The Biot–Savart Law

In April 1820, Ørsted's compass twitched beside a wire in Copenhagen, and physics learned that currents make magnetic fields. The news reached Paris that autumn, and within weeks two Frenchmen — Jean-Baptiste Biot and Félix Savart — had done what Paris does best: turned a sensation into a measurement. Hanging a small magnetized needle near a long wire and timing its oscillations, they extracted exactly how the magnetic effect weakens with distance, and on 30 October 1820 they announced the law that now carries both their names. What they handed physics is the magnetic twin of a law you already trust. Coulomb's law lets you compute the electric field of any charge arrangement: chop it into point charges, apply the inverse-square recipe to each, add. The Biot–Savart law does precisely that for currents: chop the wire into tiny slices — current elements — let each slice contribute a tiny field by an inverse-square recipe, add the slices. Same $1/r^2$ , same strategy of divide-and-conquer. But the recipe carries one twist Coulomb never had: each slice's field does not point away from it. It circles it. That single 90° twist is why magnetic field lines loop and close instead of streaming outward, and it is the deep signature of everything magnetic. For CBSE, ISC, JEE and NEET, this law is the toolmaker of the whole chapter: the wire and loop formulas you are about to meet in Fields of Straight Wires & Circular Loops are all built from it. This article gives you the recipe itself — its anatomy, its direction rule, its angle factor, and the exam craft of reading a current element correctly.

Key takeaways

The Biot–Savart law gives the tiny magnetic field of a tiny slice of current: $\Delta B = \dfrac{\mu_0}{4\pi}\,\dfrac{I\,\Delta l\,\sin\theta}{r^2}$ , where $\theta$ is the angle between the element and the line to the field point.
It is Coulomb's law with a twist: same inverse-square, same add-up-the-pieces strategy, but each element's field wraps around the current instead of pointing away from it.
Direction: right hand, thumb along the current slice — the field circles the way your fingers curl. Broadside the field is strongest ( $\sin 90° = 1$ ); along the element's own line it is exactly zero.
The constant: $\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\,m/A}$ , so $\dfrac{\mu_0}{4\pi} = 10^{-7}$ exactly — the single best arithmetic saver in the chapter.
A lone current element is a thought-tool, not a real object — real fields always come from complete circuits, summed slice by slice.
Key idea: every magnetic field of every circuit is the sum of inverse-square, angle-weighted contributions from its current slices — master the one-slice recipe and the whole chapter's formulas become add-ups.

The core insight

The Biot–Savart law is the per-slice recipe for magnetic fields: a current element $I\,\Delta l$ contributes $\Delta B = (\mu_0/4\pi)\,I\,\Delta l\,\sin\theta/r^2$ , circling the element by the right-hand rule — and the field of any circuit is the sum of its slices.

Where this lives today

When engineers need the magnetic field of anything that is not a textbook shape — the bent, jointed, spiralled conductors of real machines — they do exactly what this article taught: chop the conductor into slices, apply equation (1) to each, and let a computer add millions of contributions. The magnet-design software behind MRI scanners, particle-accelerator steering coils and fusion-reactor windings is, at its core, a Biot–Savart summation engine. Nothing cleverer has replaced the recipe; computers just made the add-up effortless. The law also moonlights outside magnetism entirely. The swirling air around a wingtip vortex obeys a mathematically identical rule — replace "current" with the strength of the swirl and "magnetic field" with air velocity — so aerodynamicists computing how one aircraft's wake buffets another are, verbatim, "using Biot–Savart". A law measured with a swinging needle in 1820 now schedules take-off separations at every major airport.

The physical picture

Picture a wire carrying current, and mentally snip out one tiny slice of it — a piece of length $\Delta l$ with the current $I$ flowing through. That snippet is a current element, written $I\,\Delta l$ : the amount of current times the length of the slice, the magnetic analogue of a point charge $q$ . The Biot–Savart law answers one question: what magnetic field does this one slice contribute at some point P a distance $r$ away? The answer has three ingredients, and each one is worth owning separately. Distance: the contribution falls as $1/r^2$ — double your distance from the slice and its effect drops to a quarter, exactly like Coulomb's law, exactly like gravity. Angle: the contribution is proportional to $\sin\theta$ , where $\theta$ is the angle between the slice's direction and the line from the slice to P. Sit broadside to the element ( $\theta = 90°$ ) and you feel its full strength; slide around toward its axis and the effect fades, reaching exactly zero on the element's own line ( $\theta = 0$ or $180°$ ). A current element is loud sideways and silent along its length — it has a blind spot fore and aft. Direction: here is the twist. The field at P does not point from the slice to P, and not from P to the slice either. It points perpendicular to both — wrapping in circles around the current's line, the way your right hand's fingers curl when the thumb rides the current.

