Maxwell’s Equations – Basic Ideas

A beginner-friendly introduction to Maxwell’s equations and how they predict the existence of EM waves.

1. Why Maxwell’s equations are important

Maxwell’s equations bring together all the basic laws of electricity and magnetism into one unified framework. These equations show how electric and magnetic fields behave, how they interact, and how they can create electromagnetic waves.

Even without solving them fully, understanding their basic meaning gives a clear picture of how changing electric and magnetic fields support each other and travel as waves.

1.2. Prediction of electromagnetic waves

Maxwell’s equations mathematically predict that electric and magnetic fields can form waves that move through space at a fixed speed. When he calculated this speed, it turned out to be the same as the measured speed of light. This led to the conclusion that light is an electromagnetic wave.

2. Gauss’s law for electric fields

This equation describes how electric charges create electric fields.

2.1. Meaning of Gauss’s law

Gauss's law states that electric field lines begin on positive charges and end on negative charges. If you place a closed surface around some charge, the total electric flux through the surface is proportional to the enclosed charge.

2.1.1. Mathematical form

\(\nabla \cdot \vec{E} = \dfrac{\rho}{\varepsilon_0}\)

Here, \(\rho\) is the charge density. This simply means: more charge → stronger electric field spreading out.

3. Gauss’s law for magnetic fields

This equation tells us something unique about magnetic fields: magnetic monopoles do not exist. In other words, you cannot have a “single magnetic charge.”

3.1. No isolated magnetic poles

Every magnet has both a north and a south pole. If you cut a magnet into two pieces, each piece still has both poles. Magnetic field lines always form closed loops.

3.1.1. Mathematical form

\(\nabla \cdot \vec{B} = 0\)

This means magnetic field lines do not start or end anywhere—they always loop back.

4. Faraday’s law of electromagnetic induction

This equation explains how changing magnetic fields create electric fields. It forms the basis of generators, induction cookers, transformers, and many more technologies.

4.1. Changing magnetic field → induced electric field

A changing magnetic field causes an electric field to form around it. This electric field can drive current in a wire loop even without touching the magnet.

4.1.1. Mathematical form

\(\nabla \times \vec{E} = -\dfrac{\partial \vec{B}}{\partial t}\)

This shows that a time-varying magnetic field produces a circulating electric field.

5. Ampère–Maxwell law

This equation originally related magnetic fields to electric currents, but Maxwell added a crucial term to fix a gap in the theory. His addition revealed that a changing electric field produces a magnetic field.

5.1. Changing electric field → magnetic field

Even in regions with no physical current, a changing electric field acts as a kind of “effective current,” producing a magnetic field. This was Maxwell’s major contribution.

5.1.1. Mathematical form

\(\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \dfrac{\partial \vec{E}}{\partial t}\)

The term \(\mu_0 \varepsilon_0 \dfrac{\partial \vec{E}}{\partial t}\) is the displacement current term added by Maxwell.

6. How Maxwell’s equations predict electromagnetic waves

The most important outcome of Maxwell’s equations is the prediction that electric and magnetic fields can support self-propagating waves.

6.1. Electric and magnetic fields regenerate each other

Two of Maxwell’s equations state that:

  • A changing magnetic field produces an electric field (Faraday).
  • A changing electric field produces a magnetic field (Ampère–Maxwell).

This mutual generation allows the disturbance to move forward without needing any material medium.

6.2. Wave equation prediction

When the equations are combined mathematically, they produce the standard form of a wave equation for both \(\vec{E}\) and \(\vec{B}\).

\(c = \dfrac{1}{\sqrt{\mu_0 \varepsilon_0}}\)

This shows that electromagnetic waves travel at speed \(c\), which matches the speed of light. This led to the conclusion that light is an electromagnetic wave.

7. Simple intuitive picture of Maxwell’s theory

You can imagine an electromagnetic wave as a disturbance in the electric field that triggers a disturbance in the magnetic field, which again triggers an electric disturbance, and so on. This chain of events moves smoothly through space, forming a wave.

7.1. Why no medium is required

Because the wave is carried by the fields themselves, it does not require particles to push or pull. This explains how sunlight reaches Earth through the vacuum of space.

7.2. Direction of fields and propagation

In a simple electromagnetic wave:

  • The electric field oscillates in one direction.
  • The magnetic field oscillates in a direction perpendicular to it.
  • The wave moves in a direction perpendicular to both.

This makes EM waves transverse waves.