Maxwell's equations represent one of the greatest achievements in physics. Before Maxwell, electricity and magnetism were understood as separate phenomena. James Clerk Maxwell unified these forces into a single electromagnetic theory through four fundamental equations. These equations describe how electric and magnetic fields are generated by charges and currents, and how they interact with each other to create electromagnetic waves.
Gauss's law for electricity is the first Maxwell equation. It states that the electric flux through any closed surface is proportional to the electric charge enclosed within that surface. We can visualize this with electric field lines emanating radially from a positive charge. The Gaussian surface, shown in blue, encloses the charge. The electric flux is represented by the field lines passing through this surface. The purple arrows show the direction of the electric field at the surface. This law applies to any charge distribution and any closed surface.
Gauss's law for magnetism is the second Maxwell equation. It states that the magnetic flux through any closed surface is always zero. This is because magnetic monopoles do not exist in nature. Unlike electric charges which can exist independently, magnetic poles always come in pairs. We can see this with a bar magnet, where magnetic field lines emerge from the north pole and return to the south pole, forming closed loops. When we draw a Gaussian surface around the magnet, the same number of field lines that exit the surface also enter it, making the total flux zero.
Faraday's law of induction is the third Maxwell equation. It states that a changing magnetic flux through a loop induces an electric field around that loop. We can demonstrate this with a magnet moving near a conducting loop. As the magnet approaches, the magnetic flux through the loop increases, inducing a current that flows in a direction to oppose this change. This is Lenz's law, represented by the negative sign in the equation. The induced current creates its own magnetic field that opposes the change in flux, which is fundamental to how generators and transformers work.
The Ampère-Maxwell law is the fourth and final Maxwell equation. Originally, Ampère's law related magnetic fields only to electric currents. However, Maxwell realized this was incomplete for time-varying situations. He added the displacement current term, which accounts for changing electric fields also producing magnetic fields. We can see this in a charging capacitor, where current flows to the plates but not between them. The changing electric field between the plates acts as a displacement current, creating a magnetic field around the capacitor. This insight was crucial for predicting electromagnetic waves.