James Maxwell is a name that every physics student will encounter, and with good reason. He made one of modern science’s most important discoveries, unifying electricity and magnetism into a single theory. Prior to Maxwell, scientists considered electricity and magnetism to be separate forces. Maxwell was the first to recognise that these two forces are inextricably linked. The discovery prompted him to write down four famous equations, now known as Maxwell’s equations, which describe how electric and magnetic fields work. These equations transformed the world by laying the groundwork for technologies such as radio waves, electricity, and modern computers.
To truly understand how James Maxwell came up with these equations, we must consider the scientific world he lived in. By the middle of the nineteenth century, many significant discoveries had been made about electricity and magnetism. Scientists such as Charles-Augustin de Coulomb, Michael Faraday, and Hans Christian Ørsted conducted experiments to demonstrate how electric charges interact and generate magnetic fields. However, there was no unified theory to explain how these forces were related.
Maxwell was born in 1831 in Scotland and was a gifted child who excelled in mathematics and physics. He attended Edinburgh Academy and then Cambridge University, where he developed a strong interest in electromagnetism. It’s important to remember that during Maxwell’s time, physics was about experiments rather than mathematical theories. But James Maxwell saw it differently. He was good with numbers and could spot patterns that others couldn’t. This combination of mathematical ability and physical intuition enabled him to make the breakthrough.
Maxwell did not start from scratch. He built on the work of others, especially Michael Faraday, who discovered electromagnetic induction in 1831. Faraday demonstrated that changing magnetic fields could generate electric currents, which was a revolutionary concept at the time. Faraday’s work was primarily experimental, and he used visualisations such as lines of force to explain his findings. These lines of force became extremely important to Maxwell. Even though Faraday was not particularly mathematically inclined, James Maxwell translated Faraday’s ideas into mathematical form, allowing him to construct a comprehensive theory.
Maxwell’s journey to the equations was long and did not occur overnight. He began by investigating the properties of magnetic fields and their relationship to electric current. In 1820, Ørsted demonstrated that a current-carrying wire generates a magnetic field around itself. But Maxwell wanted to dig deeper. He asked himself questions like,
“How does the magnetic field behave around various materials?” What happens as the electric current varies over time?
One of Maxwell’s key insights was the concept of displacement current. Prior to James Maxwell, scientists knew that an electric current could generate a magnetic field. But Maxwell realised there was more going on. He proposed that even if there was no electric current, a changing electric field could generate a magnetic field. This was groundbreaking because it explained how light could travel through empty space. In other words, Maxwell’s equations predict that changing electric and magnetic fields will travel through space as electromagnetic waves. This was a profound concept because it implied that light was itself an electromagnetic wave.
So, how did James Maxwell get from these ideas to his famous four equations? It’s actually a fascinating story about intuition and calculation. Maxwell began by studying existing laws, such as Coulomb’s law, which describes how electrically charged objects interact, and Ampère’s law, which describes how currents generate magnetic fields. He realised that these laws alone couldn’t explain all electromagnetic phenomena. So he modified them by including his displacement current term, and then combined them into a set of four equations.
The first equation is known as Gauss’s law. This shows how electric charges create electric fields. The equation ∇ · E = ρ/ε₀ tells us that the electric field (E) coming out of a closed surface is proportional to the charge (ρ) inside, divided by a constant (ε₀) called the permittivity of free space. The idea is that the strength of the electric field is proportional to the amount of charge in a region. This concept wasn’t new; Coulomb had already described it. But Maxwell wrote it down in a clear mathematical form that made it easier to use in other calculations.
The second equation is Gauss’s law for magnetism. This tells us there are no magnetic charges in the same way that there are electric charges. The equation ∇ · B = 0 means that the magnetic field (B) flowing into a closed surface always equals the amount flowing out. there’s no buildup of “magnetic charge” anywhere. In other words, while you can have isolated positive or negative electric charges, you can’t have isolated magnetic poles. Magnets always come in pairs of north and south poles. This was also known before Maxwell, but again, he was the first to write it down as a neat equation.
