The Magnetic Field & Field Lines

Push the like poles of two fridge magnets toward each other and you can feel it: a spongy, invisible cushion that resists harder the closer they get, and slips sideways if your grip wavers. You are squeezing a physical object — one with no mass you can weigh and no surface you can see, but real enough to hold a maglev train off its rails. In 1269, a soldier-scholar named Peter Peregrinus, camped at a siege in southern Italy, rolled a compass needle across a polished sphere of lodestone, inked in the directions it pointed, and watched the marks assemble into curves converging on two special spots. He had drawn the first map of that invisible object. He named the special spots poles, after the pivots of the sky. This module is about the map and the thing it maps: the magnetic field, B⃗\vec{B} — a vector stationed at every point of space, waiting to act. You have already met this style of thinking twice, in The Electric Field and Electric Field Lines, and the fastest way to master B⃗\vec{B} is to run the comparison deliberately: same field logic, same tangent-and-density rules for the lines, but different sources, different geometry, and one rule flipped so completely that it becomes the exam's favourite trick question — magnetic field lines never begin or end. They close on themselves. Everything in the four chapters ahead — forces on beams and wires, motors, generators, transformers, the entire machinery of the electrical world — happens inside this field. Learn to draw it, read it, and rank it before you learn to compute with it. The computing starts next module; the seeing starts now.

Key takeaways
  • The magnetic field B⃗\vec{B} is a vector at every point of space: it has a strength (measured in tesla, T) and a direction (the way a compass needle's north end points there).
  • Field lines are the map: tangent gives direction, crowding gives strength, and the lines never cross.
  • The flipped rule: magnetic field lines form closed loops — outside a magnet they run N → S, inside they continue S → N. Nothing in magnetism plays the role electric charges play as line-endings.
  • The tesla is enormous. Earth's field ≈ 5×10−5 T5 \times 10^{-5}\ \mathrm{T}, a fridge magnet ≈ 5×10−3 T5 \times 10^{-3}\ \mathrm{T}, an MRI scanner 1.5–3 T. The old unit gauss survives in labs: 1 G=10−4 T1\ \mathrm{G} = 10^{-4}\ \mathrm{T}.
  • Third-dimension notation: a dot ⊙\odot marks field out of the page (arrowhead toward you), a cross ⊗\otimes marks field into the page (arrow fletching away). Every module after this one uses it.
  • Key idea: B⃗\vec{B} is space pre-loaded with an instruction — and a correctly drawn field-line diagram is a calibrated instrument for reading that instruction's direction and strength at a glance.
The core insight

The magnetic field is a vector B⃗\vec{B} defined at every point of space, mapped by field lines that run tangent to it, crowd where it is strong, never cross, and — unlike electric field lines — always close into loops, because magnetism has no isolated charges for lines to start or end on.

Where this lives today

The field map you learned to draw is a working engineering document. Designers of EV motors and wind-turbine generators live inside field-line plots — software fills the machine's cross-section with Peregrinus-style lines, and the designer reads crowding as "this iron is saturating" the way you read "strong field here." MRI engineering is the tesla ladder made into a business: a 3 T scanner's field must be uniform to parts per million across a human torso (remember what even spacing means). The quietest entries on the ladder now talk too. Your phone's magnetometer — three tiny sensors reading μT-scale components — is what turns the maps app when you rotate at a junction; it is Peregrinus's compass, sampled two hundred times a second. At the other extreme, magnetoencephalography reads the 10−13 T10^{-13}\ \mathrm{T} whispers of the brain's own currents through superconducting detectors — field reading pushed thirteen orders of magnitude below the fridge magnet.