Figure

One current slice — field wraps around it, full strength broadside, zero along its own line

Why should you believe in slices at all? Because superposition — the same principle you used in Superposition: Forces & Fields of Many Charges — works for magnetic fields too. The field of a whole circuit at any point is the vector sum of the fields of its parts. The Biot–Savart law is what makes that sum computable: it prices each part. In Fields of Straight Wires & Circular Loops, next, the add-up is actually performed for the two geometries every board examines; here we make sure the per-slice price is understood so well that those results feel inevitable. One honesty note before the formula. You cannot go to a shop and buy an isolated current element — a centimetre of wire carrying current with nothing attached to either end would pile up charge at its ends and stop in an instant. Current elements are a bookkeeping device: real, measurable fields belong only to complete circuits. The law is written per-slice because per-slice is how you compute; nature only ever shows you the finished sum. Exam questions about "the field due to a small element" are asking about the bookkeeping — and that is fine, as long as you remember which is the scaffolding and which is the building.

The law and its anatomy

Here is the recipe in the form the boards use. A current element — current $I$ through a slice of length $\Delta l$ — contributes at a point P, distance $r$ away, with $\theta$ the angle between the element's direction and the line to P:

\Delta B = \frac{\mu_0}{4\pi}\,\frac{I\,\Delta l\,\sin\theta}{r^2} \qquad (1)

where: $\Delta B$ — the element's field contribution, in tesla; $I\,\Delta l$ — the current element, in ampere·metre (the "source strength", playing the role $q$ plays in Coulomb's law); $r$ — distance from slice to field point; $\theta$ — the angle between slice and sight line; $\mu_0$ — the permeability of free space, $4\pi \times 10^{-7}\ \mathrm{T\,m/A}$ . Read the structure: source strength on top, distance-squared below, and the angle factor deciding how much of the slice's full voice reaches you. The direction of $\Delta B$ is perpendicular to both the element and the sight line — circling the element by the right-hand rule, into or out of your page in the flat drawings every exam uses (the dot-and-cross notation from The Magnetic Field & Field Lines). The constant deserves its own paragraph, because it hides the chapter's best arithmetic gift. The permeability $\mu_0$ measures how readily empty space lets currents make magnetic fields, and its value is

\frac{\mu_0}{4\pi} = 10^{-7}\ \mathrm{T\,m/A}\ \text{ exactly} \qquad (2)

where: the $4\pi$ in the law's denominator and the $4\pi$ inside $\mu_0 = 4\pi \times 10^{-7}$ cancel perfectly. Never multiply out $4\pi \times 10^{-7}$ and then divide by $4\pi$ again — every Biot–Savart computation starts with a clean $10^{-7}$ , and (as you will see next module) wire problems start with $\mu_0/2\pi = 2 \times 10^{-7}$ . Examiners set numbers assuming you know this; the students who don't are the ones who run out of time. Set the two great inverse-square recipes side by side and both the kinship and the mutation stand out. This table is the concept map of the whole chapter:

feature	Coulomb's law (electric)	Biot–Savart law (magnetic)
source	point charge $q$	current element $I\,\Delta l$
falloff	$1/r^2$	$1/r^2$ — identical
constant	$k = 9 \times 10^9$	$\mu_0/4\pi = 10^{-7}$ — exactly
angle factor	none — all directions equal	$\sin\theta$ — silent along the element's axis
field direction	along the sight line (radial)	perpendicular to sight line — circles the element
field-line shape	start and end on charges	closed loops around currents
total field	add the pieces (superposition)	add the pieces (superposition)

Two rows carry all the physics. The $\sin\theta$ row says a current element is not a point source with equal voice in all directions — it is a little directed object with a blind axis. And the direction row is the 90° twist: electric fields radiate, magnetic fields circulate. Every strange and wonderful thing about magnetism downstream — field lines with no ends, forces that steer without working, the absence of magnetic charges — traces back to that one row.

