magnetic flux
Magnetic flux measures how much magnetic field passes through a loop or a surface. Picture holding a wire hoop up to a magnet and counting how many magnetic field lines thread through it, or think of how much rain falls through an open window. It answers a simple question: how much magnetism does a loop actually catch?
Precisely, for a uniform magnetic field B passing through a flat area A, the magnetic flux is Phi = B A cos(theta), where theta is the angle between the field and the line drawn perpendicular to the surface (the normal). When the field goes straight through the loop (theta = 0) the flux is largest, B A; when the field only skims along the surface (theta = 90 degrees) the flux is zero. The unit is the weber (Wb), and 1 Wb = 1 T m^2. More generally, flux is the sum (integral) of the part of B that is perpendicular to the surface, added up over the whole area.
Flux is the star of electromagnetic induction: a changing flux is exactly what creates a voltage. You can change it three ways — change the field strength, change the loop's area, or change the angle. One honest caveat: although B is a vector, flux itself is a single number (a scalar), and its sign depends on which way you choose the surface's normal, so flux can come out positive or negative.
A loop of area 0.02 m^2 lies flat in a 0.5 T field pointing straight through it, so the flux is 0.5 * 0.02 = 0.01 Wb. Tilt the loop until the field makes 60 degrees with its normal and cos(60) = 0.5, so the flux halves to 0.005 Wb.
Flux = B A cos(theta); tilting the loop away from the field cuts the flux.
Flux is not the field: a strong field can give zero flux if it runs parallel to the surface (theta = 90 degrees). A coil of N turns links N times the flux of a single loop.