Short Answer

Maxwell’s equations show that changing electric and magnetic fields create each other and spread out in the form of a wave. When Maxwell combined these equations, he found a wave equation whose speed depends on two constants: the permittivity and permeability of free space.
Using these values, he calculated the speed of the wave and found it to be 3 × 10⁸ m/s, which is exactly the speed of light.

This led Maxwell to conclude that light is an electromagnetic wave, and its speed is determined by the properties of free space.

Detailed Explanation :

How Maxwell’s Equations Relate to the Speed of Light

Maxwell’s equations are among the most powerful and meaningful results in physics. They describe how electric and magnetic fields behave, how they are generated, and how they interact. When James Clerk Maxwell analyzed these equations, he made an extraordinary discovery: the speed of electromagnetic waves predicted by his equations is equal to the speed of light. This was the first clear evidence that light is an electromagnetic wave.

Before Maxwell, no one fully understood why light travelled at the speed it did. Many scientists believed that some invisible medium must carry light, just like air carries sound. Maxwell showed that no such medium was needed. Light could travel through vacuum because it is made of electric and magnetic fields that support each other.

1. Maxwell’s Wave Equation

From two of Maxwell’s equations:

• Faraday’s law (changing magnetic field produces electric field)
• Ampere-Maxwell law (changing electric field produces magnetic field)

Maxwell derived a wave equation. It describes how electric and magnetic fields can travel as waves through space.

The wave equation for the electric field is:

∇²E = (1/c²)(∂²E/∂t²)

A similar equation applies to the magnetic field:

∇²B = (1/c²)(∂²B/∂t²)

The important part is the constant c, which appears in the denominator.

2. How Maxwell Calculated the Speed of Light

Maxwell discovered that the speed of the wave predicted by his equations is given by:

c = 1 / √(μ₀ ε₀)

Where:

• μ₀ is the permeability of free space
• ε₀ is the permittivity of free space

These values were already known from electrical experiments. So Maxwell calculated:

c ≈ 3 × 10⁸ m/s

This value was exactly the same as the known speed of light measured by scientists like Fizeau and Foucault. This was not a coincidence. It meant that light must be an electromagnetic wave.

3. Light as an Electromagnetic Wave

Maxwell’s discovery was revolutionary. It proved that:

• Light, radio waves, microwaves, and X-rays all come from the same source: oscillating electric and magnetic fields.
• Light does not need a medium to travel.
• The speed of light is fixed by the physical properties of space.

This solved many scientific problems and connected electricity, magnetism, and optics into one unified theory.

4. Why the Speed of Light Is Constant

The expression c = 1 / √(μ₀ ε₀) shows that the speed of light depends only on:

• How electric fields behave in vacuum (ε₀)
• How magnetic fields behave in vacuum (μ₀)

Since these values are constants and do not change, the speed of light is also a constant. This became one of the most important principles in physics and later influenced Einstein’s theory of relativity.

5. Electromagnetic Waves Predicted by Maxwell

Maxwell’s equations suggested that electromagnetic waves can have many wavelengths and frequencies. Even though he did not experimentally observe all of them, he predicted their existence.

Later discoveries confirmed his prediction:

• Hertz discovered radio waves
• Infrared and ultraviolet waves were studied
• X-rays and gamma rays were found later

All of them travel at the speed of light in vacuum, proving Maxwell’s theory.

6. Role in Modern Science and Technology

Maxwell’s discovery about the speed of light has shaped many scientific fields:

• Communication systems: Radio, Wi-Fi, Bluetooth
• Astronomy: Measuring distance using light speed
• Electronics: Understanding wave propagation
• Fiber optics: Transmission of light signals
• Medical imaging: X-rays, MRI
• Space exploration: Light from distant stars used for study

Everything that uses electromagnetic waves relies on Maxwell’s insight.

7. Relationship Between Fields and Speed

Maxwell showed that the speed of the electromagnetic wave depends on how quickly the fields respond to changes:

• A stronger electric property (ε₀) slows down the field response
• A stronger magnetic property (μ₀) also slows down the response

Together, these determine how fast the wave travels.

This relationship explains why light travels faster in vacuum and slower in materials like glass or water.

Conclusion

Maxwell’s equations relate to the speed of light by predicting a wave formed by changing electric and magnetic fields that travels at a speed given by 1 / √(μ₀ ε₀). This predicted value matches the measured speed of light. Thus, Maxwell concluded that light is an electromagnetic wave, and its speed is controlled by the electrical and magnetic properties of space. His work unified optics with electromagnetism and became one of the foundations of modern physics.