Ampère's law
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Ampère's law is a shortcut for finding the magnetic field around currents when the situation has nice symmetry. Instead of adding up the field from every tiny piece of wire, it relates the magnetic field looping around a closed path to the total current threading through that path. It answers, is there a quick, exact way to get the field of a straight wire, a coil, or a solenoid without a hard sum?
The law says that if you walk once around any closed loop and add up the magnetic field component along your path (a quantity called the circulation of B), the total equals mu_0 times the current I_enclosed passing through the loop: the circulation of B around the loop = mu_0 times I_enclosed. Here mu_0 is the permeability of free space. Choose a loop that follows the field's natural symmetry (a circle around a straight wire, a rectangle through a solenoid) and B comes straight out of the algebra. For a long straight wire this instantly gives B = mu_0 I / (2 pi r).
Ampère's law is the magnetic sibling of Gauss's law for electricity: both trade a hard sum for a simple relation, but only pay off when symmetry is high. In its complete form, Maxwell added an extra term (the displacement current) to account for changing electric fields, making it one of the four Maxwell's equations that unify electricity, magnetism, and light.
To find the field near a long straight wire, take a circular loop of radius r around it. By symmetry B is the same all around, so B times (2 pi r) = mu_0 I, giving B = mu_0 I / (2 pi r) in one line.
Chosen with the right symmetry, one Ampère loop gives B directly.
Ampère's law is always true, but it only lets you solve for B easily when the symmetry is high enough that B is constant along the loop. For a lopsided current, you still need the Biot-Savart law.