Definition

A magnetic field is the invisible space around a magnet where its pull or push can be felt, strongest near the poles and weaker farther away, and the field exists whether or not anything is in it: a magnet on a fridge is surrounded by its field all the time, waiting for a paper clip to come near. It is the region around a magnet or a current-carrying wire in which magnetic forces act on other magnets, magnetic materials, and moving charges, drawn as field lines that leave the north pole, curve around, and enter the south pole, with closer lines showing a stronger field; strength is measured in teslas, Earth's field is about $5 \times 10^{-5}\ \text{T}$, and every electric current makes a magnetic field just as every changing magnetic field pushes charges to make a current, which is how generators produce electricity. Formally, $\vec{B}$ is defined by the force on a moving charge, $\vec{F} = q\vec{v} \times \vec{B}$, so it does no work and only redirects motion, producing circular orbits of radius $r = mv/(qB)$; steady currents create fields by the Biot-Savart law and Ampere's law, $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$, a long straight wire giving $B = \mu_0 I/(2\pi r)$ with $\mu_0 = 4\pi \times 10^{-7}\ \text{T}\cdot\text{m/A}$, and changing flux induces an emf by Faraday's law, $\mathcal{E} = -d\Phi_B/dt$, with $\vec{E}$ and $\vec{B}$ mixing under Lorentz transformations as components of one electromagnetic field.