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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

 \frac{\partial \mathbf{v}}{\partial t} = \frac{q}{m c} \mathbf{v} \times \mathbf{B}

where  \mathbf{v} is the velocity,  q is the electric charge of the particle,  m is the mass of the particle,  c is the speed of light and  \mathbf{B} is the magnetic field.

We can write this equation in a different form

 \frac{\partial \mathbf{v}}{\partial t} = \mathbf{G} \mathbf{v}

where

 \mathbf{G} = \frac{q}{m c} \varepsilon \mathbf{B}

is a generator for a rotation and

 \varepsilon \mathbf{B} = \begin{pmatrix} 0 & B_z & -B_y\\ -B_z & 0 & B_x\\ B_y & -B_x & 0 \end{pmatrix}

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

 \mathbf{v} \left(t\right) = \exp \left(t \mathbf{G} \right) \mathbf{v} \left( 0 \right)

where a matrix exponent is implied. Because  \mathbf{G} is anti-symmetric, then  \exp \left( t \mathbf{G} \right) 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  \nabla \mathbf{B} = 0 . 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

 \mathbf{B} = \nabla \times \mathbf{A}

Thus, any vector function chosen for  \mathbf{A} 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

 \mathbf{B} = \nabla \alpha \times \nabla \beta

One benefit of this representation is that the magnetic field lines are surfaces where both  \alpha and  \beta are constant.

Another way is the poloidal - toroidal decomposion

 \mathbf{B} = \nabla \times \left( \mathbf{r} \Psi \right) + \nabla \times \left( \nabla \times \left( \mathbf{r} \Phi \right) \right)