Magnetism · RHR · Ampère's Law · Torque · Solenoid

This is the most fundamental formula in magnetism. Every piece of it has a meaning you should understand, not just memorize.

• B = magnetic field strength at a point, measured in Tesla (T). One Tesla is a HUGE field — MRI machines use 1-3 T, Earth's field is 50 microtesla (about 20,000× weaker).
• μ₀ = permeability of free space = 4π × 10⁻⁷ T·m/A. This is a universal constant of nature, like c (speed of light) or G (gravitational constant). It's an exact value by definition (since 2019) and represents "how much magnetic field you get per amp in empty space."
• I = current in the wire, in amps. More current → more field, linearly.
• 2π = geometric factor from the circular geometry. The field wraps around the wire in circles, so the "total B times circle length" follows Ampère's Law. The 2π comes from the circumference of a circle (2πr).
• r = perpendicular distance from the wire to the point where you're measuring B, in meters. NOT the distance traveled along the wire.

The 1/r dropoff: doubling your distance from the wire halves the field. Tripling distance thirds it. This is slower than a point-charge's 1/r² electric field but still pretty rapid. At 10 cm, the field is 10× weaker than at 1 cm.

Linear in current: doubling the current doubles the field at any fixed point. Multiply by 10 → field multiplies by 10. This is why high-current applications (MRI coils, electromagnets in junkyards) can create such strong fields — just pump in more amps.

Why μ₀ has 4π built in: the "4π" comes from solid-angle geometry in 3D space. Putting 4π into μ₀ makes the 2π in the denominator of B = μ₀I/(2πr) come out naturally from the geometry of a circle. It's a historical choice that cleans up other formulas.

This formula only works for long straight wires. If you have a short wire or a curved one, you need the Biot-Savart law (more general). But since most problems involve long wires, this formula handles 90% of exam questions.

Every field-from-a-wire problem has two parts: magnitude (use the formula) and direction (use RHR #1). Always solve direction first — it's fast and gives you confidence before you start crunching numbers.

The RHR for this problem: current points south (downward). Grip the wire with your right hand, thumb pointing down. Your fingers curl into the page on the left side of the wire and out of the page on the right side. Since the observer is 2 cm to the right, they see B pointing out of the page.

Cross-product notation (on the diagram):

• ⊙ (dot in circle) = vector pointing OUT of the page (toward you). Think of it as looking at the tip of an arrow flying at you.
• ⊗ (X in circle) = vector pointing INTO the page (away from you). Think of it as looking at the feathers of an arrow flying away.

You'll see these symbols in virtually every 2D magnetism diagram. Memorize them.

Magnitude calculation shortcut: B = μ₀I/(2πr) = (4π×10⁻⁷ × 45)/(2π × 0.02). The π in numerator and denominator cancel: B = (4 × 10⁻⁷ × 45)/(2 × 0.02) = 180×10⁻⁷/0.04 = 4500×10⁻⁷ = 4.5×10⁻⁴ T.

How big is 4.5 × 10⁻⁴ T? About 9× Earth's magnetic field — still tiny in absolute terms, but strong enough that a compass 2 cm from the wire would align with the wire's field and point in the "new" direction, ignoring Earth. This is exactly what Ørsted first noticed in 1820.

Always carry units: meters (for distance), amps (for current), teslas (for field). Mixing cm with meters is the #1 magnitude mistake in all of magnetism.

The algebra here is just rearranging B = μ₀I/(2πr) to solve for r: multiply both sides by r, divide both by B, and you get r = μ₀I/(2πB). Plug in numbers and you get 2.5 mm.

The 4π / 2π shortcut: notice that μ₀ contains 4π and the formula has 2π in the denominator. The ratio 4π/(2π) = 2, so you can rewrite the formula as r = 2I × 10⁻⁷ / B. Much cleaner: r = 2(10)(10⁻⁷)/(8×10⁻⁴) = 20×10⁻⁷/(8×10⁻⁴) = 2.5×10⁻³ m. Same answer, less typing.

Ørsted's original experiment (1820): the last quote is a reference to Hans Christian Ørsted's famous discovery. He was lecturing about how electric currents produce heat, when he noticed that a compass needle happened to twitch whenever he closed the battery circuit. That single observation — a moving compass next to a current — was the first experimental evidence that electricity and magnetism are connected. Before Ørsted, they were thought to be completely independent phenomena. His discovery kicked off a decade of rapid progress, culminating in Faraday's induction discovery and then Maxwell's equations.

Try it yourself (for real): take any compass, hold it near a wire connected to a battery, and watch the needle swing as you make and break the circuit. The effect is strongest near a coiled wire (solenoid), where the field is concentrated. This was the first experimental verification of Ampère's Law — you can literally recreate it in 30 seconds with a pocket compass and a 9V battery.

Why the result is so small: 2.5 mm is uncomfortably close to a wire. That means an 8×10⁻⁴ T field isn't very strong. Most electronic devices keep their magnetic fields even smaller than Earth's — fields drop off fast (1/r), so staying a few centimeters away gets you down to negligible levels.

F = ILB sin θ is the equation that turns electricity into motion. Every electric motor, every speaker, every magnetic-drive hard disk, every maglev train — they all work because current-carrying wires in magnetic fields feel forces. This single formula underlies trillions of dollars of global industry.

What's going on physically: a current is a stream of charges moving along a wire. Each charge feels F = qv×B. When you sum those individual forces over all the charges in the wire, you get F_total = IL × B (vector form), with magnitude ILB sin θ. The wire "feels" the force as a bulk push because it's bolted together — individual charges push the lattice of the metal.

Stationary charges feel no magnetic force. This is a defining property of B fields — unlike E fields, which push any charge regardless of motion, B only interacts with moving charges. A bar magnet next to a motionless pile of electrons does nothing. The second the electrons move, they suddenly feel a force.

The θ = 0° case: a wire running parallel to a field feels zero force. Think about it: if all the charges are moving along the field direction, there's no "crosswind" for them to experience. The field just slides past without interacting. This is why helical particle motion in a magnetic field has a straight-line component along B (no force) and a circular component perpendicular to B (max force).

