thanks for your answer. Yeah, I know the variation is different from the value of the magnetic flux, but I tought that, since the flux is always equal to 0, then (B1 - B0)/ Δt = 0. Isn't it?

 

[etotheipi]

For a qualitative picture you can imagine magnetic field lines. There are some deficiencies to this representation (e.g. dimensional-analysis-wise), but it is quite useful. Essentially, you can identify the number of magnetic field lines passing through a given surface as the magnetic flux, and the number of magnetic field lines through that surface per unit area as the magnetic flux density, i.e. a measure of the magnetic field strength.

Now imagine a closed surface (you can visualise this as a 'bubble' of an arbitrary shape, although note that a surface is a purely mathematical concept!). Magnetic field lines have the property that they form closed curves, i.e. if you traced one for long enough you'd end up right back where you started. It follows that every magnetic field line that enters the surface must also exit the surface (so that it can end up back where it started!).

That means, that for any closed surface you choose, the number of magnetic field lines going in exactly equals the number of field lines coming out. In other words, the next flux through the surface is always zero!

 

[Ibix]

Roughly speaking, if you draw magnetic field lines, you can think of magnetic flux as the number of such lines passing through some (imaginary) surface. Note that field lines have a direction, so lines passing through the surface in one direction count as +1 line and lines passing through in the opposite direction count as -1 line. If you imagine your imaginary surface being a closed surface (like a balloon) then the total flux must be zero: magnetic field lines are always closed, so any line that crosses the surface in one direction (+1 line) must leave again, crossing the surface in the other direction (-1 line, total zero lines).

Note that this is a very qualitative picture. There are actually infinitely many field lines, so "counting them" isn't really plausible, and you need to do the calculus properly, which is what @etotheipi posted in the post you quoted. Edit: he apparently types quicker than I do, too.

 

[leo9999]

Well, for a qualitative picture you can imagine magnetic field lines. There are some deficiencies to this representation, but it is quite useful. Essentially, you can identify the number of magnetic field lines passing through a given surface as the magnetic flux, and the number of magnetic field lines through that surface per unit area as the magnetic flux density, i.e. a measure of the magnetic field strength.

Now imagine a closed surface (you can visualise this as a 'bubble' of an arbitrary shape, although note that a surface is a purely mathematical concept!). Magnetic field lines have the property that they form closed curves, i.e. if you traced one for long enough you'd end up right back where you started. It follows that every magnetic field line that enters the surface must also exit the surface (so that it can end up back where it started!).

That means, that for any closed surface you choose, the number of magnetic field lines going in exactly equals the number of field lines coming out. In other words, the next flux through the surface is always zero!

Thanks! So in the third Maxwell's equations what is changing is the magnetic flux density, not the magnetic flux since it's always zero? I don't know if that is a valid question...

 

[etotheipi]

Thanks! So in the third Maxwell's equations what is changing is the magnetic flux density, not the magnetic flux since it's always zero? I don't know if that is a valid question...

Magnetic flux is only necessarily zero for a closed surface, but most of the time you will be considering open surfaces when applying Maxwell III. In that case, changes in magnetic flux give rise to EMFs!

Magnetic flux density is just another name for the $\boldsymbol{B}$ field. It can change if you like, I don't really understand your question

 

[leo9999]

Magnetic flux is only zero for a closed surface, but most of the time you will be considering open surfaces when applying Maxwell III. In that case, changes in magnetic flux give rise to EMFs!

Magnetic flux density is just analagous to $\boldsymbol{B}$. It can change if you like, I don't really understand your question

I mean, I'm thinking about a spinning coil (close surface) in a magnetic field. When the angle changes the number of field lines that pass through the coil changes, so the magnetic flux should change too. But since every line that enters the coil exits it, then the flux is always zero... so it's like there are less( or more depending on the angle variation) lines through the coil, but at the end the magnetic flux is always zero, then its variation should be equal to zero too

 

[etotheipi]

No, when we say flux 'enters' or 'leaves' a "surface", it's implied that we're dealing with a closed surface enclosing some (3-dimensional) volume. Then, convention dictates the normal vector to the surface points outward, so if the magnetic field points out of the surface at some point on the boundary that counts as positive flux, and vice versa.

If you just have a circular ring, then it's easiest to consider a flat open surface with a bounding curve along the ring. If the magnetic field is constant in the vicinity of the ring, then the flux is just$$\Phi_B = \boldsymbol{B} \cdot A\boldsymbol{n} = BA\cos{\theta}$$where $\boldsymbol{n}$ is a unit normal to the surface. Clearly, this will be non-zero so long as $\boldsymbol{B}$ is non-zero and not orthogonal to $\boldsymbol{n}$. For an open surface, there's no useful notion of 'entering' or 'leaving' the surface, not least because there's no a priori preferred direction for $\boldsymbol{n}$.

 

[leo9999]

No, when we say flux 'enters' or 'leaves' a "surface" (we should, really, say "region"), it's implied that we're dealing with a closed surface enclosing some (3-dimensional) volume. Then, convention dictates the normal vector to the surface points outward, so if the magnetic field points out of the surface at some point on the boundary that counts as positive flux, and vice versa.

If you just have a circular ring, then it's easiest to consider a flat open surface with a bounding curve along the ring. If the magnetic field is constant in the vicinity of the ring, then the flux is just$$\Phi_B = \boldsymbol{B} \cdot A\boldsymbol{n} = BA\cos{\theta}$$where $\boldsymbol{n}$ is a unit normal to the surface. Clearly, this will be non-zero so long as $\boldsymbol{B}$ is non-zero and not orthogonal to $\boldsymbol{n}$. For an open surface, there's no useful notion of 'entering' or 'leaving' the surface, not least because there's no a priori preferred direction for $\boldsymbol{n}$.

Thank you, now I've got it! I didn't understand the difference between close and open surface. I thought an open surface was a surface without boundaries and that a flat circle without volume was a close surface. Now it's clear. Again I'm very glad you've helped me to understand

 

[etotheipi]

Ah, yes as you noticed, that's not right. Skirting around a few mathematical subtleties (compactness), a closed surface (left) is a two-dimensional manifold that doesn't have a boundary, and an open surface (right) is a two-dimensional manifold that does have a boundary:

 

[leo9999]

Now everything make sense!
In my free time I'll study integral calculus and hopefully in a few months I'll be able to understand your first post to gain a more complete understanding of this subject!

 

[A.T.]

I've tried to to google the divergence theorem and Stoke's theorem but I've no fundation to understand them. Could you try to explain it qualitatively? My lecturer only gave us the equations without any explanation, so I'm trying to understand how it works