Magnetism & Magnetic Forces

Magnetic Field Fundamentals

• Any moving electric charge generates a magnetic field (B-field).
  • Applies to a single electron flying through space, a bulk flow of charge (current) in a copper wire, or atomic currents inside a permanent magnet.
• SI unit of magnetic-field strength: tesla (T).
  • 1\,\text{T}=1\,\dfrac{\text{N·s}}{\text{m·C}}
  • Large unit; laboratory & Earth-field values often expressed in gauss (G).
  • $1\,\text{T}=10^4\,\text{G}$

Classification of Materials

• All substances respond to magnetic fields and are placed in three categories:

  Diamagnetic Materials

  • No unpaired electrons → no permanent dipole moments.
  • Weakly repelled by external B-fields ("anti-magnetic").
  • Everyday examples: wood, plastic, water, glass, skin, etc.

  Paramagnetic Materials

  • Possess unpaired electrons → each atom has a permanent dipole.
  • In absence of field, dipoles are randomly oriented → net B = 0.
  • In an external B-field: dipoles align slightly → material becomes weakly magnetized.
  • Alignment is temporary; thermal motion randomizes dipoles once field is removed.
  • Examples: aluminum, copper, gold.

  Ferromagnetic Materials

  • Also contain unpaired electrons and permanent dipoles.
  • Distinguishing feature: can become strongly magnetized when exposed to a B-field or when cooled below a critical temperature (Curie temperature).
  • Dipoles organize into large, cooperative regions called domains; external field or domain alignment yields permanent magnets.
  • Common examples: iron (Fe), nickel (Ni), cobalt (Co).
  • Bar magnets: ferromagnetic rods with a permanent domain alignment producing distinct north (N) and south (S) poles.
  • Field lines exit N and enter S.
  • Because B-field lines form closed circles, magnetic monopoles do not exist.
  • Interaction rule: opposite poles attract; like poles repel.

Magnetic Field from Current-Carrying Conductors

Because a current is just moving charge, a wire generates a surrounding magnetic field.

Key Geometries (MCAT favorites)

• Infinitely long, straight wire

  • Magnitude at perpendicular distance $r$:
    $B = \dfrac{\mu_0 I}{2\pi r}$
  • $\mu_0$ (permeability of free space) =4\pi \times 10^{-7}\,\text{T·m/A}.
  • Reveals an inverse relationship $B \propto 1/r$.
  • Field lines are concentric circles around the wire.

• Circular loop of radius $r$

  • Magnitude at the center of the loop:
    $B = \dfrac{\mu_0 I}{2r}$
  • Note the missing $\pi$ and that the formula only applies at the center, not at arbitrary points.

Right-Hand Rules (RHR)

Two distinct RHRs appear throughout magnetism:

1. RHR for Field Direction around a Current

   • Thumb ➜ direction of conventional current $I$ (positive charge flow).
   • Curl fingers ➜ direction of the resulting magnetic-field loops.

2. RHR for Magnetic Force on a Moving Charge / Current Segment

   • Thumb ➜ velocity $\vec v$ of positive charge (or current direction $\vec I$).
   • Fingers ➜ external magnetic field $\vec B$.
   • Palm ➜ magnetic-force direction $\vec F_B$ for positive charges / currents.
   • Back of hand ➜ $\vec F_B$ for negative charges (e.g., electrons).

Example: Circular Current Loop

• Conditions
  • Wire forms one loop, carries $I = 0.25\,\text{A}$ clockwise (as viewed).
  • Diameter $=1\,\text{m} \Rightarrow r = 0.5\,\text{m}$.
• Directions (RHR-1)
  • Orient thumb tangent to current; curling fingers point into the page inside the loop and out of the page outside.
• Magnitude at center (loop formula) $B = \dfrac{\mu_0 I}{2r} = \dfrac{(4\pi \times 10^{-7})\,(0.25)}{2\,(0.5)} \approx 3.14 \times 10^{-7}\,\text{T}$
  • Converting: $3.14 \times 10^{-7}\,\text{T} = 3.14 \times 10^{-3}\,\text{G}$ (≈3.1 mG).

