Magnetism & Electromagnetic Induction

Magnetism

Part 1 of 4 — Magnetic Fields & Their Sources

The Magnetic Field

The magnetic field $\vec{B}$ is measured in tesla: $1\;\text{T} = 1\;\text{N}/(\text{A}\cdot\text{m})$. A tesla is huge — everyday fields are quoted in gauss ($1\;\text{G} = 10^{-4}\;\text{T}$). Earth's field is about $0.5\;\text{G}$; a clinical MRI magnet is $1.5$ to $3\;\text{T}$, tens of thousands of times stronger.

No Magnetic Monopoles

Cut a bar magnet in half and you get two smaller magnets, never an isolated north or south pole. Magnetic field lines therefore have no start or end — every line is a closed loop (out of the north pole, around, and back in the south). Contrast electrostatics, where lines begin on $+$ and end on $-$ charges.

Moving Charge Makes Magnetic Field

All magnetism traces back to moving charge — currents in wires, or electron motion inside atoms.

Long straight wire carrying current $I$, at distance $r$:

$B = \frac{\mu_0 I}{2\pi r}$

with $\mu_0 = 4\pi \times 10^{-7}\;\text{T}\cdot\text{m/A}$, so the handy lump is $\frac{\mu_0}{2\pi} = 2 \times 10^{-7}$ in SI units.

• Field lines are concentric CIRCLES around the wire.
• Right-hand curl rule: thumb along the current, fingers curl the way $\vec{B}$ circulates.
• Note the $1/r$ falloff — NOT inverse-square. Doubling distance halves the field.

Loops and solenoids: a current loop makes a field like a tiny bar magnet (a magnetic dipole) through its center; stacking loops into a solenoid gives a strong, nearly uniform field inside — the architecture of MRI magnets.

Magnetic Materials

• Ferromagnetic (Fe, Ni, Co): atomic dipoles lock into aligned domains — strongly attracted to magnets and can stay magnetized.
• Paramagnetic (materials with unpaired electrons, e.g. deoxyhemoglobin): weakly ATTRACTED while a field is applied; alignment vanishes when it is removed.
• Diamagnetic (paired electrons only — water, most tissue, oxyhemoglobin): very weakly REPELLED by fields.

The paramagnetic/diamagnetic switch between deoxy- and oxyhemoglobin is the physical basis of the fMRI BOLD signal — a favorite passage hook.

Worked Example — Field Near a Wire

A power cable carries $I = 10\;\text{A}$. Find the magnetic field $5\;\text{cm}$ away, and compare it to Earth's field.

Step 1 — Plug into the wire formula, using the lump.

$B = \frac{\mu_0 I}{2\pi r} = (2 \times 10^{-7})\frac{10}{0.05}$

Step 2 — Clean arithmetic. $\frac{10}{0.05} = 200$, so

$B = (2 \times 10^{-7})(200) = 4 \times 10^{-5}\;\text{T} = 0.4\;\text{G}$

Step 3 — Interpret. Comparable to Earth's $0.5\;\text{G}$ — enough to nudge a compass, far too weak to affect tissue.

Worked Example — Scaling, Not Plugging

Two long parallel wires, P and Q, each carry current $I$. The field due to P alone at distance $d$ is $B_0$. What is the field due to P at $2d$, and what happens if P's current is also tripled?

Step 1 — Distance factor. $B \propto 1/r$ (single power, unlike Coulomb's $1/r^2$): doubling $r$ gives $\frac{1}{2}B_0$.

Step 2 — Current factor. $B \propto I$: tripling gives $3 \times \frac{1}{2}B_0 = 1.5\,B_0$.

The MCAT tests exactly this discrimination: wire fields fall off as $1/r$, point-charge forces as $1/r^2$. Keep the exponents straight.

