Hertz vector

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.

[citation needed] 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

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.

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

can be found, however here the Hertz vectors treat the electric and magnetic polarizations as sources.

for the magnetic Hertz potential can be derived using the wave equation for each.

, and the result is non-zero due to the polarizations that are present.

This provides a direct pathway between easily determined properties such as current density

These wave equations yield the following solutions for the Hertz vectors: where

[1] The electric and magnetic fields can then be found using the Hertz vectors.

For simplicity in observing the relationship between polarization, the Hertz vectors, and the fields, only one source of polarization (electric or magnetic) will be considered at a time.

vector is used to find the fields as follows: 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 ℓ with an oscillation frequency

We will define the polarization vector where t is evaluated at the retarded time

Inserting this into the electric Hertz vector equation knowing that the length ℓ is small and the polarization is in one dimension it can be approximated in spherical coordinates as follows Continuing directly to taking the divergence quickly becomes messy due to the

This is readily resolved by using Legendre Polynomials for expanding a

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.

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.

, so if the loop lies in the x-y plane and has the previously defined time-varying current, the magnetic moment is

, and then into Equation (10), the magnetic Hertz vector is found in a simple form.

As in the electric dipole example, the Legendre polynomials can be used to simplify the derivatives necessary to obtain

, it is significantly simpler to express the Hertz vector in spherical coordinates by transforming from the sole

Due to the shape, the field appears as if it were a dipole.