Magnetism Part 2: Wires, Spectrometers & Induction

Magnetism Part 2: Wires, Spectrometers & Induction

Part 1 of 2 — Wires, Velocity Selectors & the Mass Spectrometer

Where this fits. The Magnetism Part 1: Fields & Forces on Charges lesson gave you the field of a current (a long wire's field circles it, $B = \mu_0 I/2\pi r$) and the force on a single moving charge, $F = qvB\sin\theta$, pointed by the right-hand rule (fingers along $\vec{v}$, curl toward $\vec{B}$, thumb gives the force on a positive charge). This lesson scales that force up to whole currents and ion beams: the push on a wire, the force between parallel wires, the velocity selector and the mass spectrometer. Part 2 adds induction, how a changing field makes an EMF, and the MRI passage.

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 as for a single charge: fingers along $I$ (conventional current), curl toward $\vec{B}$, thumb-side push.
• Wire parallel to the field → zero force; perpendicular → maximum $F = BIL$, the working principle of electric motors.

• The same push swings the galvanometer needle inside analog meters.

Parallel Wires

Each wire sits in the other's circular field (the $B = \mu_0 I/2\pi r$ field that curls around any current), so each feels $F = BIL$ from the other:

• 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:

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

The combined force $\vec{F} = q\vec{E} + q\vec{v} \times \vec{B}$ is the Lorentz force; the selector is the special case where its two parts cancel.

• 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 (the magnetic force supplies the centripetal force, $qvB = mv^2/r$):

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

With $v$, $q$, and $B$ known, the radius reads out the MASS: heavier ions sweep a larger radius (harder to turn), and higher charge a smaller one (stronger force).

• Strictly, the machine separates ions by mass-to-charge ratio $m/q$ — an ion twice as heavy with twice the charge lands on the same spot as the lighter, singly charged one. MCAT passages exploit exactly this ambiguity.

Worked Example — Force on a Motor Segment

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

• (a) What magnetic force acts on the segment?
• (b) What would the force be if the wire were instead at $30^\circ$ to the field?

(a) Apply the wire-force law. $F = BIL = (0.5)(3.0)(2.0) = 3.0\;\text{N}$

(b) Angle check. With the full $3.0\;\text{N}$ from (a): $F = 3.0\sin 30^\circ = 1.5\;\text{N}$. Only the perpendicular geometry gives the full force.

Worked Example — Selector plus Spectrometer (Passage Style)

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

• (a) What speed passes through the selector undeflected?
• (b) What is the radius of the ions' path in the analyzer?
• (c) Where does an isotope one atomic mass unit heavier, with the same charge, land?

(a) Selected speed. $v = \dfrac{E}{B} = \dfrac{2000}{0.4} = 5000\;\text{m/s}$

(b) Radius in the analyzer. With $v = 5000\;\text{m/s}$ from (a):

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

A $2.5\;\text{mm}$ radius.

(c) Isotope logic. The selector passes it at the same $v$, and with the same charge and field $r \propto m$, so it lands at a proportionally larger radius. That spatial spread IS the mass spectrum.

Wires, Selectors & Spectrometers 🎯

Key Takeaways — Part 1

• Force on a wire. $F = BIL\sin\theta$: zero when the wire lies along the field, maximum when perpendicular; motors.
• Parallel currents. Currents in the same direction attract; currents in opposite directions repel.
• Velocity selector. Crossed $E$ and $B$ fields pass only $v = E/B$, whatever the particle's mass or charge.
• Mass spectrometer. $r = mv/qB$ turns mass into radius; strictly, it sorts ions by $m/q$.

Common traps

• Like currents repel. Parallel currents ATTRACT, the reverse of like charges, which repel.
• Selector sorts by mass. A velocity selector picks only a speed: mass and charge cancel from $qE = qvB$.
• Radius reads mass alone. An ion of twice the mass and twice the charge lands where the lighter, singly charged one does.
• Full force at any angle. Only a perpendicular wire feels the full $BIL$; at $30^\circ$ to the field it feels half.

• Low-yield extras: the galvanometer needle in analog meters as an application of the wire force.