The physical picture

Start with what a field is, because exams punish loose usage. In The Electric Field you learned the two-step move that saved physics from action-at-a-distance: sources modify the space around them, and objects placed in that space respond to the modification at their own location. Magnetism runs on the same architecture. A magnet — or, as Ch. E4 will keep revealing, any moving charge or current — dresses every point of the surrounding space with a vector B⃗\vec{B}. The field is there at three in the morning with nobody watching. It is a property of the place. How do you read the instruction stationed at a point? The honest probe at this stage is the one Peregrinus used: a tiny compass. Park it at the point; its north-marked end swings to rest pointing along B⃗\vec{B} there. That settles the direction. Strength is what the next module makes quantitative — for now, rank it by how stiffly the needle snaps back when you nudge it, or by how densely the field lines crowd on a correct map. And the map is drawn exactly the way Faraday taught for electricity: thread curves through space so that each curve is everywhere tangent to the local field, with arrowheads along the local B⃗\vec{B}. Around a bar magnet the curves emerge near one end (the north pole), arc through the surrounding space, and dive back in near the other (the south pole). That much looks like a positive and negative charge in disguise. The disguise fails inside the magnet — and that failure is the deepest fact on this page.

Figure
The field of a bar magnet — closed loops, mapped by compass needles
S N inside: S → N — the loop never breaks compass rests tangent lines exit N, return to S outside — and continue through the magnet: every line is a closed loop crowding near the poles = strongest field; wide spacing far away = weak field tangent of the line at any point = direction of B there

Run the comparison with Electric Field Lines while both are fresh. There, lines begin on positive charges and end on negative ones — fountains and drains. Cut an electric dipole in half and you can carry the fountain away in one hand, the drain in the other. Now saw a bar magnet in half, hoping to isolate the north pole: you get two complete magnets, each with a fresh pair of poles. Saw again — four. All the way down to the atoms, nature refuses to hand you a loose line-ending. So the lines have nowhere to start or stop, and must close on themselves: poles always come in pairs. (Why nature refuses is Ch. E5's Gauss's Law for Magnetism; the pattern is yours from today.)

ruleelectric field linesmagnetic field lines
tangent gives direction✓✓
crowding gives strength✓✓
never cross✓✓
where they start and endbegin on ++, end on −-nowhere — every line is a closed loop
can you isolate a source endyes: a single chargeno: poles only come in pairs
what the probe issmall positive test chargesmall compass needle

The tesla, and the third dimension

Before any formula, two pieces of equipment: a unit, and a notation. The field's strength — how hard the instruction at a point acts — is measured in the SI unit named after Nikola Tesla:

[ B ]=1 tesla=1 T(1)[\,B\,] = 1\ \mathrm{tesla} = 1\ \mathrm{T} \qquad (1)

where: the tesla's operational meaning (what one tesla does to one coulomb crossing at one metre per second) is delivered by the force law next module — this module's job is calibration. The headline: one tesla is a huge field. You live inside Earth's 5×10−5 T5 \times 10^{-5}\ \mathrm{T}; a fridge magnet manages a few millitesla at its surface; an MRI scanner surrounds a patient with 1.5–3 T and must be wound from superconducting wire to do it. Because everyday fields are small decimals, labs still quote the older gauss: 1 G=10−4 T1\ \mathrm{G} = 10^{-4}\ \mathrm{T} — Earth is "about half a gauss." Boards want SI: convert on sight.

1 G=10−4 T(2)1\ \mathrm{G} = 10^{-4}\ \mathrm{T} \qquad (2)

where: the conversion runs both ways and is a one-mark gift when it appears: 50 μT=0.5 G50\ \mathrm{\mu T} = 0.5\ \mathrm{G}, and a 0.35 G0.35\ \mathrm{G} reading is 3.5×10−5 T3.5 \times 10^{-5}\ \mathrm{T}. The scale ladder below is worth memorising to one significant figure — competitive papers love answer options that differ by factors of a thousand, and knowing that a lab electromagnet cannot plausibly produce 103 T10^{3}\ \mathrm{T} kills wrong options instantly.