Magnetic Fields & Their Sources 🎯

Key Takeaways — Part 1

• $B$ in tesla; $1\;\text{G} = 10^{-4}\;\text{T}$; Earth about $0.5\;\text{G}$, MRI $1.5$ to $3\;\text{T}$
• No monopoles: field lines are closed loops, never starting or ending
• Straight wire: $B = \mu_0 I/2\pi r$ — circles around the wire, $1/r$ falloff (not $1/r^2$); right-hand curl rule
• Loops are dipoles; solenoids give uniform interior fields (MRI architecture)
• Ferro = strong and persistent; para = weakly attracted (deoxyhemoglobin); dia = weakly repelled (water, oxyhemoglobin)

Magnetism

Part 2 of 4 — Forces on Moving Charges

The Magnetic Force Law

A charge $q$ moving with speed $v$ in a field $B$ feels

$F = qvB\sin\theta$

where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. Read the three switches this builds in:

• $v = 0$ → no force. Magnetic fields ignore stationary charges (electric fields do not).
• $\vec{v}$ parallel or antiparallel to $\vec{B}$ ($\theta = 0$ or $180^\circ$) → no force.
• Maximum force at $\theta = 90^\circ$: $F = qvB$.

Direction: the Right-Hand Rule

Point fingers along $\vec{v}$, curl toward $\vec{B}$ (or: fingers along $\vec{B}$, thumb along $\vec{v}$, palm pushes) — the force on a POSITIVE charge pops out perpendicular to both. For a NEGATIVE charge, take the right-hand answer and flip it (or use your left hand).

With standard axes, remember one anchor case: $\hat{x} \times \hat{y} = \hat{z}$. A positive charge moving along $+x$ in a field along $+y$ is pushed along $+z$.

Magnetic Forces Do No Work

The force is always perpendicular to the velocity, so it never has a component along the displacement:

$W = 0 \quad\Rightarrow\quad$ speed and kinetic energy are CONSTANT.

A magnetic field can steer a charge but never speed it up or slow it down. Only electric fields change a charge's kinetic energy — the single most-tested conceptual point in this chapter.

Circular Motion in a Uniform Field

A charge moving perpendicular to a uniform $\vec{B}$ is steered in a circle: the magnetic force IS the centripetal force.

$qvB = \frac{mv^2}{r} \quad\Rightarrow\quad r = \frac{mv}{qB}$

• Faster or heavier particles sweep LARGER circles; stronger fields or bigger charges force TIGHTER ones.
• The period $T = \frac{2\pi m}{qB}$ is independent of speed — fast particles just travel bigger circles in the same time.
• Velocity partly along $\vec{B}$? That component is untouched, and the path becomes a helix around the field lines.

Worked Example — A Proton in a Clinical-Scale Field

A proton ($m = 1.67 \times 10^{-27}\;\text{kg}$, $q = 1.6 \times 10^{-19}\;\text{C}$) moves at $2 \times 10^6\;\text{m/s}$ perpendicular to a $0.5\;\text{T}$ field.

Step 1 — Force.

$F = qvB = (1.6 \times 10^{-19})(2 \times 10^6)(0.5) = 1.6 \times 10^{-13}\;\text{N}$

Tiny in absolute terms, but acting on a proton's tiny mass it bends the path hard.

Step 2 — Radius of the circle.

$r = \frac{mv}{qB} = \frac{(1.67 \times 10^{-27})(2 \times 10^6)}{(1.6 \times 10^{-19})(0.5)} = \frac{3.34 \times 10^{-21}}{8 \times 10^{-20}} \approx 0.04\;\text{m}$

About $4\;\text{cm}$. MCAT arithmetic style: divide the powers of ten first ($10^{-21}/10^{-20} = 10^{-1}$), then $3.34/8 \approx 0.4$.

Step 3 — Energy check. The proton's speed after any amount of circling: still $2 \times 10^6\;\text{m/s}$. Zero work done.

Worked Example — Electron vs. Proton, Same Field

An electron and a proton enter the same field with the same velocity. Compare their paths.

Charge sign: opposite forces → they curve in OPPOSITE rotational senses.

Radius: $r = mv/qB$ with equal $v$, equal $|q|$ — the radius scales with mass. The electron's circle is smaller by $m_e/m_p \approx 1/1836$: the proton makes a wide arc while the electron whips around a tight loop.

Forces on Moving Charges 🎯

Key Takeaways — Part 2

• $F = qvB\sin\theta$: zero for stationary charges and for motion along the field; max when perpendicular
• Right-hand rule for positive charges; flip the answer for negative ones ($\hat{x} \times \hat{y} = \hat{z}$ anchor)
• Magnetic force does NO work — it steers, never speeds up or slows down; only $E$ fields change KE
• Perpendicular entry → circle with $r = mv/qB$; period $2\pi m/qB$ is speed-independent; oblique entry → helix
• Same speed, same field: electron and proton curve opposite ways, electron's circle about 1836 times tighter

Magnetism

Part 3 of 4 — Wires, Velocity Selectors & the Mass Spectrometer

Force on a Current-Carrying Wire

A wire is a stream of moving charges, so a field pushes on it:

$F = BIL\sin\theta$

where $L$ is the length of wire in the field and $\theta$ the angle between the current direction and $\vec{B}$.

