# Maxwell I: Fundamentals¶

Purpose

Here, the fundamentals of electromagnetism are presented. For those who do not have a strong background in electromagnetism, the content provided here is an excellent place to start within EM GeoSci. After reading through *Maxwell I: Fundamentals*, you will be able to understand the physical principles which govern static problems, frequency domain electromagnetics and time domain electromagnetics as well as the survey methods used in electromagnetic geoscience.

**Introduction**

Electromagnetic theory is the basis for all materials presented within EM GeoSci. As a result, a sufficient understanding of electromagnetic theory is needed before reading more advanced materials. Here, we will cover the basics of electromagnetic theory. You will learn about the formative laws which govern electromagnetism and Maxwell’s equations. Next, you will learn how electromagnetic signals behave in matter and how they behave at interfaces. You will then learn about the fields which are generated by electromagnetic sources. In the final section, we present the required steps for solving Maxwell’s equations in practice.

**Organization**

The content in *Maxwell I: Fundamentals* is organized as follows:

Formative Laws: The formative laws of electromagnetics characterize the physical processes which occur when electromagnetic signals interact with matter. As a result, they are of fundamental importance when considering applied problems in electromagnetic geoscience. Here, formal definitions are provided for each formative law along with an appropriate mathematical description.

Overview of Maxwell’s Equations: Maxwell’s equations are a concise way of characterizing all of the physics pertaining to electromagnetic theory. Here, four common ways of representing Maxwell’s equations are shown. This page is designed to be a quick access to Maxwell’s equations with proper units and notation.

Interface Conditions: To solve Maxwell’s equations in general cases, we must consider the conditions on fields and fluxes at physical interfaces. Here, the appropriate interface conditions for \(\mathbf{e}\), \(\mathbf{h}\), \(\mathbf{j}\), \(\mathbf{d}\) and \(\mathbf{b}\) are presented.

Maxwell’s Equations: Frequency Domain: Here, we show that within a homogeneous media, frequency-dependent electromagnetic signals behave according to the vector Helmholtz equation. The physical understanding of this equation is very important when considering frequency-domain electromagnetic methods.

Maxwell’s Equations: Time Domain: Here, we show that within a homogeneous media, time-dependent electromagnetic signals behave according to the diffusive wave equation. Diffusive and wave behaviours are very important when considering time-domain electromagnetic methods.

Harmonic Plane Waves in Homogeneous Media: For harmonic planewaves in the frequency domain, we discuss the skin depth, phase velocity, wavelength and apparent resistivity. Numerical apps are also provided to visualize plane waves in various cases.

Transient Plane Waves in Homogeneous Media: For plane waves in the time domain, we discuss diffusion distance, peak time and peak velocity. Numerical apps are also provided to visualize plane waves in various cases.

Reflection and Refraction of Plane Waves: When plane waves reach an interface characterized by an abrupt change in physical properties, the signal is altered. Here, the physics pertaining to plane waves as they reach an interface is presented. Content includes: Snell’s law, the Fresnel equations and Brewster’s angle. Numerical apps are provided to simulate electromagnetic signals.

Maxwell’s Equations with Electromagnetic Sources: For applications in electromagnetic geoscience, a source is frequently used to generate electromagnetic signals. Here, we demonstrate how Maxwell’s equations must be altered to accommodate electromagnetic sources.

Dipole Sources in Homogeneous Media: The fields generated by dipole sources and of great fundamental important to electromagnetic geoscience, as they do very well at approximating the electromagnetic sources used for many geophysical applications. Here, the fields generated by dipole sources within a homogeneous medium are presented. Numerical apps are provided for visualizing the fields.

Solving Maxwell’s Equations: Here, we discuss what is needed to solve Maxwell’s equations in practice.

**Contents**

- Formative Laws
- Overview of Maxwell’s Equations
- Interface Conditions
- Maxwell’s Equations: Time Domain
- Maxwell’s Equations: Frequency Domain
- Harmonic Planewaves in Homogeneous Media
- Transient Plane Waves in Homogeneous Media
- Plane Waves at Interfaces
- Maxwell’s Equations with Electromagnetic Sources
- Dipole Sources in Homogeneous Media
- Solving Maxwell’s Equations
- Appendix