Figure
The tesla ladder — thirteen orders of magnitude, one significant figure each
10⁻¹³10⁻¹⁰10⁻⁵10⁻³10⁻¹10⁰–10¹10⁸ field strength in tesla (log scale) brainsignals heart'sfield atthe chest Earth,50 μT≈ 0.5 G fridgemagnet~5 mT strongNd magnet~0.3 T MRI1.5–3 Tlab record~45 T neutronstar~10⁸ T one tesla is a big field — your exam answers should usually land in μT, mT, or single-digit T

Now the notation. Magnetic problems are stubbornly three-dimensional, so physics borrows an archer's-eye view of an arrow. Field coming out of the page toward you shows the arrowhead: a dot, ⊙\odot. Field going into the page shows the fletching at the tail: a cross, ⊗\otimes. A region peppered with evenly spaced crosses is a uniform field into the page. This is the working language of every diagram to the end of Ch. E7; make it automatic now.

Figure
Into and out of the page — the archer's-eye notation
⊙⊙⊙⊙ ⊙⊙⊙⊙ ⊙⊙⊙⊙ B out of the page — arrowhead coming at you ×××× ×××× ×××× B into the page — fletching flying away even spacing = uniform field; both regions here have constant strength and direction

Why the map can be trusted

The line rules are not conventions to memorise — each one is forced. The derivation below proves the two that examiners test as "explain why" questions.

Derivation AWhy field lines never cross and why crowding measures strength

We want to establish two properties of any correct field-line map: that two lines can never intersect, and that the local density of lines is a genuine measure of field strength rather than an artistic choice. The strategy is proof by contradiction for the first (assume a crossing, extract an absurdity) and a conserved-count argument for the second, using the fact that lines cannot appear or vanish in empty space. Watch how both arguments lean only on the definition "a field line is everywhere tangent to B⃗\vec{B}" — that is what makes them work for electric and magnetic maps alike.

Step 1. Suppose two field lines crossed at a point P. Each line, by definition, is tangent to the field where it passes — so the field at P would have to point along line 1's tangent and along line 2's tangent: two different directions at once.

Step 2. But B⃗\vec{B} at a point is a single vector. A compass parked at P settles into one orientation, not two. The assumption fails: field lines can never intersect. (The one loophole is a point where B=0B = 0 exactly — with no field, there is no direction to contradict. Null points like the one between two like poles are the only places maps show lines meeting, and there they meet without arrows.) Why this matters for marks: "the field cannot have two directions at the same point" is the complete one-line answer boards want — no mention of "lines repelling" or any force between lines, which is a physics error.

Step 3. For density: draw the fixed quota of lines a map assigns to a source, and follow the bundle that threads some small window. In empty space lines neither start nor end, so the same number of lines that enters any imaginary tube of them must leave it — the count through the bundle is conserved along its length.

Step 4. Where the tube widens — the bundle spreading over a larger area — the fixed count is shared across more space, so lines per unit area drop exactly as the area grows. Near a pole the same count squeezes through a small area: high density. The map is forced to show strong field as crowding and weak field as spacing; the artist has no freedom left. Verified: this is the same conserved-count logic that made electric line density reproduce the inverse-square law in Electric Field Lines; here it needs no charge at all, only "no starts or ends in empty space" — which for magnetism holds everywhere, since every line is a closed loop.

Reading real patterns

Three source shapes dominate this track, and you should recognise their field maps on sight the way you recognise letters.

The bar magnet you have already read: loops out of N, around, into S, continuing inside from S to N. Densest at the poles — that is where iron filings cling in bristling tufts.

The straight current-carrying wire produces circles centred on the wire, stacked along its length like rings on a curtain rod — no poles anywhere, just concentric loops that tighten (strengthen) near the wire. The flat circular loop of current produces a pattern that, from a distance, is a dead ringer for a short bar magnet's. Both are previews: Ørsted's compass caught the wire's circles in 1820, and Fields of Straight Wires & Circular Loops delivers their formulas and right-hand rules three modules on. Absorb today the shapes and their astonishing headline — currents make magnetic fields, with no magnet in sight — the bar magnet is the special case; the moving charge is the general one.