The θ = 90° case: maximum force. This is what every motor and speaker is designed for — make the current perpendicular to B so you get the biggest possible mechanical output per amp.

Mnemonic: "I Love Babies" = I, L, B. The sin θ is the only angular modifier. If in doubt, draw the angle between the current direction and the B-field direction, and measure that angle.

This course uses two different right-hand rules that are easy to mix up. Here's the cheat sheet:

• RHR #1 — B field around a wire: thumb = current direction, curled fingers = B field direction (field circles around the wire).
• RHR #2 — Force on a current in a field: thumb = current, fingers = field, palm push = force.

When to use which: if you're asking "where does the field POINT?" → use RHR #1 (field created by the wire). If you're asking "which way does the wire FEEL a force?" → use RHR #2 (wire in an external field).

Why physics uses two different setups: they answer different questions. RHR #1 connects current → field (Ampère/Biot-Savart). RHR #2 connects current + external field → force (Lorentz law). They're independent pieces of physics.

A common student mistake: trying to use "curl-the-fingers" RHR #1 on force problems, or "thumb-palm" RHR #2 on field problems. If you're wrestling with your hand position and nothing feels right, stop and ask: "Am I looking at the field a current creates, or the force a current feels from an external field?" Choose the rule that matches the question.

Quick check for this slide: current north, field east, force... thumb up, four fingers to the right, palm pushes away from you (into the page). That's correct — force into page.

Test yourself: current up, field into page, force = ? (Answer: east, by RHR #2. Thumb up, fingers into page, palm opens east.)

This example hammers in the same right-hand rule from Slide 6, but with different input directions. Every problem is basically: "thumb = current, fingers = field, palm push = force." The three must be mutually perpendicular, like the edges of a corner of a cube.

The xyz relationship: if current I is along the x-axis and field B is along the y-axis, the force F is along the z-axis. Mathematically, F = IL × B. The "×" is the cross product, which produces a vector perpendicular to both inputs.

Think of it as a corner of a cube: pick any corner of a cube. Three edges meet there, each perpendicular to the other two. The right-hand rule assigns I, B, F to those three edges in a specific order: I (thumb) → B (index) → F (middle). If you curl from I to B, your thumb gives F. This is the "right-hand coordinate system."

For current east + B into page:

• Thumb points east (current)
• Fingers point into the page (field)
• Palm opens northward (force direction)

Alternative method — the FBI rule (not recommended): some teachers use "F, B, I" assignment with different fingers. This introduces confusion with the RHR for charges. Stick with the "palm push" method — one rule for everything.

Check with the xyz test: east = +x. Into page = −z (with out-of-page = +z). So F should be IL×B = x̂ × (−ẑ) = +ŷ, which points north. ✓ The cross-product formalism confirms the palm rule.

Practice advice: do this RHR test on every problem for the first ~20 problems until it becomes muscle memory. After that, you'll see a picture and the force direction will "appear" automatically in your head without conscious effort.

The problem shows a wire at 30° to a magnetic field, but there's actually two angles between them: 30° and 150° (the supplement). Which one do you use in F = ILB sin θ?

Answer: it doesn't matter. sin 30° = sin 150° = 0.5. This isn't a coincidence — it's because of a trigonometric identity: sin(180° − θ) = sin θ. The supplementary angle identity guarantees that however you measure the angle between wire and field, you'll get the same force magnitude.

Why the identity exists: the sine function is symmetric around 90°. It rises from 0 at 0° to 1 at 90°, then falls symmetrically back to 0 at 180°. Any pair of angles that are mirror images around 90° (like 30° and 150°) give the same sine value.

Practical upshot: always pick the smaller of the two angles — it's the one that's usually drawn and easier to read. But if you accidentally pick the larger one, your answer is still correct.

Arithmetic detail — "sin 30° = 0.5" is exact: (5)(2.5)(0.002)(0.5) = 0.0125 N. Some common sine values to memorize:

• sin 0° = 0
• sin 30° = 0.5
• sin 45° = √2/2 ≈ 0.707
• sin 60° = √3/2 ≈ 0.866
• sin 90° = 1

You'll see these over and over in physics problems. Having them memorized saves time and reduces calculator dependence.

Why the answer is so small: B = 0.002 T is about 40× Earth's field — unimpressive. Combined with a modest 5 A current and 2.5 m wire, you get 0.0125 N, or about the weight of a paperclip. That's consistent — you don't feel wires in your house jumping around from magnetic forces because Earth's field is tiny.

Whenever you see "F/L" (force per length) instead of just "F" in a magnetism problem, that's a hint that the length of the wire doesn't matter — you can work directly with the ratio. Rewriting F = ILB sin θ as F/L = IB sin θ removes L entirely and makes everything simpler.

Why this is useful: for infinitely long wires (or very long ones), the total force would be infinite. The force per meter, though, is a perfectly well-defined finite number. Same logic applies to any problem where you don't know (or don't care about) the length.

Units check: F/L is in newtons per meter (N/m), which equals tesla × amp (T·A). Rearranging: B = (F/L)/I → (N/m)/A = N/(A·m) = T. ✓

Backwards-solving for B: this problem is an exam-style twist — instead of giving you B and asking for F, it gives you F/L and asks for B. Algebra: B = (F/L) / I = 0.75 / 35 ≈ 0.0214 T. Notice the answer is small (~21 milliTesla), because a modest force on a strong current doesn't need a huge field.

Direction of the force — even when B is unknown in magnitude: you still use the RHR based on the directions given. Current west + field south → force out of the page. The RHR doesn't care about the magnitudes, only the vectors.

A good exam habit: always check your answer's magnitude for plausibility. 0.021 T is about 40× Earth's field, which is strong but not absurd — it's the kind of field you'd get near a small permanent magnet. If you got B = 21 T or 21 μT, you'd know to recheck your setup.

A fully enclosed current loop in a uniform magnetic field experiences ZERO net force. Every segment of the loop has a partner on the opposite side where the current flows the other way, and those forces cancel in pairs. So if the whole loop is inside B, F_net = 0 — guaranteed. (You may still get a torque — see Slides 29-34 — but no linear force.)

