### 22.1 Magnets

- Magnetism is a subject that includes the properties of magnets, the effect of the magnetic force on moving charges and currents, and the creation of magnetic fields by currents.
- There are two types of magnetic poles, called the north magnetic pole and south magnetic pole.
- North magnetic poles are those that are attracted toward the Earth’s geographic north pole.
- Like poles repel and unlike poles attract.
- Magnetic poles always occur in pairs of north and south—it is not possible to isolate north and south poles.

### 22.2 Ferromagnets and Electromagnets

- Magnetic poles always occur in pairs of north and south—it is not possible to isolate north and south poles.
- All magnetism is created by electric current.
- Ferromagnetic materials, such as iron, are those that exhibit strong magnetic effects.
- The atoms in ferromagnetic materials act like small magnets (due to currents within the atoms) and can be aligned, usually in millimeter-sized regions called domains.
- Domains can grow and align on a larger scale, producing permanent magnets. Such a material is magnetized, or induced to be magnetic.
- Above a material’s Curie temperature, thermal agitation destroys the alignment of atoms, and ferromagnetism disappears.
- Electromagnets employ electric currents to make magnetic fields, often aided by induced fields in ferromagnetic materials.

### 22.3 Magnetic Fields and Magnetic Field Lines

- Magnetic fields can be pictorially represented by magnetic field lines, the properties of which are as follows:

- The field is tangent to the magnetic field line.
- Field strength is proportional to the line density.
- Field lines cannot cross.
- Field lines are continuous loops.

### 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

- Magnetic fields exert a force on a moving charge
*q*, the magnitude of which is$$F=\text{qvB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta ,$$where $\theta $ is the angle between the directions of $v$ and $B$. - The SI unit for magnetic field strength $B$ is the tesla (T), which is related to other units by
$$\mathrm{1\; T}=\frac{\text{1 N}}{\mathrm{C}\cdot \text{m/s}}=\frac{\text{1 N}}{\mathrm{A}\cdot \mathrm{m}}.$$
- The
*direction*of the force on a moving charge is given by right hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of $v$, the fingers in the direction of $B$, and a perpendicular to the palm points in the direction of $F$. - The force is perpendicular to the plane formed by $\mathbf{\text{v}}$ and $\mathbf{\text{B}}$. Since the force is zero if $\mathbf{\text{v}}$ is parallel to $\mathbf{\text{B}}$, charged particles often follow magnetic field lines rather than cross them.

### 22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications

- Magnetic force can supply centripetal force and cause a charged particle to move in a circular path of radius
$$r=\frac{\text{mv}}{\text{qB}},$$where $v$ is the component of the velocity perpendicular to $B$ for a charged particle with mass
*$m$*and charge*$q$*.

### 22.6 The Hall Effect

- The Hall effect is the creation of voltage $\epsilon $, known as the Hall emf, across a current-carrying conductor by a magnetic field.
- The Hall emf is given by
$$\epsilon =\text{Blv}\phantom{\rule{0.25em}{0ex}}(B,\phantom{\rule{0.25em}{0ex}}v,\phantom{\rule{0.25em}{0ex}}\text{and}\phantom{\rule{0.25em}{0ex}}l,\phantom{\rule{0.25em}{0ex}}\text{mutually perpendicular})$$for a conductor of width
*$l$*through which charges move at a speed $v$.

### 22.7 Magnetic Force on a Current-Carrying Conductor

- The magnetic force on current-carrying conductors is given by
$$F=\text{IlB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\mathrm{\theta ,}$$where
*$I$*is the current, $l$ is the length of a straight conductor in a uniform magnetic field*$B$*, and*$\theta $*is the angle between*$I$*and*$B$*. The force follows RHR-1 with the thumb in the direction of*$I$*.

### 22.8 Torque on a Current Loop: Motors and Meters

- The torque $\tau $ on a current-carrying loop of any shape in a uniform magnetic field. is
$$\tau =\text{NIAB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta ,$$where $N$ is the number of turns, $I$ is the current, $A$ is the area of the loop, $B$ is the magnetic field strength, and $\theta $ is the angle between the perpendicular to the loop and the magnetic field.

### 22.9 Magnetic Fields Produced by Currents: Ampere’s Law

- The strength of the magnetic field created by current in a long straight wire is given by
$$B=\frac{{\mu}_{0}I}{2\mathrm{\pi r}}(\text{long straight wire}),$$where $I$ is the current, $r$ is the shortest distance to the wire, and the constant ${\mu}_{0}=\mathrm{4\pi}\phantom{\rule{0.15em}{0ex}}\times \phantom{\rule{0.15em}{0ex}}{\text{10}}^{-7}\phantom{\rule{0.25em}{0ex}}\text{T}\cdot \text{m/A}$ 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 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 Ampere’s law.
- The magnetic field strength at the center of a circular loop is given by
$$B=\frac{{\mu}_{0}I}{2R}\text{}(\text{at center of loop}),$$where $R$ is the radius of the loop. This equation becomes $B={\mu}_{0}\text{nI}/(2R)$ for a flat coil of $N$ loops. RHR-2 gives the direction of the field about the loop. A long coil is called a solenoid.
- The magnetic field strength inside a solenoid is
$$B={\mu}_{0}\text{nI}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}(\text{inside a solenoid}),$$where $n$ is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.

### 22.10 Magnetic Force between Two Parallel Conductors

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

### 22.11 More Applications of Magnetism

- Crossed (perpendicular) electric and magnetic fields act as a velocity filter, giving equal and opposite forces on any charge with velocity perpendicular to the fields and of magnitude
$$v=\frac{E}{B}\text{.}$$