In this entry we present some lesser known mathematical identities related to magnetic fields.
Rotation Matrix Edit
Let us consider the non relativistic motion of a charged particle in a uniform magnetic field
where is the velocity, is the electric charge of the particle, is the mass of the particle, is the speed of light and is the magnetic field.
We can write this equation in a different form
is a generator for a rotation and
is the tensor product of the magnetic field vector and the Levi Civita antisymmetric tensor. The last term is the same as the magnetic field block in the anti-symmetric electromagnetic tensor.
The solution for the equation of motion is
where a matrix exponent is implied. Because is anti-symmetric, then is a rotation matrix whose angle increases linearly with time.
Divergence - less Field Edit
Due to the absence of magnetic monopoles, the magnetic field is divergence free . A general vector field can be described by three scalar functions (one for each component). A divergence - less field can be described by just two.
There are several ways in which a divergence - less field can be represented. The most well known example is using a vector potential
Thus, any vector function chosen for is guaranteed to yield a diverge - less field. One difficulty with this approach is that it is redundant, meaning that different vector potential yield the same magnetic field. This is sometimes called gauge degree of freedom.
Another way is using the Euler scalar potentials
One benefit of this representation is that the magnetic field lines are surfaces where both and are constant.
Another way is the poloidal - toroidal decomposion