Figure

Two inverse-square recipes side by side — Coulomb radiates, Biot–Savart circulates

Derivation AReading the recipe — units, angles and the two sanity checks

The goal here is not to derive the Biot–Savart law from something deeper — at this level it is an experimental starting point, exactly as Coulomb's law was — but to take equation (1) apart and verify it behaves the way a law of nature must: correct units, sensible limits, and a direction rule that never contradicts itself. This is the checking discipline you will reuse on every formula this chapter builds from the law. Watch two things: how the units of $\mu_0$ make the tesla come out right, and how the $\sin\theta$ factor encodes the blind spot.

Step 1. Check the units. From (1), $[\Delta B] = [\mu_0] \cdot \mathrm{A \cdot m / m^2} = \mathrm{(T\,m/A)(A/m)} = \mathrm{T}$ . The permeability's odd-looking unit, tesla·metre per ampere, is exactly what is needed to turn "amps times metres over metres squared" into tesla. The formula is dimensionally airtight.

Step 2. Check the distance behaviour. Fix $\theta = 90°$ and double $r$ : $\Delta B$ falls by $2^2 = 4$ . A slice of current is an inverse-square source, like a point charge, like a point mass. Numerically: a 1.0 cm slice carrying 5.0 A gives, broadside at 10 cm, $\Delta B = 10^{-7} \times \dfrac{5.0 \times 0.01}{(0.10)^2} = 5.0 \times 10^{-7}\ \mathrm{T}$ ; move to 20 cm and it reads $1.25 \times 10^{-7}\ \mathrm{T}$ — a quarter, on the nose.

Step 3. Check the angle behaviour. Keep $r = 10$ cm and slide from broadside ( $\theta = 90°$ , $\sin\theta = 1$ ) to $\theta = 30°$ ( $\sin\theta = 0.5$ ): the same slice now contributes $2.5 \times 10^{-7}\ \mathrm{T}$ — half. Slide all the way to the element's own line ( $\theta = 0$ ): $\sin 0 = 0$ , contribution exactly zero. The blind spot is not an approximation; it is built into the law. Why the blind spot makes physical sense: the field must circle the current. Directly ahead of or behind the slice, "circling the current" gives the field no consistent direction to point — every direction around the axis is equally entitled. The only self-consistent value there is zero. The sine factor is geometry keeping the direction rule honest.

Step 4. Check the direction rule for consistency. Take a slice with current flowing up the page and a field point to its right: right hand, thumb up, fingers curl — at that point they plunge into the page. Move the field point to the left of the slice: the same curled fingers now come out of the page. Opposite sides, opposite senses, one rule — the two sides join smoothly into the closed rings of the hero diagram, with no seam anywhere. Verified: units close to tesla in Step 1; the numbers in Steps 2–3 recomputed directly from equation (1) with the exact constant 10⁻⁷; the direction check in Step 4 matches the right-hand thumb rule that Fields of Straight Wires & Circular Loops uses for whole wires.

Limiting cases

case	result	meaning
$\theta = 90°$ (broadside)	$\Delta B = \dfrac{\mu_0}{4\pi}\dfrac{I\,\Delta l}{r^2}$ — maximum	the slice at full voice
$\theta = 0°$ or $180°$ (on the element's line)	$\Delta B = 0$ exactly	the blind spot fore and aft
double the distance	$\Delta B$ falls to a quarter	inverse-square, same as Coulomb
double the current or the slice length	$\Delta B$ doubles	field is proportional to source strength $I\,\Delta l$
add up slices around a full circuit	wire and loop formulas	next module's business — the add-ups
isolated element on its own	not physically realizable	scaffolding for computation, never an observable

Common trapThe field of an element points from the slice to the point, like Coulomb

This is the single most common Biot–Savart error, and it imports the wrong intuition wholesale. In Coulomb's law the field is radial — along the line joining source and point. In the Biot–Savart law the field is perpendicular to that line and perpendicular to the current slice: it wraps around the current. If your answer to "which way does $\Delta B$ point?" is ever "toward P" or "away from P", stop and redo the right hand: thumb along the current, and your curled fingers at P give the direction — which in a flat exam diagram is always into or out of the page. The second half of the same trap: forgetting the blind spot and assigning a nonzero field to a point straight ahead of the element. On the element's own line, $\sin\theta = 0$ and the contribution vanishes — no exceptions.

