Theorem 1. If is a scalar field and a vector field both of class , then
Proof is just direct calculation, using .
Theorem 2. Let be a vector field of class . Then . That is, is an incompressible vector field.
2. What can be said about Magnetic Field?
As we've seen in previous Physics & EE Talk #2, Gauss's Law gives us information about electric field in electostatics. However, it does not tell us about magnetic fields.
2-1. What is a magnetic field?
Well, the source of magnetic field is movement of charged particles. To be specific, if a point charge of coulombs is at position and is moving with velocity , then the magnetic field it induces is
In SI units, is measured in teslas. The constant
is known as the permeability of free space.
2-2. Magnetic field induced from continuous charge
In the case of a magnetic field that arises from a continuous charged medium (such as electric current moving through a wire), rather than from a single moving charge, we replace by a suitable charge density function and the velocity of a single particle by the velocity vector field of the charges. Then we define the current density field by
Analogous to definition of electric field in continuous charge distribution, we use the following definition for the magnetic field resulting from moving charges in a region in space:
The convergence of on bounded region is NON-TRIVIAL (of course), but it is indeed TRUE and well defined. (If you want to see the proof, follow the instructions on Physics & EE Talk #2, Section 4-1.)
Before continuing our calculations, we comment further regarding the current density field . The vector field at a point is such that its magnitude is the current per unit area at that point, and its direction is that of the current flow. It is not hard to see then that the total current across an oriented surface is given by the flux of ; that is
3. Vector Potential of Magnetic field
Returning to the magnetic field , we show that it can be identified as the curl of another vector field . First by direct calculation,
Therefore, the magnetic field in section 2-2 can be restated as
Using the Theorem 1 in Section 1,
since is independent of . Hence,
as does not contain any of the variables of integration. Consequently,
where
The vector field is called a vector potential of the field . Thus using Theorem 2 of Section 1
4. Meaning of Divergence
The divergence of a vector field can be interpreted using the divergence theorem.
The magnetic flux entering the closed surface is equal to the flux flowing out that same surface. So, no matter how small the magnet, it can not be a monopole, which the net flow of magnetic flux on a closed surface enclosing the magnet is non-zero. Therefore
No magnetic monopoles exist!!
5. Conclusion
A magnetic field is equal to curl of another vector field.
Therefore, divergence is always zero, which magnetic flux on any closed surface is zero.
There is no magnetic monopole.
[Physics and EE #3] Does Magnetic Monopole Exist? | Ecency
