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 qp = +1.6\times10^{-19}\,\text{C}, mass mp = 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.