The uniform field — parallel, evenly spaced lines, all one direction — is the theatre in which nearly every exam problem is staged. Between the flat poles of a large horseshoe magnet, or (the engineering favourite, coming in Solenoids & Toroids) inside a long coil, the field is the same vector everywhere over a working region. When a problem says "a charged particle enters a uniform field B=0.3 TB = 0.3\ \mathrm{T} into the page," this is the picture: a rectangle of evenly spaced crosses. A practical ICSE and IGCSE papers ask you to describe: lay a card over a bar magnet, scatter iron filings, tap — the sheet self-organises into the line pattern (arrows not included). Plotting compasses add the arrows one patient dot at a time: place, mark where the needle points, move to the mark, repeat. Peregrinus's method, seven and a half centuries on.

Limiting cases

caseresultmeaning
very far from any magnet, on Earthlines settle into Earth's own pattern, ≈ 5×10−55\times10^{-5} Tevery lab experiment sits inside a background field — usually negligible, occasionally the point
lines drawn evenly spaced and paralleluniform fieldthe standard exam arena: same B⃗\vec{B} at every point of the region
line spacing doublesfield roughly halvesdensity reads strength — ranking questions live here
two like poles facing each othera point between them where B=0B = 0the null point: the one legal place for lines to meet, arrowless
magnet cut in halftwo complete magnets, four polesno isolated pole; lines stay closed loops through each piece
compass at a pointneedle rests along B⃗\vec{B}the operational meaning of "direction of the field"

Worked examples

Board level — reading, converting, and comparing (CBSE/ICSE/IGCSE pattern, 3 marks)

A student's magnetometer app reads Earth's field at her desk as 0.35 G0.35\ \mathrm{G}. (a) Convert this to tesla. (b) An MRI scanner operates at 1.5 T1.5\ \mathrm{T}. How many times stronger is the scanner's field than the field at her desk? (c) She sketches the field between the flat poles of a large horseshoe magnet as parallel, evenly spaced lines. What two facts about the field is her sketch asserting?

Step 1. Part (a), the conversion 1 G=10−4 T1\ \mathrm{G} = 10^{-4}\ \mathrm{T}: B=0.35×10−4=3.5×10−5 T=35 μTB = 0.35 \times 10^{-4} = \boxed{3.5 \times 10^{-5}\ \mathrm{T}} = 35\ \mathrm{\mu T}. (Sanity: Earth's field should land between 25 and 65 μT. ✓)

Step 2. Part (b), a straight ratio: 1.53.5×10−5=4.3×104\dfrac{1.5}{3.5 \times 10^{-5}} = 4.3 \times 10^{4}. The scanner's field is about 43,000 times stronger\boxed{43{,}000\ \text{times stronger}} — which is why MRI rooms screen for steel implants at the door.

Step 3. Part (c): parallel lines assert the direction is the same everywhere in the gap; even spacing asserts the strength is the same everywhere. Together: the field is uniform — one vector, copied across the region.

Competitive level — superposing two fields at a point (JEE-Main/NEET pattern, ~2 minutes)

At a point P, the horizontal component of Earth's magnetic field is 30 μT30\ \mathrm{\mu T} due north. A nearby installation adds a second horizontal field of 40 μT40\ \mathrm{\mu T} due east at P. (a) Find the magnitude and direction of the resultant horizontal field. (b) A plotting compass is placed at P. Where does its north end point? (c) Express the resultant in gauss.

Step 1. Fields are vectors, so they add like vectors — one field per point, built by superposition. The two contributions are perpendicular, so the magnitude is Pythagorean: B=302+402=900+1600=50 μTB = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \boxed{50\ \mathrm{\mu T}}.

Step 2. Direction: tan⁡θ=4030=43\tan\theta = \dfrac{40}{30} = \dfrac{4}{3}, so θ=53∘\theta = 53^{\circ} east of north (the 3–4–5 triangle — spot it and skip the calculator).

Step 3. Part (b): the compass needle rests along the local total field. Its north end points 53∘53^{\circ} east of true north — the compass is now a liar about geography, and an honest reporter of physics.