Worked examples

Board level — one slice, three positions (CBSE/ISC pattern, 3 marks)

A straight conductor carries $I = 5.0\ \mathrm{A}$ . Consider a small element of it, of length $\Delta l = 1.0\ \mathrm{cm}$ . Find the magnetic field contribution of this element at a point (a) 10 cm from the element, broadside on ( $\theta = 90°$ ); (b) 10 cm away but at $\theta = 30°$ to the element's direction; (c) 10 cm away on the element's own line. State the direction in case (a) if the current flows up the page and the point lies to its right.

Step 1. Part (a), straight from equation (1) with the clean constant: $\Delta B = 10^{-7} \times \dfrac{5.0 \times 0.010 \times \sin 90°}{(0.10)^2} = 10^{-7} \times \dfrac{0.050}{0.010} = \boxed{5.0 \times 10^{-7}\ \mathrm{T}}$ .

Step 2. Part (b): only the angle changes, so scale by $\sin 30° = 0.5$ : $\Delta B = \boxed{2.5 \times 10^{-7}\ \mathrm{T}}$ . No need to recompute from scratch — ratios are faster and safer.

Step 3. Part (c): on the element's own line $\theta = 0$ , so $\sin\theta = 0$ and $\Delta B = \boxed{0}$ — the blind spot.

Step 4. Direction for (a): right hand, thumb up the page along $I$ ; at a point to the right, the fingers curl into the page (⊗). Writing "into the plane of the paper, by the right-hand rule" is the full mark; an arrow toward or away from the wire is the classic zero.

Competitive level — the angle factor under pressure (JEE-Main pattern, ~2.5 minutes)

A current element of magnitude $I\,\Delta l = 4.0\ \mathrm{A \cdot mm}$ sits at the origin, pointing along the $+x$ axis. Compare its field contributions at three points in the $xy$ -plane, each 5.0 cm from the origin: $P_1$ at $(0, 5.0\ \mathrm{cm})$ , $P_2$ at $(3.0\ \mathrm{cm}, 4.0\ \mathrm{cm})$ , and $P_3$ at $(5.0\ \mathrm{cm}, 0)$ . Find $\Delta B$ at each and give the direction at $P_1$ and $P_2$ .

Step 1. All three points share the same $r = 5.0$ cm — the only thing that differs is $\theta$ , the angle each sight line makes with the $+x$ direction. Read the angles from the coordinates: at $P_1$ the sight line is straight up the $y$ -axis, $\theta = 90°$ , $\sin\theta = 1$ . At $P_2$ the sight line has components $(3, 4)/5$ , so $\sin\theta = 4/5 = 0.8$ (the sine is the perpendicular share — the $y$ -part). At $P_3$ the sight line lies along the element itself: $\theta = 0$ , $\sin\theta = 0$ .

Step 2. Price the full-voice point first: $\Delta B_1 = 10^{-7} \times \dfrac{(4.0 \times 10^{-3})(1)}{(0.050)^2} = 10^{-7} \times \dfrac{4.0 \times 10^{-3}}{2.5 \times 10^{-3}} = \boxed{1.6 \times 10^{-7}\ \mathrm{T}}$ .

Step 3. Scale for the others: $\Delta B_2 = 0.8 \times \Delta B_1 = \boxed{1.28 \times 10^{-7}\ \mathrm{T}}$ , and $\Delta B_3 = \boxed{0}$ (blind spot — no computation needed, and writing one wastes exam seconds).

Step 4. Directions: at both $P_1$ and $P_2$ the field must be perpendicular to the $xy$ -plane (perpendicular to both the element and the sight line). Right hand, thumb along $+x$ : above the axis the fingers come out of the page (⊙) at both points. Note what the question is really testing: the sine reads off the coordinates as $\text{(perpendicular offset)}/r$ — the 3-4-5 triangle is the angle factor. Timing note: about 2.5 minutes. The efficient route prices one point and scales the rest by their sines — the moment you see equal distances, the problem is a $\sin\theta$ comparison in disguise. Spotting the $\theta = 0$ blind spot instantly (zero, no arithmetic) is a JEE-Main classic; option-setters love hiding it among plausible small numbers.

