11.1 Magnetism and Its Historical Discoveries

Magnets have two types of magnetic poles, called the north magnetic pole and the south magnetic pole. North magnetic poles are those that are attracted toward Earth’s geographic North Pole.
Like poles repel and unlike poles attract.
Discoveries of how magnets respond to currents by Oersted and others created a framework that led to the invention of modern electronic devices, electric motors, and magnetic imaging technology.

11.2 Magnetic Fields and Lines

Charges moving across a magnetic field experience a force determined by $\vec{F} = q \vec{v} \times \vec{B} .$ The force is perpendicular to the plane formed by $\vec{v}$ and $\vec{B} .$
The direction of the force on a moving charge is given by the right hand rule 1 (RHR-1): Sweep your fingers in a velocity, magnetic field plane. Start by pointing them in the direction of velocity and sweep towards the magnetic field. Your thumb points in the direction of the magnetic force for positive charges.
Magnetic fields can be pictorially represented by magnetic field lines, which have the following properties:
1. The field is tangent to the magnetic field line.
2. Field strength is proportional to the line density.
3. Field lines cannot cross.
4. Field lines form continuous, closed loops.
Magnetic poles always occur in pairs of north and south—it is not possible to isolate north and south poles.

A magnetic force can supply centripetal force and cause a charged particle to move in a circular path of radius $r = \frac{m v}{q B} .$
The period of circular motion for a charged particle moving in a magnetic field perpendicular to the plane of motion is $T = \frac{2 π m}{q B} .$
Helical motion results if the velocity of the charged particle has a component parallel to the magnetic field as well as a component perpendicular to the magnetic field.

An electrical current produces a magnetic field around the wire.
The directionality of the magnetic field produced is determined by the right hand rule-2, where your thumb points in the direction of the current and your fingers wrap around the wire in the direction of the magnetic field.
The magnetic force on current-carrying conductors is given by $\vec{F} = I \vec{l} \times \vec{B}$ where I is the current and l is the length of a wire in a uniform magnetic field B.

The net force on a current-carrying loop of any plane shape in a uniform magnetic field is zero.
The net torque τ on a current-carrying loop of any shape in a uniform magnetic field is calculated using $τ = \vec{μ} \times \vec{B}$ where $\vec{μ}$ is the magnetic dipole moment and $\vec{B}$ is the magnetic field strength.
The magnetic dipole moment $μ$ is the product of the number of turns of wire N, the current in the loop I, and the area of the loop A or $\vec{μ} = N I A \hat{n} .$

Perpendicular electric and magnetic fields exert equal and opposite forces for a specific velocity of entering particles, thereby acting as a velocity selector. The velocity that passes through undeflected is calculated by $v = \frac{E}{B} .$
The Hall effect can be used to measure the sign of the majority of charge carriers for metals. It can also be used to measure a magnetic field.

A mass spectrometer is a device that separates ions according to their charge-to-mass ratios by first sending them through a velocity selector, then a uniform magnetic field.
Cyclotrons are used to accelerate charged particles to large kinetic energies through applied electric and magnetic fields.