Magnetism Part 2: Wires, Spectrometers & Induction

Part 2 of 2 — 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 (the energy per charge a source supplies, from the Circuits Part 3: EMF & Capacitance lesson) appears in a loop of $N$ turns whenever the flux through it CHANGES; a steady flux, however enormous, induces nothing. Three knobs, straight from $\Phi = BA\cos\theta$:

• Change the field strength $B$.
• Change the loop's area $A$. A rod of length $L$ sliding at speed $v$ along rails across a field sweeps out area, giving $\varepsilon = BLv$.
• Rotate the loop (change $\theta$): this one is the electric generator.

Lenz's Law — the Minus Sign

With its sign, Faraday's law reads $\varepsilon = -N\,\Delta\Phi/\Delta t$; the minus sign is Lenz's law. 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:

• A superconducting solenoid (zero resistance, persistent current) makes a uniform $1.5$ to $3\;\text{T}$ field.
• Protons act as tiny dipoles that align with it; an RF pulse tips them, and as they relax they emit the detected signal.
• Switched gradient coils change the flux fast, inducing eddy currents.
• Safety: ferromagnetic objects become projectiles and implanted wires can pick up induced EMFs; the static field does not harm tissue.

• The loud banging during a scan comes from the gradient coils themselves: each time their current switches, the main field shoves on the windings ($F = BIL$).

Strategy — Answer Induction Questions with a Checklist

• First: is the flux actually changing? If not, the EMF is zero and you are done.
• Then: which way is it changing (in or out, growing or shrinking)?
• Lenz: the induced current opposes THAT change.
• Magnitude: $N\,\Delta\Phi/\Delta t$ — watch for turns $N$ quietly given in the stem.

Worked Example — Field Collapse in a Coil

Problem. A $50$-turn coil of area $0.2\;\text{m}^2$ has its plane perpendicular to a $0.5\;\text{T}$ field (field along the normal, $\cos\theta = 1$). The field is switched off over $0.1\;\text{s}$; the coil's circuit has $25\;\Omega$.

• (a) What is the initial flux through one turn?
• (b) What EMF is induced while the field collapses?
• (c) Which way does the induced current circulate?
• (d) How large is the induced current?

(a) Initial flux. $\Phi = BA = (0.5)(0.2) = 0.1\;\text{Wb}$

(b) EMF magnitude. With $\Delta\Phi = 0.1\;\text{Wb}$ per turn from (a):

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

(c) Direction by Lenz. The flux is DYING, so the induced current circulates to prop it up: its own field points the same way as the original one. That is counterclockwise viewed with the field pointing at you.

(d) Induced current. With $|\varepsilon| = 50\;\text{V}$ from (b): $I = 50/25 = 2\;\text{A}$ while the collapse lasts.

Worked Example — Magnet and Loop (Sign Logic Only)

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

• (a) How does the flux through the loop change, and what pole does the loop present to the magnet?
• (b) Where does the energy dissipated in the loop come from, and what changes if the magnet is pulled away?

(a) Flux and Lenz. Field lines exit north poles, so as the pole approaches, the flux through the loop increases. The loop opposes the increase: its induced current makes a NORTH pole facing the incoming magnet, repelling it.

(b) Energy and reversal. The work you do pushing against that repulsion is exactly the electrical energy dissipated in the loop. Pull the magnet away and everything flips: the flux decreases, so the loop shows a south face and tugs the magnet back.

MCAT Integration: Induction & MRI 🎯

Key Takeaways — Part 2

• Magnetic flux. $\Phi = BA\cos\theta$ in webers, with $\theta$ from the loop's normal; zero when $\vec{B}$ lies in the plane.
• Faraday's law. $|\varepsilon| = N\,\Delta\Phi/\Delta t$: only CHANGING flux induces an EMF, by changing $B$, $A$ or $\theta$.
• Lenz's law. The induced current opposes the change in flux; that is energy conservation, so pushing a magnet in takes work.
• The pole rule. An approaching north pole meets a north face and is repelled; a retreating one meets a south face and is attracted.

Common traps

• Steady flux induces EMF. A huge but constant flux induces nothing; only its rate of change matters.
• Angle from the plane. $\theta$ is measured from the normal, so a field lying in the loop's plane gives zero flux.
• Opposing the field itself. The current opposes the CHANGE: a dying flux is propped up, not reversed.
• Forgetting the turns. Multiply by $N$: a $100$-turn coil gives $100$ times the EMF of a single loop.

• Low-yield extras: the gradient coils' banging during a scan.

Next up: the Optics lesson — light as an electromagnetic wave, reflection and refraction, and image formation by mirrors and lenses.