Exam traps

Radial-direction imports from Coulomb. The element's field circles the current; it never points along the line joining element and point. In flat diagrams the correct answer is almost always ⊙ or ⊗ — an in-plane arrow from the slice toward P is an automatic zero for the direction mark.

Forgetting the blind spot. On the element's axis ( $\theta = 0°$ or $180°$ ), $\Delta B = 0$ exactly. MCQ setters place a field point "directly ahead of the element" and offer four nonzero options plus zero; zero wins.

Multiplying out $\mu_0$ and dividing by $4\pi$ again. Use $\mu_0/4\pi = 10^{-7}$ as one object. Every second spent computing $4\pi \times 10^{-7} = 1.257 \times 10^{-6}$ and then dividing by $12.57$ is a second donated to the paper-setter.

Measuring $\theta$ from the wrong reference. $\theta$ is the angle between the element's direction and the line from element to field point — not the angle at the field point, not the angle with any axis of the diagram. Mark the element's arrow and the sight line, and read the angle between those two.

Unit slips in $I\,\Delta l$ . The element length arrives in cm or mm; convert to metres before substituting, or carry the power of ten explicitly. With $\mu_0/4\pi = 10^{-7}$ , answers land in the $10^{-7}$ – $10^{-5}$ T range for bench-scale numbers — an answer near a full tesla means a dropped conversion.

Claiming an element's field is measurable. Conceptual questions (ISC loves them) ask why the Biot–Savart law is stated for an element that cannot exist alone. The expected answer: an isolated element would accumulate charge at its ends and cannot carry a steady current; the law is a calculational device, and only the summed field of a complete circuit is observable.

Boards and constants. CBSE/ISC/JEE/NEET all supply or expect $\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\,m/A}$ ; this module never needs $g$ . IB and IGCSE do not examine the Biot–Savart law itself — if those are your boards, this article is enrichment that makes the next one obvious.

Origins · 1 minJean-Baptiste Biot and how this was discovered

Jean-Baptiste Biot

1774–1862 · Paris

Biot was already famous when Ørsted's news reached Paris — he had ascended to 4,000 metres in a balloon with Gay-Lussac to test whether Earth's magnetism weakens with altitude, and had convinced the French Academy that meteorites really do fall from the sky, after the village of L'Aigle was showered with three thousand stones in 1803. In the autumn of 1820 he teamed up with Félix Savart, a physician who had drifted into physics through the acoustics of the violin. Their instrument was disarmingly simple: a small magnetized needle hung near a long vertical wire. The needle's oscillation period revealed the strength of the magnetic influence, and by timing swings at different distances they nailed the law by which a current's magnetism falls off. Announced on 30 October 1820 — barely six months after Ørsted's discovery — and sharpened into the per-element form with the analytical help of Laplace, the result made magnetism computable. Biot never stopped ranging: the "Biot" in optics (the rotation of polarized light by sugar solutions, the ancestor of every saccharimeter) is the same man.

History

1820

— April: Ørsted's compass twitches in Copenhagen. September: the news electrifies the Paris Academy. 30 October: Biot and Savart announce their distance law for the magnetic action of a current.

1820

— Laplace helps recast the measurements as a per-element formula — the law as every textbook now writes it.

1826

— Ampère's great Mémoire folds the element law into a complete mathematical theory of currents.

1873

— Maxwell's Treatise absorbs Biot–Savart as the steady-current limit of the full field equations.

1920

s — Aerodynamics reuses the identical formula for the swirling flow around vortex lines — the Biot–Savart law of aircraft wings.

Going further

Texts. NCERT Physics Class 12, Ch. 4 (Moving Charges and Magnetism, §4.5) · HC Verma, Concepts of Physics Vol. 2, Ch. 35 · ISC Nootan Physics XII, magnetic effects chapter.

Provenance. All results standard. A verifier should recompute: Board example — $10^{-7} \times 5.0 \times 0.01/0.01 = 5.0 \times 10^{-7}$ T, half at $\sin 30°$ , zero on-axis; Competitive example — $\Delta B_1 = 10^{-7} \times 4.0\times10^{-3}/2.5\times10^{-3} = 1.6 \times 10^{-7}$ T, $\times 0.8 = 1.28 \times 10^{-7}$ T, zero at $P_3$ ; the derivation's quarter-at-double-distance check.