Magnetic Force on Moving Charges (Lorentz Force)

• Total electromagnetic force = Lorentz force.
• For purely magnetic part on a single charge: $F_B = q v B \sin\theta$
  • $q$: charge.
  • $v$: speed.
  • $B$: magnetic-field magnitude.
  • $\theta$: smallest angle between $\vec v$ and $\vec B$.
• Key consequences
  • Force exists only if motion has a component perpendicular to $\vec B$ (because of $\sin\theta$).
  • $\vec F_B$ is always perpendicular to both $\vec v$ and $\vec B$ → causes uniform circular or helical motion, not work/energy change.

Example: Proton in Uniform Field

• Given
  • Proton (charge $q<em>p = +1.6\times10^{-19}\,\text{C}$, mass $m</em>p = 1.67\times10^{-27}\,\text{kg}$)
  • Velocity: $v = 15\,\text{m/s}$ toward top of page.
  • Field: $B = 3.0\,\text{T}$ into page.
• Magnitude
  $F_B = q v B \sin90^{\circ} = (1.6\times10^{-19})(15)(3.0) \approx 7.2\times10^{-18}\,\text{N}$
• Direction (RHR-2)
  • Thumb ↑ (velocity), fingers ✋ into page (field) ⇒ palm points left ⇒ force leftward.
• Resulting motion
  • $\vec v$ ⟂ $\vec F_B$ ⇒ uniform circular motion.
  • Equate magnetic force to centripetal: $m v^2/r = q v B \Rightarrow r=\dfrac{m v}{q B}$.
    $r = \dfrac{(1.67\times10^{-27})(15)}{(1.6\times10^{-19})(3.0)} \approx 5.2\times10^{-8}\,\text{m}$ (≈52 nm).

Magnetic Force on Current-Carrying Wire

• Straight segment in uniform $\vec B$ experiences: $F_B = I L B \sin\theta$
  • $I$: current.
  • $L$: length of wire within field (vector points with current).
  • $\theta$: angle between $\vec L$ and $\vec B$.
• Direction: use same RHR-2, replacing $\vec v$ with current direction $\vec I$ (positive charge convention).

Example: 2-m Wire in 30-G Field

• Parameters
  • $L = 2.0\,\text{m}$, $I = 5.0\,\text{A}$ up the page.
  • $B = 30\,\text{G} = 30\times10^{-4}\,\text{T} = 3.0\times10^{-3}\,\text{T}$ into page.
  • $\theta = 90^{\circ}$ (perpendicular).
• Magnitude
  $F_B = (5.0)(2.0)(3.0\times10^{-3})\sin90^{\circ} = 3.0\times10^{-2}\,\text{N}$.
• Direction (RHR-2)
  • Thumb ↑ (current), fingers into page ⇒ palm points left ⇒ force leftward on wire.

Broader Connections & Recap (Electricity + Magnetism)

• Electric charge comes in + and – varieties; unlike gravity, electrostatic forces can be attractive or repulsive.
• Conductors vs. insulators determine charge mobility.
• Electric fields created by charges exert forces $F_E = qE$; concept parallels gravitational fields.
• Coulomb’s law mirrors Newton’s law of gravitation but with possibility of repulsion.
• Charges carry electric potential energy that changes as they move through an electric potential difference (voltage).
• An electric dipole features separated +q and –q; its potential distribution obeys $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p\cos\theta}{r^2}$ (where $p = qd$ is dipole moment).
• Magnetism adds a second field type influencing moving charges and currents but doing no work (force ⟂ motion).
• Upcoming exploration: detailed study of electric circuits, where moving charges interact with resistors, capacitors, sources, etc.