Fields of Straight Wires & Circular Loops

On 21 April 1820, in a Copenhagen lecture room, Hans Christian Ørsted switched on a current with a compass lying nearby — and the needle moved. Not toward the wire, not away from it: sideways, as if trying to circle it. That sideways twitch was the first human glimpse of the pattern this article puts numbers on: electric currents wrap magnetic fields around themselves, and the two shapes civilisation actually builds — the straight wire and the circular loop — wrap them in the two patterns every examination board on this list wants you to command. The straight wire's field is rings: concentric circles centred on the wire, weakening as you move out. The loop's field threads through its own hole: straight along the axis at the centre, fountaining around the outside in a pattern that — seen from a distance — is indistinguishable from a small bar magnet. Between them, these two geometries power the chapter ahead: stack loops and you get the solenoid; put two wires side by side and you get forces between currents and the very definition of the ampere; spin a loop in a field and you get the electric motor. This article states the straight wire's formula honestly (its full derivation needs calculus — the add-up is done in the UG deep dive), derives the loop-centre field with nothing beyond algebra — the one place the Biot–Savart recipe adds up in your head — and drills the right-hand rules that turn every "which way does the field point?" question into free marks. The numbers matter too: by the end you will know that the wire feeding a phone charger wraps the room in a field comparable to Earth's, and why that ratio decides real engineering.

Key takeaways

Long straight wire, distance $d$ : $B = \dfrac{\mu_0 I}{2\pi d}$ — field lines are concentric circles, direction by the right-hand thumb rule (thumb = current, curl = field).
Circular loop, centre: $B = \dfrac{\mu_0 I}{2R}$ , and $N$ turns multiply it: $B = \dfrac{\mu_0 N I}{2R}$ — direction by the curl rule (fingers curl with the current, thumb = field through the hole).
On the loop's axis at distance $x$ : $B = \dfrac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$ — reduces to the centre value at $x = 0$ and dies as $1/x^3$ far away (the bar-magnet look).
Working constants: $\dfrac{\mu_0}{2\pi} = 2 \times 10^{-7}$ for wires, $\mu_0 = 4\pi \times 10^{-7}$ for loops — clean numbers, no $\pi$ arithmetic.
Anchor number: 10 A at 5 cm gives $4.0 \times 10^{-5}$ T — the same size as Earth's field, which is why compasses misbehave near cables.
Key idea: wires wrap circles, loops thread their holes — two Biot–Savart add-ups, two right-hand rules, and most of the chapter's machinery.

The core insight

A long straight wire wraps concentric field circles of strength $B = \mu_0 I/2\pi d$ around itself, and a circular loop drives $B = \mu_0 N I/2R$ through its centre — the two canonical Biot–Savart sums, with directions given by the right-hand thumb and curl rules.

Where this lives today

The loop-centre formula is quietly embedded in every device that needs a known field from scratch. Calibration laboratories generate reference fields with large flat coils computed straight from equation (3); the "degaussing" coils that wipe magnetic memory, the coils in metal detectors at airport security, and the transmit loop in your phone's wireless charger are all equation (4) running at engineering tolerances. Double the archaeology: the classroom "tangent galvanometer" — a compass at the centre of a vertical coil, measuring current by how far the needle swings between the coil's field and Earth's — is the Board-level worked example rebuilt as an instrument, and it was how currents were measured for half a century. The wire's 40-μT anchor number runs the other way: it sets interference budgets. Engineers laying out aircraft avionics, hospital MRI suites and particle-physics beamlines must route power cabling so that stray $\mu_0 I/2\pi d$ fields stay below the sensitivity of nearby instruments — and the $1/d$ falloff, so much gentler than inverse-square, is precisely why the problem is hard and why heavy currents get twisted pairs and shielded returns. Ørsted's compass-twitch, at industrial scale, is now a line item in every electromagnetic-compatibility audit.