What makes partial-field problems interesting: if only part of the loop is inside the field, the cancellation fails. Only the segments inside the field feel a force; the ones outside feel nothing. So whatever force the inside segments produce becomes the net force on the whole loop.

This problem's logic:

• Top side (outside field): no force (no B there)
• Left side and right side (partially inside, both vertical): the parts inside the field experience equal and opposite forces — they cancel.
• Bottom side (fully inside field): horizontal current in B → force = ILB, directed south (by RHR). This is the ONLY unbalanced force, so it's the net force on the loop.

Real-world relevance — motors and pumps: this is exactly how a linear motor or electromagnetic pump works. Put a fluid-carrying pipe in a magnetic field, pass current through the fluid, and the partial-field asymmetry creates a net force that pumps the fluid. Liquid metal cooling systems in nuclear reactors use this trick to move molten sodium without any moving parts.

Also — induction stoves: the partial-coupling of a coil's field with a metal pan creates eddy currents that feel this exact type of force, which in turn generate friction and heat. All from F = IL×B applied to geometry where symmetry doesn't save you.

This derivation bridges two different formulas — one for wires, one for single charges — and shows they're actually the same physics in disguise. A current-carrying wire is just a huge number of moving charges, so a force on a "current" must reduce to a force on individual charges when you look at each one.

The substitutions, step by step:

• Start with F = ILB sin θ (force on a wire of length L carrying current I in field B)
• Current = charge passing per unit time: I = Q/t, where Q is total charge and t is time.
• In time t, the charges move a distance L = vt (where v is their speed).
• Substitute: F = (Q/t)(vt)B sin θ. The t's cancel!
• Simplify: F = QvB sin θ.
• For a single charge, Q = q (one particle). So: F = qvB sin θ.

Why this is beautiful: two independent formulas, derived from completely different physical setups, turn out to be the same equation. Physics is full of these "Aha, they're the same thing" moments — they're what make learning the subject rewarding.

Which formula to use when:

• Problem mentions wires and currents → use F = ILB sin θ
• Problem mentions single charges, protons, electrons, ions → use F = qvB sin θ
• Problem has both → use both; the wire formula is the "sum" of the single-charge formulas over all the charges in the wire

One equation, two faces. The underlying vector form is F = qv×B for a single charge or F = IL×B for a wire segment. These are the same equation, just with different bookkeeping for how many charges are moving.

Every force equation in magnetism has the same sin θ factor: F = ILB sin θ for currents, F = qvB sin θ for charges, τ = NIAB sin θ for torques on loops. It's not a coincidence — it comes from the cross product in the underlying vector equation F = qv×B. A cross product's magnitude is |v||B|sin θ.

The three canonical cases (memorize these):

• θ = 0° (parallel): sin 0 = 0 → NO force. Moving parallel to B is like moving along a field line; nothing happens magnetically.
• θ = 90° (perpendicular): sin 90° = 1 → MAXIMUM force, F = qvB. This is the "clean" case that most textbook problems use.
• Intermediate angle: partial force F = qvB sin θ, with the component parallel to B doing nothing.

Decomposing the velocity: you can also think of it as splitting v into two components: v_∥ (parallel to B) and v_⊥ (perpendicular to B). The parallel component contributes nothing to the force; only v_⊥ = v·sin θ matters. So F = q·v_⊥·B = qvB sin θ. Same answer, different viewpoint.

Why "perpendicular is strongest" makes intuitive sense: think of water running through a paddle wheel. Water flowing parallel to the wheel's axle does nothing — it slides past without pushing anything. Water flowing perpendicular to the axle hits the paddles head-on and transfers maximum momentum. The magnetic force is basically the same — only the "sideways" component of v does work against B.

Helical paths: in the real world, particles rarely move exactly perpendicular to B. The parallel part flies straight forward while the perpendicular part circles around B, producing a helical trajectory — like a corkscrew traveling down a field line. Auroras, Van Allen belts, solar wind in Earth's magnetosphere — all helices.

The right-hand rule for a moving charge is: point your fingers along v (velocity), curl them toward B, and your thumb gives the direction of F on a POSITIVE charge. For a negative charge, flip the answer.

Alternative "palm" method (what the transcript uses): thumb along v, fingers along B, palm pushes in the direction of F on a positive charge. Same rule, different finger positions — pick whichever your hand memorizes better.

Why the negative charge flips the answer: the formula F = qv×B has q explicitly multiplying the cross product. If q is negative (like an electron, q = −e), the sign of F is reversed. So you compute the cross product for a positive charge with the RHR, then flip if the charge is negative. Never use the "left hand rule" — it leads to confusion and errors. Just use the right hand and mentally flip the answer for electrons.

Why this matters physically: in a wire, current is conventionally defined as the direction positive charges would flow. But in reality, the moving charges in a copper wire are negative electrons flowing the opposite direction. Both views give the same force on the wire because: (negative charge) × (opposite velocity) = (positive charge) × (original velocity). The two wrongs make a right. This is why F = IL×B works regardless of which charges are actually moving.

Hall effect: if you pass current through a metal strip in a magnetic field, positive and negative charges experience force in opposite perpendicular directions. This sorts charge across the strip, creating a tiny voltage difference — the Hall voltage. By measuring the sign of the Hall voltage, you can determine whether the charge carriers in a material are electrons (most metals) or holes (some semiconductors). This is how we actually know electrons are the current carriers in most materials.

1.28 × 10⁻¹⁶ N sounds laughably small — and it is, on human scales. For perspective, the force of gravity on a single grain of sand is about 10⁻⁶ N, which is already a billion times larger than this force. So why does it matter?

Because the proton is REALLY tiny. With a mass of 1.67 × 10⁻²⁷ kg, the acceleration that tiny force produces is:

a = F/m = 1.28 × 10⁻¹⁶ / 1.67 × 10⁻²⁷ ≈ 7.7 × 10¹⁰ m/s²

That's 77 billion m/s² — about 8 billion g's. A human would be vaporized instantly. But the proton, being so tiny, barely notices. Newton's 2nd law doesn't care about absolute size, only the ratio F/m.

