5.1 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 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.

5.2 Magnetic Fields and Magnetic Field Lines

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

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 are continuous loops.

5.3 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
  $$F=\text{qvB}\text{sin}\theta ,$$
  where $\theta$ is the angle between the directions of $v$ and $B\text{.}$
• 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 $v\text{,}$ the fingers in the direction of $B\text{,}$ and a perpendicular to the palm points in the direction of $F\text{.}$
• The force is perpendicular to the plane formed by $\mathbf{\text{v}}$ and $\mathbf{\text{B}}\text{.}$ Because the force is zero if $\mathbf{\text{v}}$ is parallel to $\mathbf{\text{B}}\text{,}$ charged particles often follow magnetic field lines rather than cross them.

5.4 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 $q$.

5.5 The Hall Effect

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

5.6 Magnetic Force on a Current-Carrying Conductor

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

5.7 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
  $$\tau =\text{NIAB}\text{sin}\theta ,$$
  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.

5.8 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
  $$B=\frac{{\mu }_{0}I}{2\mathrm{\pi r}}(\text{long straight wire}),$$
  $I$ is the current, $r$ is the shortest distance to the wire, and the constant ${\mu }_0=\mathrm{4\pi }\times {\text{10}}^{-7}\text{T}\cdot \text{m/A}$ 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
  $$B=\frac{{\mu }_{0}I}{2R}(\text{at center of loop}),$$
  $R$ 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
  $$B={\mu }_0\text{nI}(\text{inside a solenoid}),$$
  where $n$ is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.

5.9 Magnetic Force between Two Parallel Conductors

• The force between two parallel currents $I_1$ and $I_2\text{,}$ separated by a distance $r\text{,}$ 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.

5.10 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
  $$v=\frac{E}{B}\text{.}$$