Step 4. Part (c): 50 μT=5.0×10−5 T=0.50 G50\ \mathrm{\mu T} = 5.0 \times 10^{-5}\ \mathrm{T} = \boxed{0.50\ \mathrm{G}}. Timing note: under two minutes — the intended path is recognising the 3–4–5 triangle and the one-field-per-point principle. If you caught yourself wondering whether the two fields "push each other," reread the derivation: fields superpose; they do not interact. The compass feels only the single resultant vector.

Exam traps

Line-endings at the poles. Drawing magnetic lines that stop dead at S (or sprout from N with no return) is the most common diagram error on CBSE, ICSE and IGCSE papers. Lines are closed loops; show the continuation or leave the interior clean, but never draw an endpoint.

Crossing lines. In sketches of combined fields (magnet + Earth), letting two lines touch loses the mark instantly — unless you are deliberately marking the arrowless null point between like poles.

Lines are not trajectories. Same myth as in electricity, same verdict: a field line is not the path a particle follows. What the magnetic field actually does to a moving charge — a force perpendicular to both the field and the velocity — is the next module's headline, and it makes "particle slides along the line" even more wrong for magnetism than it was for electricity.

Gauss left in the answer. Boards mark in SI. Convert gauss to tesla on sight (×10−4\times 10^{-4}), and watch the reverse conversion in lab-flavoured JEE questions.

Density misread. Line density compares points within one diagram; quotas differ between diagrams — never compare crowding across two maps.

Compass wording. The needle's north-marked end points along B⃗\vec{B} — on the page, a compass is a field-direction meter, full stop (the geographic can of worms waits for Ch. E5).

Units and symbols. BB in tesla (T); 1 T=104 G1\ \mathrm{T} = 10^{4}\ \mathrm{G}; μT and mT prefixes do heavy duty. The name for BB varies by board — CBSE/NEET say "magnetic field," A-Level and ICSE often say "magnetic flux density" — same quantity, same tesla; do not let the longer name spook you before flux itself arrives in Ch. E6.

Origins · 1 minPeter Peregrinus and how this was discovered
PP
Peter Peregrinus
fl. 1269 · Lucera, southern Italy

Petrus Peregrinus de Maricourt — "Peter the Pilgrim" — was a military engineer in the army besieging Lucera in the summer of 1269, and evidently a man who could concentrate anywhere. In camp, on the 8th of August, he finished the Epistola de magnete, the first surviving experimental treatise on magnetism. He shaped a lodestone into a sphere, laid a needle on its surface, marked the direction it settled, moved it, marked again — and watched the marks join into great circles converging at two opposite points, which he named poles after the celestial poles. He showed that like poles repel and unlike attract, and that a broken magnet yields two complete magnets, never an isolated pole. Roger Bacon called him one of the only two perfect mathematicians. His compass-dot method is still how schoolchildren map fields, and his two central findings — fields mapped by lines, poles only in pairs — open this chapter 757 years later.

History
1269
— Peter Peregrinus writes the Epistola de magnete at the siege of Lucera: the first field map, and poles named and paired.
1600
— William Gilbert's De Magnete argues Earth itself is a great magnet, explaining the compass.
1820
— Ørsted's compass twitches beside a current-carrying wire: magnetism without magnets, the field's sources generalised forever.
1852
— Faraday, defending his "lines of force," insists the field is physically real — space itself altered.
1960
— The SI names its unit of magnetic flux density the tesla; Earth's familiar half-gauss becomes 50 μT.

Going further

Texts. NCERT Physics Class 12, Ch. 4 (Moving Charges and Magnetism, opening sections) and Ch. 5 (Magnetism and Matter, field-line rules) · HC Verma, Concepts of Physics Vol. 2, Ch. 34 · CIE 9702 Topic 20.1 · IGCSE 0625 Section 4.5.1 (magnetic field patterns and plotting).