The physical picture

Both patterns follow from one sentence of the previous module: every slice of current wraps its field around itself. The Biot–Savart Law priced each slice; geometry decides what the slices' contributions build. Around a straight wire, every slice wraps rings around the same line, so the total field is rings too: closed circles centred on the wire, lying in planes perpendicular to it. No beginnings, no ends — the pattern iron filings draw when you sprinkle them on a card threaded by a wire. Strength falls as $1/d$ : gentler than inverse-square, because as you back away from the wire, more of its length sits effectively "broadside" to you and keeps contributing. Direction is the right-hand thumb rule: grip the wire with your right hand, thumb riding the current, and your fingers circle the way the field does. Point the thumb up and the field runs anticlockwise seen from above; flip the current and the rings reverse. For a circular loop, wrap the wire into a ring and watch what the slices' wrapped fields do in the hole: every slice, all the way around the ring, pushes its field the same way through the centre. The contributions gang up. Outside the ring they disagree and partially cancel; through the hole they cooperate perfectly — which is why a loop concentrates field in its middle and why coils, not straight wires, are how you build anything magnetic on purpose. The direction shortcut is the curl rule (the thumb rule's mirror twin): curl your right-hand fingers along the current's circulation, and your thumb points the field through the hole. A loop seen with its current running anticlockwise pushes field toward you — that face behaves like a north pole, and the far face like a south: the loop is a flat magnet, a fact the module on Current Loops as Magnetic Dipoles will grow into a whole theory.

Figure

The two patterns — rings around a wire, a thread through a loop

A scope note for IGCSE candidates: your syllabus examines exactly this section — the patterns around a wire, a flat coil and (next module) a solenoid, plus the right-hand rules — and none of the formulas below. Master the two drawings and the two hand rules and the 0625 questions are fully covered; the algebra that follows is for the boards that demand numbers.

The straight wire, priced

The result first, because for the wire the honest derivation is beyond an algebra-only ceiling. At perpendicular distance $d$ from a long straight wire carrying current $I$ :

B = \frac{\mu_0 I}{2\pi d} \qquad (1)

where: $B$ — field strength in tesla; $I$ — current in amperes; $d$ — perpendicular distance in metres; and the working constant is $\dfrac{\mu_0}{2\pi} = 2 \times 10^{-7}$ T·m/A exactly. Read the structure: field proportional to current (double the amps, double the field everywhere), inverse in the first power of distance (double your distance, halve the field — not quarter). Each individual slice of wire contributes by inverse-square, but backing away recruits a longer stretch of wire into effective broadside range, and the two effects trade to leave $1/d$ . That per-slice-versus-total distinction was the closing warning of The Biot–Savart Law; here it becomes a formula you can feel. Where does (1) come from? Adding the Biot–Savart contributions of every slice of an infinite line — a genuine integral, performed in the UG deep dive (Biot-Savart Law). What this module owes you is the sanity of the result, and a check it passes beautifully: the units work ( $\mathrm{T\,m/A} \times \mathrm{A/m} = \mathrm{T}$ ), the symmetry works (rings, by the thumb rule, since every slice wraps the same line), and — next module — Ampère's Circuital Law will re-derive equation (1) in two lines of algebra, no integral anywhere. Nothing in this chapter rests on unverified trust for long. Now the anchor number that makes the formula real. A wire carrying $I = 10$ A — a toaster-scale current — at $d = 5.0$ cm:

B = \frac{2 \times 10^{-7} \times 10}{0.050} = 4.0 \times 10^{-5}\ \mathrm{T} = 40\ \mathrm{\mu T}.

Earth's magnetic field is 30–60 μT depending on where you live. A hand-span from an ordinary appliance cable, the wire competes head-to-head with the entire planet — which is why a compass beside a live wire swings (Ørsted's discovery, on your desk), and why magnetically sensitive equipment is routed away from power cabling. By a metre away the wire's field has fallen to 2 μT and Earth wins comfortably.

The circular loop, earned

For the loop's centre, the Biot–Savart add-up needs no calculus at all — the ring's geometry makes every slice contribute identically, so the "integral" collapses to a multiplication. This is the lane's one full tour of the recipe becoming a result, and boards from CBSE to JEE-Advanced ask for exactly this argument.

Derivation AThe field at the centre of a circular loop

The goal is the field at the exact centre of a circular loop of radius $R$ carrying current $I$ , starting from the per-slice Biot–Savart recipe and using nothing beyond algebra. The strategy is to check that, for this one special point, every slice of the ring contributes the same magnitude in the same direction — once that is established, the total is just one slice's contribution times how many slice-lengths fit around the ring, and the answer assembles itself. Watch how the two potential complications, the angle factor and the direction, both turn out perfectly cooperative at the centre; that cooperation is what makes this the one add-up you can do in your head.

