The first of the four Maxwell’s Equations is called Ampère’s Law, named after the Frenchman André-Marie Ampère. It has to do with an electric current creating a magnetic field. If you ever wrapped an insulated wire around a nail and connected a battery to it you have experienced Ampère’s Law. In fact, electric current is today measured in amperes or amps for short.
Ampère developed this equation by experimenting with magnets. If you pass an electric current through a wire, perhaps wound around an iron core (like a nail), you create a magnetic field. Ampère found that sometimes a magnet attracted another magnet, but other times it repelled it. Then he found that attraction or repulsion depends on the direction that current flows in the wire and in what orientation you hold the magnets. By trying different experiments and different equations, he gradually found that this equation (without the last term, that is, everything after the + sign) could be used to calculate the results of his experiments.
Ampère’s Law has many practical applications. It can be used to know what magnetic field is generated by an electric current. This is useful in building electromagnets, motors, generators, transformers, and more. This equation has to do with magnetic field, H, and current, I. To use this equation, you must first pick a closed curve. It can be any curve at all, anywhere you want. The only important restriction is that it be closed. A circle is a good example (notice the little circle in the very first symbol).
The left side of the equation says to first calculate the magnetic field along the entire length of the curve that you picked. Then, you must add up all the magnetic field that is parallel to the curve. Magnetic field that is perpendicular to the curve is excluded.
The right side of the equation (ignoring the complicated looking last term, the displacement current, which is discussed elsewhere) is the total current flowing through the same curve that we used for the left hand side.
What this equation says is that if we increase the current flowing through a closed curve, the total magnetic field around that curve also increases. If you care to actually calculate numbers, you can add up the magnetic field around the curve and tell exactly how much current is flowing through the curve (ignoring that pesky last term). There are actually meters that clamp around a wire and sum up the magnetic field. The meter then reads out how much current is flowing in the wire.
What is amazing is that we can pick any curve we want, anywhere, any shape, any time. As long as it is a closed curve, this equation works perfectly.
In practical terms, this law says that if you want a stronger magnet, put another battery on that coil of wire. Just don’t put too many batteries on it or it will go up in smoke and the magnetic field will be gone. But even if that happens, Ampère’s Law still works, anytime, anywhere, and with any closed curve.