Provenance. All results are standard. A verifier should recompute: 0.35 G=3.5×10−5 T0.35\ \mathrm{G} = 3.5\times10^{-5}\ \mathrm{T}; MRI ratio 1.5/(3.5×10−5)=4.3×1041.5/(3.5\times10^{-5}) = 4.3\times10^{4}; resultant field 302+402=50 μT\sqrt{30^2+40^2} = 50\ \mathrm{\mu T} at tan⁡−1(4/3)=53∘\tan^{-1}(4/3) = 53^{\circ} east of north =0.50 G= 0.50\ \mathrm{G}; ladder anchors Earth 5×10−55\times10^{-5} T, fridge ∼5×10−3\sim5\times10^{-3} T, MRI 1.5–3 T, 1 G=10−41\ \mathrm{G} = 10^{-4} T.

Prerequisites. The Electric Field · Electric Field Lines. Leads to. Force on a Moving Charge: qv×B · Fields of Straight Wires & Circular Loops · Solenoids & Toroids (coming up in E4) · Bar Magnets & the Dipole Picture and Gauss's Law for Magnetism (coming up in E5).

Are magnetic field lines actually real, physical things?

The field is real — it stores energy, carries momentum, and (as Ch. E8 will show) travels through empty space as light. The lines are a brilliant sampling scheme: curves threaded tangent to the field, drawn in fixed quota so their crowding encodes strength. There is field between the lines, at every point; the map just cannot ink infinitely many curves. Treat them like contour lines on a hiking map: not painted on the mountain, yet utterly trustworthy about where it is steep.

Why do magnetic field lines close into loops when electric ones don't?

Because nothing magnetic plays the role electric charge plays. Electric lines begin and end on charges — fountains and drains. Magnetism, as far as every experiment can tell, has no fountains: cut a magnet forever and you always get pole pairs, never a loose north. With nowhere to start and nowhere to end, lines can only close on themselves. Whether isolated poles (monopoles) could exist is a live question — detectors keep watch — but "none found, lines must loop" is the working truth, given exact form in Ch. E5's Gauss's Law for Magnetism.

If I place a stationary charge in a magnetic field, what happens to it?

Nothing. This is the cleanest single contrast with electricity: an electric field grabs any charge, moving or not; a magnetic field acts only on moving charge. Park an electron between the poles of the strongest magnet on Earth and it sits there, unbothered. Set it moving and a force appears — sideways, perpendicular to both velocity and field. How that force depends on speed, angle and sign is the entire next module, where the tesla finally gets its operational definition.

Iron filings aren't magnets. Why do they draw the field map?

Each filing is a sliver of soft iron that the field temporarily magnetises — an induced miniature magnet aligned with the local field (the why belongs to Ch. E5's Dia-, Para- & Ferromagnetism). Each filing then acts as a tiny compass needle: it twists to lie along B⃗\vec{B} and clicks end-to-end with its neighbours into chains that trace the tangent curves in seconds. Filings give pattern without arrows; a plotting compass, moved dot by dot, adds the directions.

My textbook calls B "magnetic flux density" but my teacher says "magnetic field". Which is right for exams?

Both name the same vector, measured in tesla. "Magnetic field" is the everyday physics name (CBSE, NEET, JEE all use it); "magnetic flux density" is the formally precise SI name that A-Level and ICSE papers prefer, and it will make full sense once magnetic flux arrives in Ch. E6 — BB is literally flux per unit area. Use your board's habitual name: if it is measured in tesla and probed by a compass, it is our B⃗\vec{B}.

Everyday magnets barely reach millitesla. How does an MRI get to 3 T?

Not with a permanent magnet — with a current: moving charges are the true general source of magnetic fields. An MRI's field is wound from kilometres of niobium-titanium wire carrying hundreds of amperes, cooled with liquid helium until its resistance vanishes so the current circulates without loss. Solenoids & Toroids gives the geometry that makes a coil's interior field strong and uniform; the ladder says why it's worth it: 3 T is sixty thousand Earths.

Before this

Where this leads