The force is perfectly perpendicular — it never speeds the proton up. Even with this gigantic acceleration, the proton's speed stays at 4 × 10⁶ m/s forever. Only its direction changes. The acceleration is purely centripetal — pointing inward toward the center of the orbit.

Orbit radius from this force: using R = mv/(qB) = (1.67×10⁻²⁷)(4×10⁶) / (1.6×10⁻¹⁹ × 2×10⁻⁴) = 208 m. Yep — a 200-meter orbit. Weak fields make BIG circles. Strong fields like 2 T (Slide 18) give centimeter-scale orbits at the same speed.

Take-home: the absolute magnitude of F is meaningless without context. Always compare to the particle's mass to find the acceleration, or to its momentum to find the orbit size. Tiny forces on tiny particles produce huge effects.

This is one of the coolest and most surprising facts in physics: the magnetic force does ZERO work on a moving charge, ever. It can change the direction of velocity, but it can never speed the particle up or slow it down.

Why? Work is W = F·d·cos(angle between F and d). The magnetic force F is always perpendicular to the velocity v (because F = qv×B, and a cross-product is always perpendicular to both inputs). Displacement d is in the direction of v. So the angle between F and d is always 90°, and cos 90° = 0. No work, ever.

What does it do then? The magnetic force only changes direction, never magnitude. The particle's kinetic energy ½mv² stays constant, but v's direction rotates. That's exactly what happens in circular motion: the speed stays the same, only direction changes. Magnetic forces are nature's perfect steering — they bend paths without accelerating them.

The circle emerges automatically: if force is always perpendicular to velocity and constant in magnitude, the particle traces a circle. Think about it: the force is always pointing toward some center (the center of the circle), and the particle's trajectory rotates around that center forever (or until it hits something). This is the definition of circular motion.

Hybrid motion — helices: if the particle's velocity isn't fully perpendicular to B (say it has some component parallel to B), then only the perpendicular component circles, while the parallel component moves in a straight line along the field. The combined path is a helix — a spiral along the field lines. This is how charged particles travel through Earth's magnetic field, spiraling from one pole to the other along field lines.

Protons and electrons in the same field circle in opposite directions. This is a direct consequence of F = qv×B: flipping the sign of q flips the direction of F, which flips the direction of circulation.

Memory trick: in a B field pointing into the page, positive charges circle counterclockwise and negative charges circle clockwise. Reverse the field direction and both reverse.

Why this matters in real devices:

• Mass spectrometers: separate ions by mass because R = mv/(qB). Heavier ions arc wider, lighter ones arc tighter, even with the same charge. Different elements end up at different detector positions.
• Cyclotrons: accelerate both positive and negative particles, but the circulation direction differs, so the accelerating gap must switch polarity accordingly.
• Magnetic bottle / mirror: traps charged particles (both signs) by making B stronger at the ends. Used in fusion research.
• Cosmic rays in Earth's field: positively charged protons and negatively charged electrons spiral in opposite directions along Earth's field lines, which is why auroras produce the color patterns they do.

Inside a cloud chamber: particle physics detectors used to reveal tracks of charged particles — looking at how the track curves in a known B field lets you immediately tell charge sign and momentum. This is how the positron (the antimatter electron) was discovered in 1932: a curve going "the wrong way" for an electron but with the electron's mass.

Same speed → same period (almost): protons and electrons in the same B circle with different radii (R ∝ m), but the orbital period T = 2πm/(qB) is also proportional to m. The lighter electron orbits smaller and faster than the heavier proton by a factor of ~1836.

When a physics problem asks for the radius of a circular motion, the standard move is to set two expressions for force equal to each other: the force causing the circular motion on one side, and the required centripetal force on the other.

What is centripetal force? It's not a new kind of force — it's the requirement for any circular motion. Any object moving in a circle of radius R at speed v needs an inward-pointing force of magnitude mv²/R. This force can come from gravity (planets), tension (ball on a string), friction (car turning on a road), or — in this case — the magnetic force on a moving charge.

The setup: in magnetism, the "real" force is F_magnetic = qvB. The circular-motion requirement is F_centripetal = mv²/R. Set them equal: qvB = mv²/R. Cancel one v from each side: qB = mv/R. Solve for R: R = mv/(qB).

Why the v cancels: look carefully — magnetic force has one v, centripetal force has v². After cancellation, the R depends linearly on v (not quadratically). This is why faster particles don't loop in smaller circles (as intuition might suggest) — instead they loop in bigger ones, because their momentum fights the turning force harder.

"mv = BqR" as the master equation: the professor is right — write it in this form and you can rearrange to solve for any unknown: R = mv/(Bq), v = BqR/m, m = BqR/v, q = mv/(BR), or B = mv/(qR). All of these show up in different problems.

Period of orbit (bonus result): T = 2πR/v = 2πm/(qB). Notice that the period doesn't depend on speed or radius — all particles of the same charge and mass orbit with the same frequency regardless of how fast they're going! This is called the cyclotron frequency and is the key to why cyclotrons work.

For magnetism problems, you'll need to recognize a handful of fundamental constants on sight:

• Proton mass: m_p = 1.673 × 10⁻²⁷ kg
• Electron mass: m_e = 9.109 × 10⁻³¹ kg (about 1836× lighter than a proton)
• Elementary charge: e = 1.602 × 10⁻¹⁹ C (proton positive, electron negative)
• Speed of light: c = 3 × 10⁸ m/s
• μ₀ (permeability of free space): 4π × 10⁻⁷ T·m/A ≈ 1.257 × 10⁻⁶
• ε₀ (permittivity of free space): 8.854 × 10⁻¹² C²/(N·m²)
• Coulomb's k: 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C²

Dimensional analysis check for R = mv/(qB): [mv] = kg·m/s (momentum). [qB] = C·T = C·(kg/(A·s²)) = A·s·kg/(A·s²) = kg/s. So mv/qB has units (kg·m/s)/(kg/s) = m. Meters. ✓

Physical interpretation of R = mv/qB:

• Higher mass m → bigger radius (heavy particles are harder to turn)
• Higher velocity v → bigger radius (fast particles resist bending more)
• Higher charge q → smaller radius (bigger force = tighter turn)
• Higher field B → smaller radius (stronger push = tighter turn)

Comparison: for the same 5 × 10⁶ m/s speed and 2.5 T field, an electron (1836× lighter) would have R_e = R_p / 1836 ≈ 11 micrometers. This is why electrons are used in small instruments (CRT screens, mass spectrometers) while protons need bigger ones (cyclotrons).