The third equation, known as Faraday’s law of induction, is perhaps the most famous. This describes how changing magnetic fields make electric fields. The equation ∇ × E = -∂B/∂t shows that a changing magnetic field (∂B/∂t) creates a circulating electric field (∇ × E). This law is the basis for electric generators, transformers, and many other technologies. Faraday had discovered this law experimentally, but Maxwell’s equation made it a fundamental part of the theory of electromagnetism.
Finally, the fourth equation is Ampère’s law with Maxwell’s addition. This shows how electric currents and changing electric fields create magnetic fields. The equation ∇ × B = μ₀J + μ₀ε₀∂E/∂t includes both the current density (J) and the rate of change of the electric field (∂E/∂t), which was Maxwell’s big idea. This was where Maxwell’s displacement current comes in. He realised that a changing electric field acts like a current and can produce a magnetic field, even in empty space. This was the key insight that allowed him to predict electromagnetic waves.
The beauty of these equations is their ability to work together. When combined, they predict the existence of electromagnetic waves travelling at the speed of light. Maxwell did not simply stop with these equations; he calculated the speed of these waves and discovered that it matched the known speed of light. This prompted him to propose that light is an electromagnetic wave, which was a groundbreaking discovery.
Maxwell’s prediction of the speed of light was extremely accurate. He calculated it to be approximately 310,740,000 meters per second. Today, we know that the precise speed of light in a vacuum is 299,792,458 meters per second. That means Maxwell’s calculation was off by only 3.6%! When you consider the tools he had in the 1860s, this small difference is incredible. Furthermore, Maxwell’s theory predicted that light would have varying wavelengths. We now know that visible light, radio waves, X-rays, and gamma rays are all electromagnetic waves, albeit with different wavelengths. Visible light, for example, has wavelengths ranging from 380 to 700 nanometres, whereas radio waves can be a few centimetres to hundreds of meters long.
James Maxwell published his findings in 1865 in a paper titled “A Dynamical Theory of the Electromagnetic Field.” Initially, his work was not widely accepted. Many scientists struggled to understand the mathematical approach, and the idea that light was an electromagnetic wave was difficult to accept. Experiments eventually confirmed Maxwell’s predictions, and his equations became the foundation of classical electromagnetism.
In 1887, Heinrich Hertz conducted one of the most important experiments to support Maxwell’s theory. Hertz successfully generated and detected electromagnetic waves in the laboratory, demonstrating that Maxwell’s equations were correct. He generated electromagnetic waves using a spark gap transmitter and detected them with a wire loop with a small gap. When the waves struck the wire loop, they produced a small spark. Hertz determined that the wavelength of these waves was approximately 4 meters. Using Maxwell’s equations, he calculated their frequency to be approximately 75 MHz (million cycles per second). This was groundbreaking because it demonstrated that these waves behaved exactly like light, but with a much longer wavelength. Today, we use waves similar to those Hertz created for FM radio, which operates at frequencies ranging from 88 to 108 MHz.
Maxwell’s work transformed our understanding of electromagnetism. It also had a significant impact on the evolution of other branches of physics. His equations are consistent with Albert Einstein’s special theory of relativity, developed in 1905. Einstein once stated,
“The work of James Maxwell changed the world forever.”
Einstein’s theory of relativity is based on the idea that the speed of light is constant, which derives directly from Maxwell’s equations.
To summarise, Maxwell’s journey to the four electromagnetic equations was one of curiosity, intuition, and mathematical brilliance. He expanded on previous work but took it to a new level by combining electricity and magnetism into a unified theory. His equations are still used in a wide range of fields, including electric circuits and quantum mechanics. They are not only historically significant; they are also used every day in modern physics and engineering. For example, when engineers design antennas for cell phones, they use Maxwell’s equations to determine how the antenna should be shaped to best transmit and receive signals. Maxwell’s equations are used in particle accelerators, such as the Large Hadron Collider, to design powerful magnets that guide particles at nearly the speed of light. Even in everyday devices such as microwave ovens, Maxwell’s equations explain how electromagnetic waves heat food. It is estimated that more than half of today’s economy is based on technologies derived from Maxwell’s equations. From Wi-Fi to medical imaging devices such as MRI machines, Maxwell’s work continues to shape our world more than 150 years later.
