Hertz vectors, or the Hertz vector potentials, are an alternative formulation of the electromagnetic potentials. They are most often introduced in electromagnetic theory textbooks as practice problems for students to solve.[1] There are multiple cases where they have a practical use, including antennas[2] and waveguides.[3] Though they are sometimes used in such practice problems, they are still rarely mentioned in most electromagnetic theory courses, and when they are they are often not practiced in a manner that demonstrates when they may be useful or provide a simpler method to solving a problem than more commonly practiced methods.
Hertz vectors can be advantageous when solving for the electric and magnetic fields in certain scenarios, as they provide an alternative way to define the scalar potential
\phi
A
Considering cases of electric and magnetic polarization separately for simplicity, each can be defined in terms of the scalar and vector potentials which then allows for the electric and magnetic fields to be found. For cases of just electric polarization the following relations are used.
And for cases of solely magnetic polarization they are defined as:
To apply these, the polarizations need to be defined so that the form of the Hertz vectors can be obtained. Considering the case of simple electric polarization provides the path to finding this form via the wave equation. Assuming the space is uniform and non-conducting, and the charge and current distributions are given by
\rho(r,t),J(r,t)
P=P(r,t)
\rho=-\nabla ⋅ P
J=
| \partialP | |
| \partialt |
\Pi
D
H
\Pie
\Pim
This is simply done by applying the d'Alembert operator
| ||||
| \Box=\left(\nabla |
| \partial2 | |
| \partialt2 |
\right)
c2=\left(\mu\epsilon\right)-1
J
\left[P\left(r'\right)\right]
\left[M\left(r'\right)\right]
|r-r'|/v
\Pie
Similarly, in the case of only magnetic polarization being present, the fields are determined via the previously stated relations to the scalar and vector potentials.
For the case of both electric and magnetic polarization being present, the fields become
Consider a one dimensional, uniformly oscillating current. The current is aligned along the z-axis in some length of conducting material with an oscillation frequency
\omega
t'=t-|r-r'|/v
Continuing directly to taking the divergence quickly becomes messy due to the
|r-r'|
1/r
It is important to note that in the above equation,
x
x'
r
r'
\gamma
x
x'
Taking the divergence
Then the gradient of the result
Finally finding the second partial with respect to time
Allows for finding the electric field
Using the appropriate conversions to Cartesian coordinates, this field can be simulated in a 3D grid. Viewing the X-Y plane at the origin shows the two-lobed field in one plane we expect from a dipole, and it oscillates in time. The image below shows the shape of this field and how the polarity reverses in time due to the cosine term, however it does not currently show the amplitude change due to the time varying strength of the current. Regardless, its shape alone shows the effectiveness of using the electric Hertz vector in this scenario. This approach is significantly more straightforward than finding the electric field in terms of charges within the infinitely thin wire, especially as they vary with time. This is just one of several examples of when the use of Hertz vectors is advantageous compared to more common methods.
Consider a small loop of area
A
I\sin\left(\omegat\right)
| M= | dm |
| dV |
m=IA\hat{n
m=IA\sin\left(\omegat\right)\hat{z
M
As in the electric dipole example, the Legendre polynomials can be used to simplify the derivatives necessary to obtain
E
B
Due to the dependence on
r
\hat{z
\hat{r
\hat{\theta
This field was simulated using Python by converting the spherical component to x and y components. The result is as expected. Due to the changing current, there is a time dependent magnetic field which induces an electric field. Due to the shape, the field appears as if it were a dipole.