Visualization of the Electromagnetic Fields

Purpose

Here, we provide numerical modeling tools for visualizing the electric field, magnetic field and current density caused by an electrical current dipole source. A set of questions about the fields and their dependency on various parameters are then presented as a learning exercise. By completing this exercise, you will become comfortable with the numerical modeling tools provided and gain a fundamental understanding of the fields which are caused by a harmonic electrical current dipole.

Link to the modeling app

Research Questions

Here, a set of questions about the fields and their dependency on various parameters are presented as a learning exercise. The user is encouraged to answer these questions in order, as we begin with simple cases and move to cases with more challenging physics. The user is also encouraged to use analytic and asymptotic expressions as a reference when completing the exercise.

DC (Static) Case:

For the DC case, ensure that the frequency is set to 0 Hz. When answering the following questions, examine the x, y and z components of the electric field, magnetic field and current density. The vector plot is recommended.

  • Do the electric field, magnetic field or current density have quadrature (imaginary) components?

  • Does the magnetic field have any components along the orientation of the dipole?

  • When you alter the conductivity (\(\sigma\)), does the shape of the electric or magnetic field change? Does the magnitude change?

  • When you alter the conductivity (\(\sigma\)), does anything about the current density change? (note that \(\mathbf{J = \sigma E}\))

  • Do your observations make sense when considering the DC approximation?

Near-Field Approximation:

Now set the conductivity of the background to 1 S/m and set the frequency to 1 Hz. According to our asymptotic approximations, the near-field (\(| kr | \ll 1\)) should behave like the DC field.

  • Examine the electric and magnetic fields. Within the domain defined, do these fields behave like DC fields?

Now slowly increase the frequency by factors of 10. When you reach 1000 Hz, notice that at sufficient distance from the dipole, the near-field approximation is no longer valid. However near the dipole source, the fields more or less behave like DC fields.

  • At what distance is the near-field approximation no longer valid at 1000 Hz? Use this value to confirm that \(| kr | \ll 1\).

The Inductive Response:

According to Faraday’s law, the effects of EM induction increase as frequency increases. Set the conductivity to 0.1 S/m and choose a point (x,y,z) = (40m, 0m, 0m). Examine the x,y and z components of the electric and magnetic fields.

  • At what frequency do the effects of EM induction become significant?

  • Now increase the background conductivity to 1 S/m and examine the same location. At what frequency do the effects of EM induction become significant?

  • Now choose a location closer to the dipole source (x,y,z) = (10m, 0m, 0m). At what frequency do the effects of EM induction become significant compared to the primary field?

Magnetic Permeability and Dielectric Permittivity

Set the log-conductivity to .

  • Try increasing the relative permeability (\(\mu_r\)). Do you notice any significant changes in the shape and amplitude of the electric and magnetic fields?

  • Now try increasing the relative permittivity (\(\varepsilon_r\)). When you do this at low frequencies, do you notice any significant changes in the shape and amplitude of the electric and magnetic fields? How about when you do this at high frequencies?