Step 1. Set up one slice. Take a slice of the ring, length $\Delta l$ , carrying current $I$ . From the centre, this slice sits at distance $r = R$ — every slice does, because the centre is equidistant from the whole ring. First cooperation: no slice is nearer or farther than any other.

Step 2. Check the angle. The slice points along the ring (tangent); the sight line from slice to centre runs along the radius. Tangent and radius meet at exactly $90°$ , so $\sin\theta = 1$ for this slice — and by the same geometry, for every slice. Second cooperation: every slice speaks at full broadside voice.

Step 3. Check the direction. Right hand, thumb along the slice's current: at the centre, the wrapped field of this slice points straight along the loop's axis. Walk around the ring repeating the check and the answer never changes — for a consistent circulation of current, every slice pushes its field through the hole the same way. Third cooperation: no contribution cancels any other. Why the centre is so special: at any other point — even elsewhere inside the loop — slices sit at different distances, at different angles, pushing in tilted directions, and the sum needs real calculus. The centre is where all three complications switch off simultaneously.

Step 4. Add the slices. Each contributes $\Delta B = \dfrac{\mu_0}{4\pi}\dfrac{I\,\Delta l}{R^2}$ , all in the same direction, so the total is this expression with $\Delta l$ replaced by the entire circumference $2\pi R$ :

B = \frac{\mu_0}{4\pi}\,\frac{I\,(2\pi R)}{R^2} = \frac{\mu_0 I}{2R} \qquad (2)

Step 5. Stack turns. A flat coil of $N$ turns is $N$ copies of the same ring, each contributing (2) in the same direction — superposition at its laziest:

B = \frac{\mu_0 N I}{2R} \qquad (3)

Why the loop beats the wire at its own game: compare (2) with the wire formula at $d = R$ — the loop's centre field is $\mu_0 I/2R$ versus the wire's $\mu_0 I/2\pi R$ , a factor of $\pi$ larger. Bending the same wire into a ring at the same closest distance concentrates the field roughly threefold, because the ring surrounds the point instead of passing by it once. Geometry, not raw current, is the first tool of magnet design. *Verified: dimensions — $\mathrm{(T\,m/A)\,A/m = T}$ ✓. Numbers — a single loop, $R = 5.0$ cm, $I = 2.0$ A: $B = (4\pi \times 10^{-7})(2.0)/(0.10) = 2.5 \times 10^{-5}$ T, recomputed ✓. The $2\pi R/4\pi = 1/2$ cancellation is the source of the "2" in the denominator — a common recall check.*

Figure

Loop-centre add-up — every slice equidistant, broadside, and pushing the same way

One more result completes the loop's story, and boards state it rather than derive it (the derivation needs component-resolving beyond our ceiling — it lives in the UG deep dive, Magnetic Field of Steady Currents). On the loop's axis, a distance $x$ from the centre:

B = \frac{\mu_0 I R^2}{2\,(R^2 + x^2)^{3/2}} \qquad (4)

where: the denominator's $(R^2 + x^2)^{1/2}$ is just the slant distance from a point on the ring to the axis point — the 3-4-5 triangles of the worked example below. Read (4) by its checkpoints: at $x = 0$ it collapses to the centre formula (2), as it must; and for $x \gg R$ it shrinks like $1/x^3$ — much faster than the wire's $1/d$ , because from far away the loop's near and far sides nearly cancel. That $1/x^3$ tail is the calling card of a dipole: the loop's far field has exactly the shape of a small bar magnet's, which is the door into Current Loops as Magnetic Dipoles later in this chapter.