Real device check: a 2 cm radius orbit at 5 × 10⁶ m/s means the proton orbits in about 2π(0.02)/(5×10⁶) ≈ 25 nanoseconds — or 40 million revolutions per second. Cyclotrons exploit this exact periodic motion to accelerate particles at a precise frequency.

A proton at 5 × 10⁶ m/s is fast — about 1.7% of the speed of light. At this speed, relativistic corrections are still small (a few thousandths of a percent), so the classical formula KE = ½mv² still gives a good answer. Above about 10% c, you need the relativistic formula KE = (γ − 1)mc² instead.

Why this speed matters: 130 keV is characteristic of particles in sun-like stellar cores (which run at millions of kelvin) and in medical imaging machines. The protons in proton-beam cancer therapy start at around 250 MeV — nearly 2000× more energetic than this example. The LHC protons reach 7 TeV — 50 million times more.

The order-of-magnitude math: (1.67 × 10⁻²⁷)(5 × 10⁶)² = (1.67 × 10⁻²⁷)(2.5 × 10¹³) = 4.18 × 10⁻¹⁴. Half of that is 2.09 × 10⁻¹⁴ J. Doing this in scientific notation keeps the exponent tracking straightforward — multiply the mantissas, add the exponents.

Why convert to eV: 2.09 × 10⁻¹⁴ J is an abstract number with too many zeros. 130,703 eV is immediately recognizable: "about 130 keV, the range of a soft X-ray or an accelerated electron in an old CRT TV." Physicists use eV precisely because it gives human-scale numbers for sub-atomic processes.

The conversion trick: divide joules by 1.6 × 10⁻¹⁹ to get eV. That's just canceling one unit for another — a pure dimensional conversion like converting meters to feet.

The electron-volt (eV) is a unit of energy, not voltage. It's defined as "the energy gained by one electron moving through a potential difference of 1 volt." Because an electron's charge is e = 1.6 × 10⁻¹⁹ C, the energy is simply (charge)(voltage) = (1.6 × 10⁻¹⁹ C)(1 V) = 1.6 × 10⁻¹⁹ J.

Why physicists prefer eV over joules for small things: one joule is enormous compared to the energies of individual particles. An electron's rest-mass energy is about 511,000 eV ≈ 8 × 10⁻¹⁴ J — that's a 14-digit tiny number in joules but just "511 keV" in electron-volts. Particle physics uses:

• eV — atomic and molecular binding energies (hydrogen: 13.6 eV)
• keV — X-rays (10–100 keV)
• MeV — nuclear physics, particle rest masses (proton: 938 MeV)
• GeV — high-energy physics (LHC beams: 7000 GeV = 7 TeV)
• TeV — collider energies

The formula connection: kinetic energy of a charged particle accelerated through voltage V is KE = qV. If q = 1 e and V = 1 volt, then KE = 1 eV by definition. An electron accelerated by a 100,000 V potential gains 100,000 eV = 100 keV of kinetic energy. This is exactly how CRT TVs and old X-ray tubes worked.

Converting eV to J: just multiply by e = 1.6 × 10⁻¹⁹. Converting J to eV: divide by 1.6 × 10⁻¹⁹. This is why the previous slide's 2.09 × 10⁻¹⁴ J ÷ (1.6 × 10⁻¹⁹) = 130,703 eV.

Fun fact: mosquitoes flying at top speed have roughly 10¹² eV of kinetic energy — about the same as a single proton in the LHC. Scale matters!

This is one of the most counterintuitive results in electromagnetism: two wires carrying current in the same direction attract each other, while two wires carrying current in opposite directions repel. If you first learn about Coulomb's Law (like charges repel, opposites attract), this feels completely backwards.

The RHR walk-through: say both wires carry current "up" (north). Wire 1 creates a magnetic field that, at wire 2's location, points into the page (use RHR #1). Now wire 2 has current up in a field pointing into the page. Applying RHR #2 to wire 2: thumb up, fingers into page, palm pushes toward wire 1. So wire 2 is pushed toward wire 1 — attraction.

The mirror reasoning for repulsion: if wire 2 carries current in the opposite direction (down), everything else stays the same but the force reverses. Now wire 2 is pushed away from wire 1 — repulsion.

"Like charges repel, but like currents attract" is a deep result. The electric rule comes from Coulomb's law acting on static charges. The magnetic rule comes from the fact that the magnetic field encircles wires and the force is a cross-product (I × B), not a simple attraction to field lines. These are two genuinely different rules.

How to remember it: think of power cables in your house. The "hot" wire and the "neutral" wire carry currents in opposite directions (the current goes out on one and returns on the other). They repel slightly, which is why loose cables sometimes rattle. If they carried current in the same direction, they'd clamp together. This is also why DC transmission lines bundle their pair with physical separators.

Historic note: Ampère himself first demonstrated this in 1820, shortly after Oersted discovered that electric currents deflect compasses. It was part of the observation that united electricity and magnetism into one phenomenon — electromagnetism.

This derivation is really "two ideas bolted together." Wire 1 produces a magnetic field (Slide 2's formula). Wire 2 sits in that field carrying a current, and any current in a field feels F = ILB (Slide 5's formula). Combine them and you get F = μ₀I₁I₂L/(2πR).

The two-step mental picture:

• Step 1 — wire 1's field at wire 2's location: B₁ = μ₀I₁/(2πR), where R is the distance between wires.
• Step 2 — force on wire 2 in that field: F₂ = I₂·L·B₁ = I₂·L·μ₀I₁/(2πR).