Prerequisites. The Magnetic Field & Field Lines · Electric Current & Charge Flow · Coulomb's Law. Leads to. Fields of Straight Wires & Circular Loops · Ampère's Circuital Law · Biot-Savart Law (UG deep dive).

Why is the law written for a tiny element instead of a whole wire?

Because a whole wire's field depends on its entire shape, and there are infinitely many shapes — no single formula could cover them all. The element is the universal building block: state the law once for a slice, and any circuit's field follows by adding slices. It is the same strategy chemistry uses with atoms: learn the brick, build any wall. The add-ups for the two shapes your board actually examines — the long straight wire and the circular loop — are performed in Fields of Straight Wires & Circular Loops, and both drop out of the one-slice recipe you have just learned.

If an isolated current element can't exist, how can the law be tested?

Directly, it can't — you can never hold up one slice and measure its field alone. What experiment tests is the sum: compute the predicted total field of a complete circuit by adding elements, then measure that. Biot and Savart themselves worked with a long straight wire (a sum), and every prediction built from the element law since — loops, coils, bent wires, solenoids — has matched measurement. The element law is like a single term in a bank statement: you never see it in isolation, but if every balance always checks out, you trust the bookkeeping.

Where does the sine factor actually come from?

From the requirement that the field must circle the current consistently. A current element has a direction, and its field wraps around that direction like rings around an axle. Points broadside to the axle sit where the rings are "facing" them — full effect. As you move toward the axle's own line, there is less and less "around" for the field to do at your location, and exactly on the line there is no consistent circling direction at all, so the field must vanish. The sine is the smooth mathematical bridge between full voice at 90° and enforced silence at 0° — geometry, not an arbitrary decoration.

Is the field really zero straight ahead of a current element, even very close to it?

Yes — zero at every distance, as long as the point is exactly on the element's line. This feels wrong because "closer must mean stronger" is usually safe intuition, but the angle factor beats the distance factor here: $\sin 0° = 0$ kills the contribution no matter how small $r^2$ is. For a whole wire the situation differs: a point near a long wire is broadside to many elements even if it is on the line of one of them, which is why real wires have strong nearby fields. The blind spot belongs to the element, not to the circuit.

How is this law related to what a compass actually shows near a wire?

The compass shows the finished sum. Around a long straight wire, every element's wrapped contribution reinforces into neat concentric circles — the pattern The Magnetic Field & Field Lines described and Ørsted's needle first betrayed. The Biot–Savart law is the ledger behind the pattern: it explains why the circles weaken with distance (inverse-square per slice, summed) and why the direction follows the right-hand rule (each slice already circles that way, so the sum does too). Next module, that ledger is totted up into the formula $B = \mu_0 I / 2\pi d$ that puts numbers on the compass's behaviour.

Do I need calculus to use the Biot-Savart law in exams?

Not at this level. Board and JEE-Main questions use the law in exactly two ways, both algebra: pricing a single element's contribution (the worked examples above), and quoting the results of standard add-ups — straight wire, loop centre, loop axis — which the next module states and, where the geometry is kind, derives without any integral. The genuinely calculus-hungry cases (arbitrary curves, off-axis points) live at university level; the UG deep-dive article listed in Going further performs them in full if you want to see the machinery. What exams reward here is the anatomy: constant, source strength, inverse square, sine, right-hand direction.

Why does mu0 have that strange value with a 4 pi inside?

By design, and for your convenience. The SI unit system chose to define the ampere (via the force between parallel wires, a story completed in the module on Parallel Currents coming later in this chapter) in a way that made $\mu_0$ come out as exactly $4\pi \times 10^{-7}$ . Packing the $4\pi$ into the constant means it cancels against the $4\pi$ in the Biot–Savart denominator, leaving the clean $10^{-7}$ , and wire formulas inherit a tidy $2 \times 10^{-7}$ . The 2019 redefinition of SI units shifted $\mu_0$ 's status from "defined" to "measured", but its value is unchanged to better than a part in a billion — for every exam on Earth, $4\pi \times 10^{-7}$ it remains.