Limiting cases

case	result	meaning
wire, double the distance	$B$ halves	$1/d$ , not inverse-square — recruited length compensates
wire vs loop at same closest distance	loop centre is $\pi \approx 3.1$ times stronger	surrounding a point beats passing it
loop axis, $x = 0$	(4) → $\mu_0 I/2R$	axis formula contains the centre formula
loop axis, $x = \sqrt{3}\,R$	$B$ = centre value ÷ 8	slant distance doubles → cube kicks in
loop axis, $x \gg R$	$B \propto 1/x^3$	dipole tail — the bar-magnet disguise
$N$ turns	multiply by $N$	superposition; the flat-coil formula
stack loops into a tube	uniform interior field	the solenoid — next module's opening act

Common trapDoubling the distance from a wire quarters the field

It halves it. The inverse-square reflex belongs to point sources — charges, masses, single current slices — and a long wire is none of these. Each slice does obey inverse-square, but backing off recruits a longer effective stretch of wire, and the sum falls only as $1/d$ . Exams test the reflex both ways: a wire question expecting $1/d$ (halve, not quarter) and a loop-axis question expecting the cube at long range (the same loop that read $2.5 \times 10^{-5}$ T at its centre reads one-eighth of that at $x = \sqrt{3}R$ , because the slant distance merely doubled). Before applying any falloff, name your source: slice → square, wire → first power, distant loop → cube. Three geometries, three exponents, and the formula sheet cannot decide for you.

Worked examples

Board level — the compass by the cable (CBSE/ISC pattern, 3 marks)

A long straight vertical wire carries $I = 12$ A. (a) Find the magnitude of the magnetic field at a point P, 2.5 cm from the wire. (b) A flat circular coil of $N = 100$ turns and radius $R = 10$ cm carries $I = 0.50$ A. Find the field at its centre. (c) For the wire in (a), if the current flows upward and P lies to the east of the wire, state the field's direction at P.

Step 1. Part (a), equation (1) with the working constant: $B = \dfrac{2 \times 10^{-7} \times 12}{0.025} = \boxed{9.6 \times 10^{-5}\ \mathrm{T}}$ — about two Earth-fields; a compass at P would swing hard.

Step 2. Part (b), equation (3): $B = \dfrac{(4\pi \times 10^{-7})(100)(0.50)}{2 \times 0.10} = \dfrac{(4\pi \times 10^{-7})(50)}{0.20} = \boxed{3.1 \times 10^{-4}\ \mathrm{T}}$ . Note the leverage: a smaller current than the wire's, but a hundred turns and surrounding geometry buy a field three times larger than (a).

Step 3. Part (c): right-hand thumb rule — grip the wire, thumb up along the current; the fingers circle anticlockwise when viewed from above. Sketch that circle as seen from above with north at the top of the page and east at the right: at the easternmost point, anticlockwise travel is heading up the page. Direction at P: $\boxed{\text{horizontal, due north}}$ .

Step 4. The mark-scheme sentence for (c): "By the right-hand thumb rule the field lines are anticlockwise circles when viewed from above; at a point east of the wire the tangent to the circle points north." Boards award the rule-plus-tangent reasoning, not a bare compass direction.

Competitive level — one coil, two checkpoints (JEE-Main pattern, ~3 minutes)

A circular coil of $N = 50$ turns and radius $R = 6.0$ cm carries $I = 1.2$ A. (a) Find the field at its centre. (b) Find the field on the axis at $x = 8.0$ cm from the centre. (c) At what axial distance does the field fall to one-eighth of the centre value?

Step 1. Part (a), equation (3): $B_0 = \dfrac{(4\pi \times 10^{-7})(50)(1.2)}{2 \times 0.060} = \boxed{6.3 \times 10^{-4}\ \mathrm{T}}$ (calculator: $6.28 \times 10^{-4}$ ).

Step 2. Part (b): spot the triangle before substituting. $R = 6$ cm and $x = 8$ cm make the slant distance $\sqrt{R^2 + x^2} = \sqrt{36 + 64} = 10$ cm — a 6-8-10 triangle, so $(R^2 + x^2)^{3/2} = (0.10)^3 = 1.0 \times 10^{-3}\ \mathrm{m^3}$ , no calculator needed. Then with equation (4) scaled by $N$ :

B = \frac{(4\pi \times 10^{-7})(50)(1.2)(0.060)^2}{2 \times 1.0 \times 10^{-3}} = \boxed{1.36 \times 10^{-4}\ \mathrm{T}}.

Step 3. Cross-check by ratio instead of re-substitution: $\dfrac{B_0}{B} = \dfrac{(R^2+x^2)^{3/2}}{R^3} = \left(\dfrac{10}{6}\right)^3 = \dfrac{1000}{216} = 4.63$ , and $6.28 \times 10^{-4}/4.63 = 1.36 \times 10^{-4}$ T ✓. The ratio form is faster and dodges the powers of ten entirely.