Newton's third law in action: by symmetry, you could also calculate B₂ at wire 1's location and then F₁ on wire 1. You'd get the exact same magnitude. The two wires push (or pull) each other with equal and opposite forces — which is exactly what Newton's third law requires. The physics is automatically self-consistent.

The force per unit length: dividing both sides by L gives F/L = μ₀I₁I₂/(2πR). This is the more common form you'll see in textbooks because it doesn't depend on how long the wires are — it's a property of the current pair and their separation. Infinite wires are an idealization; force per meter is what physically matters.

Why the formula contains both currents multiplied: it makes sense — if either wire is carrying zero current, there's no field and no force. Doubling either current doubles the force; doubling both gives 4× the force. Current-current interactions are quadratic in scale.

Every parallel-wire problem reduces to plugging numbers into F = μ₀I₁I₂L/(2πR). Five inputs: the two currents, the length of the wires, the distance between them, and the constant μ₀. Then do arithmetic.

The canceling trick: notice that 4π in the numerator of μ₀ and 2π in the denominator leave you with 2 × 10⁻⁷. So you can rewrite the formula as F = (2×10⁻⁷)·I₁I₂·L/R. This shaves one step off the arithmetic and makes order-of-magnitude estimates faster.

Crunching this example: F = (2×10⁻⁷)(50)(50)(30)/(0.02) = (2×10⁻⁷)(2500)(30)/(0.02) = (2×10⁻⁷)(75,000/0.02) = (2×10⁻⁷)(3,750,000) = 0.75 N. ✓

Always convert cm to m first: 2 cm → 0.02 m. If you leave it as 2, you'll be off by a factor of 100. Distance and length must be in meters for μ₀ to give forces in newtons.

I²L/R scaling: the force is quadratic in current (I²), linear in wire length, and inversely proportional to separation. That means:

• Double both currents → force quadruples (I² scaling)
• Double the wire length → force doubles
• Halve the separation → force doubles

This is why coil windings in electromagnets generate such large forces: short separations × high current² gives enormous force per unit length. It's also why closely-packed power cables need robust mechanical clamps.

0.75 newtons sounds small, but spread across 30 meters of wire, that's only 0.025 N/m (2.5 grams per meter of wire length). For two wires carrying 50 A just 2 cm apart, that's surprisingly modest — but enough to cause real engineering problems at scale.

Why the ampere was defined this way: before 2019, the official definition of the ampere was: "that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 × 10⁻⁷ N per meter of length." Read that definition carefully — it's literally this problem. The ampere was defined as the current that produces a specific force between parallel wires.

In 2019 the definition changed: the SI now defines the ampere in terms of the elementary charge e = 1.602176634 × 10⁻¹⁹ C exactly, with no reference to wires. But the old wire-force definition is still a useful mental picture and a great way to see why μ₀ has its strange value of 4π × 10⁻⁷.

At industrial scales the forces are no joke: a power line might carry 1000 A, with wires 50 cm apart. The force per meter becomes F/L = μ₀I²/(2πR) = (4π×10⁻⁷)(10⁶)/(2π·0.5) = 0.4 N/m. In strong short-circuit events, currents of 100,000 A for a fraction of a second can rip power bus bars apart with explosive force. This is why substations use massive steel supports for their conductors.

MRI machines and fusion reactors — places with extremely high currents and strong B fields — must account for mechanical stress on coils from the I×B force. In tokamaks, the superconducting coils experience Lorentz forces equivalent to thousands of tons per square meter.

Ampère's Law says: take any closed loop in space, add up the magnetic field along that loop, and the result equals μ₀ times the current passing through the loop. Written mathematically: ∮ B·dl = μ₀·I_enclosed.

The analogy to Gauss's Law: for electricity, Gauss's Law says "total flux through a surface = enclosed charge divided by ε₀." For magnetism, Ampère's Law says "total B along a loop = enclosed current times μ₀." Both are about "how much field gets generated by some source."

Why it recovers B = μ₀I/(2πR): pick a circle of radius R around a long wire as your Amperian loop. By symmetry, B has the same magnitude at every point on the circle and is tangent to it. So ∮ B·dl = B·(circumference) = B·(2πR). The current enclosed is just I. Set them equal: B·(2πR) = μ₀I → B = μ₀I/(2πR). That's the formula from Slide 2!

Why Ampère's Law is more powerful than the formula: the formula B = μ₀I/(2πR) only works for a single long straight wire. Ampère's Law works for any current distribution: solenoids, toroids, current sheets, cables with a core. It's the master equation; the wire formula is just one application.

The "symmetry trick": Ampère's Law is only useful when the current distribution has high symmetry (cylindrical, planar, toroidal). Then you can argue that B is constant along certain paths, pull it out of the integral, and solve. For weird asymmetric current shapes, you'd need the Biot-Savart Law instead — which is painful and usually done with a computer.

It's also Maxwell's third equation (in the differential form, it becomes ∇×B = μ₀J). When Maxwell later added a "displacement current" correction, Ampère's Law became the equation responsible for electromagnetic waves — the same math that gives us light, radio, and WiFi.

Ampère's Law works for any closed path, but the whole trick is picking a path where the math is simple. For a solenoid, the clever choice is a rectangular path that has one side running inside the solenoid (parallel to B) and the opposite side running outside (where B ≈ 0).

Why this rectangle makes everything cancel:

• Inside segment (AC): B is strong and parallel to the path → contribution = B·L (nonzero)
• Outside segment (BD): B is nearly zero → contribution ≈ 0
• Top & bottom segments (BA and DC): B is perpendicular to the path (B points along the axis, path goes sideways) → contribution = 0 (because cos 90° = 0)

So only one segment contributes: ∮ B·dl = B·L. This is the art of applying Ampère's Law — force the integral to have only one nonzero segment.

The enclosed current: the rectangle encloses some number of solenoid turns, each carrying current I. If the rectangle has length L and the solenoid has n turns per meter, then the rectangle encloses n·L turns → total enclosed current = n·L·I.