• Same right-hand rule: fingers along $I$ (conventional current), curl toward $\vec{B}$, thumb-side push.
• Wire parallel to the field → zero force; perpendicular → maximum $F = BIL$.
• This is the working principle of electric motors and of the galvanometer needle inside analog meters.

Parallel Wires

Each wire sits in the other's circular field:

• Currents in the SAME direction: wires ATTRACT.
• OPPOSITE directions: wires REPEL.

(Memory hook: it is the opposite of charges — "likes attract" for currents.)

The Velocity Selector

Cross an electric field and a magnetic field at right angles, both perpendicular to a charged particle's velocity, arranged so the electric force ($qE$) and magnetic force ($qvB$) oppose each other. They balance only at one speed:

$qE = qvB \quad\Rightarrow\quad v = \frac{E}{B}$

• Particles at exactly $v = E/B$ fly straight through — regardless of their mass OR charge (both cancel).
• Faster particles feel excess magnetic force and deflect one way; slower ones deflect the other.

The Mass Spectrometer

After the selector fixes $v$, the beam enters a region with only a magnetic field and bends in a circle:

$r = \frac{mv}{qB}$

With $v$, $q$, and $B$ known, the radius reads out the MASS:

• Heavier ions → larger radius (harder to turn).
• Higher charge → smaller radius (stronger force).
• Strictly, the machine separates ions by mass-to-charge ratio $m/q$ — a doubly charged heavy ion can land on the same spot as a singly charged light one. MCAT passages exploit exactly this ambiguity.

Worked Example — Force on a Motor Segment

A $2.0\;\text{m}$ segment of wire carries $3.0\;\text{A}$ perpendicular to a $0.5\;\text{T}$ field.

Step 1 — Apply the wire-force law.

$F = BIL = (0.5)(3.0)(2.0) = 3.0\;\text{N}$

Step 2 — Angle check. If the wire were instead at $30^\circ$ to the field: $F = 3.0\sin 30^\circ = 1.5\;\text{N}$. Only the perpendicular geometry gives the full force.

Worked Example — Selector plus Spectrometer (Passage Style)

Ions of mass $3.2 \times 10^{-26}\;\text{kg}$ and charge $+e$ pass through a velocity selector with $E = 2000\;\text{V/m}$ and $B = 0.4\;\text{T}$, then enter an analyzing region with the same $0.4\;\text{T}$ field.

Step 1 — Selected speed.

$v = \frac{E}{B} = \frac{2000}{0.4} = 5000\;\text{m/s}$

Step 2 — Radius in the analyzer.

$r = \frac{mv}{qB} = \frac{(3.2 \times 10^{-26})(5000)}{(1.6 \times 10^{-19})(0.4)} = \frac{1.6 \times 10^{-22}}{6.4 \times 10^{-20}} = 2.5 \times 10^{-3}\;\text{m}$

A $2.5\;\text{mm}$ radius. Powers of ten first: $10^{-22}/10^{-20} = 10^{-2}$; then $1.6/6.4 = 0.25$.

Step 3 — Isotope logic. An isotope one atomic mass unit heavier, same charge and same selected $v$, lands at a proportionally larger radius — that spatial spread IS the mass spectrum.

Wires, Selectors & Spectrometers 🎯

Key Takeaways — Part 3

• Wire in a field: $F = BIL\sin\theta$ — zero when parallel, max when perpendicular; basis of motors
• Parallel currents attract, antiparallel repel
• Velocity selector: crossed $E$ and $B$ pass only $v = E/B$, independent of mass and charge
• Mass spectrometer: $r = mv/qB$ converts mass to radius; really sorts by $m/q$
• MCAT arithmetic: strip powers of ten first, then do one clean small-number division

Magnetism

Part 4 of 4 — MCAT Integration: Induction, MRI & Passage Strategy

Magnetic Flux

$\Phi = BA\cos\theta$

in webers ($1\;\text{Wb} = 1\;\text{T}\cdot\text{m}^2$), where $\theta$ is the angle between $\vec{B}$ and the NORMAL to the loop's area. Flux is maximal when the field threads straight through the loop and ZERO when the field lies in the loop's plane.

