12.1 The Biot-Savart Law

The magnetic field created by a current-carrying wire is found by the Biot-Savart law.
The current element $I d \vec{l}$ produces a magnetic field a distance r away.

12.2 Magnetic Field Due to a Thin Straight Wire

The strength of the magnetic field created by current in a long straight wire is given by $B = \frac{μ_{0} I}{2 π R}$ (long straight wire) where I is the current, R is the shortest distance to the wire, and the constant $μ_{0} = 4 π \times 10^{−7} T \cdot m/s$ is the permeability of free space.
The direction of the magnetic field created by a long straight wire is given by right-hand rule 2 (RHR-2): Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops created by it.

The force between two parallel currents $I_{1}$ and $I_{2},$ separated by a distance r, has a magnitude per unit length given by $\frac{F}{l} = \frac{μ_{0} I_{1} I_{2}}{2 π r} .$
The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions.

The magnetic field strength at the center of a circular loop is given by $B = \frac{μ_{0} I}{2 R} (at center of loop),$ where R is the radius of the loop. RHR-2 gives the direction of the field about the loop.

The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampère’s law.
Ampère’s law can be used to determine the magnetic field from a thin wire or thick wire by a geometrically convenient path of integration. The results are consistent with the Biot-Savart law.

The magnetic field strength inside a solenoid is
$B = μ_{0} n I (inside a solenoid)$
where n is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.
The magnetic field strength inside a toroid is
$B = \frac{μ_{o} N I}{2 π r} (within the toroid)$
where N is the number of windings. The field inside a toroid is not uniform and varies with the distance as 1/r.

Materials are classified as paramagnetic, diamagnetic, or ferromagnetic, depending on how they behave in an applied magnetic field.
Paramagnetic materials have partial alignment of their magnetic dipoles with an applied magnetic field. This is a positive magnetic susceptibility. Only a surface current remains, creating a solenoid-like magnetic field.
Diamagnetic materials exhibit induced dipoles opposite to an applied magnetic field. This is a negative magnetic susceptibility.
Ferromagnetic materials have groups of dipoles, called domains, which align with the applied magnetic field. However, when the field is removed, the ferromagnetic material remains magnetized, unlike paramagnetic materials. This magnetization of the material versus the applied field effect is called hysteresis.