Putting it together: B·L = μ₀(nLI) → B = μ₀nI. The L cancels, and you get the famous solenoid formula. Notice that the field strength doesn't depend on the length of your rectangle or the cross-sectional area — it only depends on n and I. That's the power of Ampère's Law applied with symmetry.

A solenoid — a tightly-wound coil of wire — has a remarkable property: the magnetic field inside is nearly uniform everywhere except near the ends. It's the same magnitude at the center, at the edges, near the wall, or along the axis. This makes solenoids incredibly useful as "magnetic bottles" for generating controlled fields.

Why it's uniform: each loop of wire contributes its own B field; when you stack many loops closely, the fields from all the loops reinforce along the axis and cancel perpendicular to it. The inside of the solenoid "sees" contributions from loops on both sides, which adds up to a constant vector.

Outside the solenoid: the field is nearly zero. All the flux from the inside comes out one end and loops around to the other end. If you could see field lines, they'd look like those of a bar magnet, with N and S poles at the two ends. In fact, solenoids and bar magnets are magnetically indistinguishable from the outside — Ampère's original insight that magnetism comes from current loops.

Three ways to make B stronger:

• Increase I — more amps → more field (but the wire heats up as I²R)
• Increase n (turns per meter) — pack more windings into the same length
• Decrease L while keeping N — same number of loops compressed tighter

The "n = N/L" relationship: what matters physically is density of turns, not total turns. A 100-turn coil that's 1 cm long has the same field strength as a 10,000-turn coil that's 1 meter long (both have n = 10,000 turns/m). Every real solenoid formula depends on n, not N.

Solenoid problems are usually the easiest on an exam because the formula B = μ₀nI has only three inputs: n (turns per meter), I (current), and μ₀ (the constant). There's no angle, no cross-product, no integration. Just multiply.

The one trap: "n" is turns per meter, not total turns. If you're given N = 800 turns and length L = 15 cm, you must divide: n = N/L = 800/0.15 ≈ 5,333 turns/m. Forgetting this conversion is the #1 mistake — students plug in 800 instead of 5333 and get a field 150× too small.

Always convert cm to m first: 15 cm = 0.15 m. If you use 15, you'll get B off by a factor of 100. SI units throughout, always.

Sanity check the answer: 0.0335 T ≈ 335 Gauss. Earth's field is about 0.5 Gauss, so this solenoid is 670× stronger than Earth's field — plausible for 800 turns at 5 A in a short cylinder. It's strong enough to visibly deflect a compass needle from many meters away, but nowhere near MRI strength (~1–3 T).

How to make a stronger solenoid: three knobs — more current (more heat), more turns per meter (tighter winding), or add an iron core (multiplies B by the iron's permeability μ, typically 100–5000× stronger). Real electromagnets combine all three. A typical electromagnet in a junkyard crane uses iron-core solenoids to generate ~1 T and lift cars.

A rectangular loop has two sides where current flows up and two sides where current flows the opposite way. When you put this loop in a uniform magnetic field, each side experiences a force F = IL×B — but because the currents go in opposite directions, the forces are also opposite.

The geometry: current up on the right side + B pointing east → F into the page (by RHR). Current down on the left side + B pointing east → F out of the page. These two forces are equal in magnitude but opposite in direction, so the net force on the loop is zero. But they're applied at different points, so they create a couple — a pair of equal-and-opposite forces offset in space that produces pure rotation with no translation.

Why this is called a "couple": in mechanics, two equal forces acting in opposite directions along different lines of action produce only torque, never translation. It's the purest form of rotational force. Every current loop in a uniform field feels a couple — the loop spins in place but doesn't drift.

What the loop does with that torque: it rotates toward an orientation where its magnetic moment M aligns with B. Once aligned, the forces switch geometry: now both forces act along the same line (and in opposite directions), so the couple's moment arm collapses to zero and the torque vanishes. That's the equilibrium position.

Galvanometer connection: this is how old analog meters work. A small coil with a pointer glued on is placed between permanent magnets. Current through the coil produces torque, the coil rotates against a restoring spring, and the pointer's angle indicates the current's magnitude. Simple, elegant, and entirely based on τ = NIAB sin θ.

Torque is the rotational version of force. Just as force makes things accelerate linearly (F = ma), torque makes things rotate (τ = Iα, with I being rotational inertia here — different I than current). The formula τ = F·r·sin θ says torque depends on three things: how hard you push, how far out you push from the pivot, and what angle you push at.

Why r = W/2 (not W): the rotation axis passes through the middle of the loop, splitting the width W in half. Each long side of the loop is therefore only W/2 away from the axis. This is the "lever arm" for each of the two forces.

Why the top and bottom sides produce NO force: if the loop is oriented so the long sides (length L) are perpendicular to B, then the short sides (length W) must be parallel to B. The force F = ILB sin 90° = ILB only works when current is perpendicular to B. When parallel, sin 0° = 0, so force = 0. The top and bottom wires just sit there doing nothing.

Both forces push in the same rotational direction: the left wire's force pushes it "out of the page," the right wire's pushes it "into the page" — but both produce torque in the same rotational sense about the center axis. So the torques add (τ_total = τ₁ + τ₂), not cancel.

Why this matters: this is the fundamental operating principle of every electric motor. Current flowing through a loop in a magnetic field produces torque; the loop spins; by continuously switching current direction with a commutator, you can keep the torque always pushing in the same rotational sense forever. From a 3-gram drone motor to a 3-ton electric train motor, they all use exactly this geometry.

The quantity M = NIA is called the magnetic dipole moment of the coil. It's a single number that summarizes "how magnetic" a current loop is — combining all three things that matter: how many turns (N), how much current (I), and how big the loop is (A).

Why bundle them together? The torque formula τ = MB sin θ looks simpler and cleaner with M in place. It also makes the analogy to electric dipoles clearer: τ_elec = pE sin θ (for electric dipole moment p in field E), τ_mag = MB sin θ. Same form, different "dipole strength."

