1
$\begingroup$

So I have a doubt regarding the equations of Magnetic flux and Gauss' Law for magnetic fields.

According to Gauss' Law for magnetic fields, integral form of div(B) = $0$ can be written as:

$\oint \mathbf{B}.d\mathbf{a} = 0$

And magnetic flux through a surface is also defined as $\iint \mathbf{B}.d\mathbf{S} = \Phi _{B}$

If we combine the two, does it mean that flux is always zero through the surface? I believe $\iint$ and $\oint$ are the same thing (correct me if I am wrong) ? What is the significance of $\Phi _{B}$ then?

$\endgroup$
  • $\begingroup$ You can think of the Gauss Law flux as a count of the number field lines leaving a closed surface. Magnetic field lines always form closed loops. So each line that comes in ( - flux ) must also go out. $\endgroup$ – R.W. Bird Sep 16 '19 at 18:40
2
$\begingroup$

I believe ∬ and ∮ are the same thing (correct me if I am wrong)?

$\oint$ is a closed contour integral symbol while $\iint$ is a double integral symbol.

Gauss's Law for magnetism in integral form is

$$\unicode{x222F}_S\,\mathbf{B}\cdot\mathrm{d}\mathbf{A}=0$$

which states that the magnetic flux through a closed (no boundary) surface is zero (there are no magnetic monopoles). The symbol $\unicode{x222F}_S$ indicates integration over the closed surface $S$.

The magnetic flux through a surface bounded by a closed countour is not necessarily zero. For example, the Maxwell-Faraday equation is (for a surface unchanging with time)

$$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{A}$$

where $\Sigma$ is the surface of integration bounded by the closed countour $\partial\Sigma$.

$\endgroup$
  • 1
    $\begingroup$ Shouldn't the surface integral over $\Sigma$ in the last equation be an $\iint$ symbol, for consistency with the notation conventions established earlier in this post? $\endgroup$ – Emilio Pisanty Sep 16 '19 at 17:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.