Faraday's Law — Changing Flux Makes EMF

$|\varepsilon| = N\frac{\Delta\Phi}{\Delta t}$

An EMF appears in a loop of $N$ turns whenever the flux through it CHANGES. Three knobs, straight from $\Phi = BA\cos\theta$:

1. change the field strength $B$,
2. change the loop's area $A$,
3. rotate the loop (change $\theta$) — this one is the electric generator.

A steady flux — however enormous — induces nothing. Rate of change is everything.

Lenz's Law — the Minus Sign

The induced current flows in the direction whose OWN magnetic field opposes the change in flux that created it.

• Flux increasing into the loop → induced current circulates to push flux back out.
• Pull a magnet's north pole AWAY → the loop makes a south-facing pole to hold on.

Lenz is conservation of energy in disguise: if the induced current instead reinforced the change, you would get runaway free energy. It is also why you must do work to shove a magnet into a coil.

Strategy — the MRI Passage

An MRI scanner is a physics anthology the MCAT keeps reusing:

• The main solenoid produces a uniform $1.5$ to $3\;\text{T}$ field; superconducting wire carries huge persistent current with zero resistance.
• Hydrogen nuclei (protons) act as tiny magnetic dipoles that align with the field and precess; a radio-frequency pulse tips them, and their relaxation emits the detected RF signal.
• Rapidly switched gradient coils change flux fast — inducing eddy currents (and the loud banging).
• Safety: ferromagnetic objects become projectiles; implanted wires can develop induced EMFs. Nothing about the STATIC field harms diamagnetic tissue.

Strategy — Answer Induction Questions with a Checklist

1. Is flux actually changing? If not, EMF is zero — done.
2. Which way is it changing (in/out, growing/shrinking)?
3. Lenz: the induced current opposes THAT change.
4. Magnitude: $N\,\Delta\Phi/\Delta t$ — watch for turns $N$ quietly given in the stem.

Worked Example — Field Collapse in a Coil

A circular coil of $50$ turns and area $0.2\;\text{m}^2$ sits with its plane perpendicular to a $0.5\;\text{T}$ field (so the field is along the normal, $\cos\theta = 1$). The field is switched off over $0.1\;\text{s}$.

Step 1 — Initial flux through one turn.

$\Phi = BA = (0.5)(0.2) = 0.1\;\text{Wb}$

Step 2 — EMF magnitude.

$|\varepsilon| = N\frac{\Delta\Phi}{\Delta t} = 50 \times \frac{0.1}{0.1} = 50\;\text{V}$

Step 3 — Direction by Lenz. Outward-through-the-coil flux is DYING, so the induced current circulates to prop it up — counterclockwise as seen from the side the field pointed toward. If the coil's circuit resistance is $25\;\Omega$, the induced current is $I = 50/25 = 2\;\text{A}$ while the collapse lasts.

Worked Example — Magnet and Loop (Sign Logic Only)

A bar magnet's north pole is pushed TOWARD a conducting loop.

Step 1 — Flux change. Field lines exit north poles, so flux through the loop (toward it) is increasing.

Step 2 — Lenz. The loop opposes the increase: its induced current makes a NORTH pole facing the incoming magnet, repelling it.

Step 3 — Energy audit. You feel that repulsion as resistance to your push; the work you do against it is exactly the electrical energy dissipated in the loop. Reverse the motion (pull away) and everything flips: the loop shows a south face and tugs the magnet back.

MCAT Integration: Induction & MRI 🎯

Key Takeaways — Part 4

• $\Phi = BA\cos\theta$ (webers); angle measured from the loop's normal, zero flux when field lies in the plane
• Faraday: $|\varepsilon| = N\,\Delta\Phi/\Delta t$ — only CHANGING flux induces; change $B$, $A$, or orientation
• Lenz: induced current opposes the change; it is conservation of energy, and why generators resist being turned
• MRI: superconducting solenoid, precessing proton dipoles, RF pulses, switching gradients inducing eddy currents
• Checklist: flux changing? which way? oppose it; then $N\,\Delta\Phi/\Delta t$ for size