Units of M: amps × meters² = A·m². A bar magnet the size of your thumb has a magnetic moment of around 0.1 A·m². A neodymium fridge magnet has about 1 A·m². The Earth, as a giant magnetic dipole, has M ≈ 8 × 10²² A·m².

Scaling with N: doubling the turns doubles the torque because each loop independently contributes NIA·B·sin θ, and they add in phase. This is why electromagnets use thousands of tightly-wound turns — it linearly multiplies the force without increasing the current (and therefore without overheating the wire).

Why the width W cancels into A: in the derivation, you see (w/2 + w/2) = w appear when adding the two torques from the two wire segments. Then LW = A, and the formula collapses to τ = IAB sin θ per turn. What looks like a rectangle-specific derivation actually works for any loop shape — the only thing that matters is the enclosed area A.

The single biggest source of errors in magnetism problems is: the angle θ is measured from the normal to the loop's face, not from the face itself. Students constantly mis-identify this angle and get signs and magnitudes wrong.

What's a "normal vector"? Imagine an arrow sticking straight out of the loop's flat surface, perpendicular to it, like a flagpole on a round table. That arrow is the normal vector (written A⊥ or n̂). If the loop is lying flat on the ground, its normal points straight up. If the loop is vertical, its normal points horizontally out of the face.

The conversion rule: if a problem tells you "B is parallel to the loop's face," that's 90° from the normal (→ max torque, sin 90° = 1). If it says "B is perpendicular to the loop's face" (= passes through it), that's 0° from the normal (→ zero torque, sin 0° = 0). Always convert "angle from face" to "angle from normal" by subtracting from 90°.

Why physics uses the normal and not the face: because the flux formula Φ = BA cos θ uses the same convention. Every formula involving a "loop's orientation in a field" — flux, induced EMF, torque, magnetic moment — is defined relative to the normal. This keeps all the equations consistent.

Quick gut check: if you set up a problem and the answer doesn't match the expected behavior (e.g., you expect maximum torque but get zero), check whether you used the correct angle. Swapping "angle from face" with "angle from normal" flips your sin to cos and your max to zero.

A current loop in a magnetic field is not in equilibrium at θ = 90° — it's at the position of maximum twist. The loop "wants" to rotate toward θ = 0°, where its normal vector lines up with B. This is exactly analogous to a compass needle, which always twists until it aligns with Earth's magnetic field.

Door handle analogy: imagine pushing a door. If you push perpendicular to the door (from the face), it swings open — maximum torque. If you push parallel to the door (along the edge), nothing happens — zero torque. The force has to have a component perpendicular to the lever arm to produce rotation. For the current loop, the lever arm is the width of the loop, and F = ILB is the force on each side.

Stable vs. unstable equilibrium: at θ = 0° (normal aligned with B), the loop is at stable equilibrium — any small rotation creates a restoring torque that pushes it back. At θ = 180° (normal anti-aligned with B), the loop is also in equilibrium, but unstable — any tiny rotation sends it flipping around to θ = 0°. This is why bar magnets flip when you put them near each other: they seek the stable N-to-S alignment.

Energy story: the potential energy of a magnetic dipole in a field is U = −M·B cos θ. At θ = 0°, U is minimum (most negative) — that's the stable position. At θ = 180°, U is maximum — unstable. At θ = 90°, U = 0 — but the torque is maximum, so the loop starts swinging immediately.

Why compasses work: the compass needle is a tiny magnetic dipole. Earth's B field applies torque until the needle aligns with it (north pole of needle pointing toward magnetic north). Same physics as this slide, just scaled down to a pocket navigation tool.

When you see "R = 30 cm" for a circular coil, the key move is computing A = πR² = π(0.3)² ≈ 0.283 m². Notice how the 30 cm must become 0.30 m before squaring — if you forget the conversion, you'll be off by a factor of 10⁴.

Why R² (not R)? Torque depends on area because the magnetic moment M = NIA counts how much "magnetic dipole strength" the loop carries. A bigger loop encloses more flux for the same current, so it interacts more strongly with the external field. Since A = πR² for a circle, doubling the radius quadruples the torque.

The order of plugging in: (N)(I)(A)(B) = (50)(8)(0.283)(5). A smart way: group (NI) = 400 first, then (NIB) = 2000, then multiply by 0.283 to get 565.5. Or any order that gives you clean intermediate products.

565.5 N·m is a LOT of torque. To put it in perspective, a typical car engine produces ~300 N·m of peak torque. So a 30-cm coil carrying 8 A in a 5 T field twists with roughly twice the torque of an engine. This is why industrial motors use coils in strong fields — you can generate huge mechanical forces from compact wire geometries.

Connecting to later: this is also exactly how electric motors work. A spinning coil in a field experiences torque — that's the fundamental mechanism. The tricky part is using a commutator to keep the torque always in the same rotational direction as the coil turns; otherwise it would oscillate and never complete a rotation.

This problem is a "find the unknown variable" twist on the torque formula. You're given the torque and asked to find B — the opposite of most problems where you're given B and asked to find τ. The trick is just algebra: rearrange τ = NIAB sin θ to solve for whatever's missing.

Why max torque means sin θ = 1: the problem specifies maximum torque, which pins the angle at 90° because sin 90° = 1 (the biggest possible value). That simplifies the equation to τ = NIAB, and now you can directly solve: B = τ / (NIA).

Order of operations matters: compute NIA first (200 × 15 × 0.2 = 600), then divide τ by that number. This is much cleaner than trying to plug everything in at once. In general, when you have a formula with many factors, evaluate constants first.

Units sanity check: [τ] = N·m, [N] = dimensionless, [I] = A, [A] = m², so [NIA] = A·m². Then [τ/(NIA)] = (N·m)/(A·m²) = N/(A·m) = T. Units work out to Tesla. Always verify units on derived answers.

Real-world context: 2 T is a strong field — about the strength of an MRI machine. For perspective, Earth's magnetic field is ~50 μT (40,000× weaker), and a typical fridge magnet is around 5 mT. Electric motors in hybrid cars use fields around 1–2 T in their rotor windings, producing torques that launch the vehicle.