Maxwell's Equations: The Physics Behind Microwave Imaging

The Four Equations That Changed the World

Before we can understand how microwave imaging detects tumors, we need to understand the physics of electromagnetic waves. That means James Clerk Maxwell.

In the 1860s, Maxwell unified electricity and magnetism into four elegant equations. They describe every electromagnetic phenomenon—from radio to light to X‑rays to the microwaves we use in THORACIS AI.

Equation 1: Gauss's Law for Electricity

∇⋅E = ρ/ε0​

What it means: Electric charges create electric fields. The more charge, the stronger the field.

Why it matters for us: The human body has electric properties. Tumors have different charge distributions than healthy tissue—which affects how microwaves travel through them.

Equation 2: Gauss's Law for Magnetism

∇⋅B = 0

What it means: Magnetic monopoles don't exist. Magnetic field lines always form closed loops.

Why it matters for us: Our antennas generate both electric and magnetic fields. Understanding both helps us design better antenna arrays.

Equation 3: Faraday's Law of Induction

∇×E = −∂B/∂t​

What it means: A changing magnetic field creates an electric field. This is how generators work—and also how antennas radiate.

Why it matters for us: Our transmitting antenna creates a changing magnetic field, which induces an electric field that propagates through the chest.

Equation 4: Ampère's Law (with Maxwell's correction)

∇×B=μ0J + μ0ε0 (∂E/∂t)

What it means: Electric currents AND changing electric fields create magnetic fields. The second term (Maxwell's addition) predicts electromagnetic waves.

Why it matters for us: This equation predicts that electromagnetic waves can travel through space—and through tissue. That's how our signal gets from the transmitting antenna to the receiving antenna.

Putting It All Together: The Wave Equation

From these four equations, Maxwell derived that electric and magnetic fields travel as waves:

This describes how microwaves propagate—and how they interact with tissue.

How This Helps Us Detect Tumors

When a microwave passes through tissue, its speed and attenuation depend on the tissue's dielectric properties. Tumors have higher water content, which means:

• Higher permittivity (ε) → slower wave speed

• Higher conductivity (σ) → more attenuation

Our VNA measures the S21 parameter—how much signal gets from antenna A to antenna B. By comparing S21 with and without a tumor, we can detect the tumor's presence.

The math behind this is complex, but the insight is simple: Maxwell's equations tell us that different tissues interact with microwaves differently. We measure that difference.

In Our System

We don't solve Maxwell's equations directly (that would take too long). Instead, we use the NanoVNA to measure the actual S21 transmission. Then we extract features and let machine learning find the patterns. But the physics is still there—underneath everything we do.

Want to Learn More?

In my next post, I'll explain how we use the Inverse Fast Fourier Transform (IFFT) to extract even more information from the S21 signal—revealing time‑domain reflections that tell us not just "is there a tumor?" but "where is it located?"