Step 4. Part (c): demand $\dfrac{(R^2+x^2)^{3/2}}{R^3} = 8$ , so $(R^2 + x^2)^{1/2} = 2R$ , giving $x^2 = 3R^2$ and $x = \sqrt{3}\,R = \boxed{10.4\ \mathrm{cm}}$ . Worth memorising as a landmark: one-eighth at $\sqrt{3}R$ — it appears in MCQs with indecent frequency. Timing note: three minutes if you hunt triangles first. The setter chose 6 and 8 to hand you a 10; JEE numbers are rarely accidents. The one-eighth landmark in (c) is a ten-second recall for the prepared and a two-minute algebra excursion for everyone else — that asymmetry is the entire reason it keeps being asked.

Exam traps

Wrong falloff for the source. Wire → $1/d$ (halve at double distance), distant loop → $1/x^3$ (eighth at double slant distance), single slice → $1/r^2$ . Name the geometry before reaching for an exponent.

$d$ is perpendicular distance. In equation (1), $d$ is measured perpendicular from the wire, never along a slant. Diagrams that give you a slant distance are testing exactly this.

Thumb rule vs curl rule. Straight wire: thumb = current, fingers = field circles. Loop: fingers = current circulation, thumb = field through the hole. Swapping them reverses your answer; saying which rule you used earns the reasoning mark on CBSE/ISC.

Radius vs diameter. Both loop formulas carry $R$ , the radius. Questions quote "a coil of diameter 12 cm" hoping you substitute 0.12 — the resulting factor-of-2 error is the most common lost mark on this topic.

Forgetting $N$ . Real coils have turns. If the stem says "coil of 50 turns", equations (2) and (4) each pick up the factor $N = 50$ . Conversely, "a circular loop" means $N = 1$ — do not import a phantom $N$ from the previous question.

Centre formula off-centre. Equation (2) holds at the exact centre only. For any axial point use (4); for off-axis points there is no board-level formula at all (and claiming $\mu_0 I/2R$ there is a physics error, not an approximation).

Constants discipline. Wires: use $\mu_0/2\pi = 2 \times 10^{-7}$ . Loops: use $\mu_0 = 4\pi \times 10^{-7}$ and keep $\pi$ symbolic until the last keystroke. CBSE/ISC/JEE/NEET all supply $\mu_0$ ; IB-SL/HL and A-Level provide these formulas on the data sheet (9702 quotes all three); IGCSE examines patterns and rules only — no formulas, as flagged above.

Origins · 1 minHans Christian Ørsted and how this was discovered

Hans Christian Ørsted

1777–1851 · Copenhagen

Ørsted believed, on frankly philosophical grounds, that nature's forces must be secretly connected — he was a friend of poets and a devotee of the idea that electricity, magnetism, light and heat were one force wearing masks. Unlike most system-builders, he kept experimenting. On 21 April 1820, during a lecture demonstration, he noticed a compass needle stir each time he closed the circuit of a voltaic pile. The deflection was tiny and the audience unimpressed, but Ørsted spent the next three months making it undeniable, and that July published a four-page Latin pamphlet that set off a continental race: within months Ampère had the mathematics of interacting currents underway in Paris, and Biot and Savart had measured the distance law. The needle's sideways settle — tangent to circles around the wire — was the first evidence that a field could circulate, the exact pattern of this article's first diagram. Denmark's scientific patriarch thereafter, Ørsted founded its technical university, coined the word "electromagnetism", and named aluminium. The CGS field unit — the oersted — carried his compass into every magnetics lab of the next century.

History

1820

— 21 April: Ørsted's lecture-room compass twitches beside a live wire; July: his pamphlet announces electromagnetism to Europe.

1820

— Autumn: Schweigger and Poggendorff wind the wire into multiple turns to multiply the effect — the coil is born as a measuring instrument.

1820

— 30 October: Biot and Savart announce the quantitative distance law behind the wire's field.

1825

— Sturgeon wraps eighteen turns around an iron bar: the first electromagnet lifts twenty times its own weight.

1882

— d'Arsonval hangs a coil in a magnet: the moving-coil galvanometer turns the loop's field-and-torque physics into the ancestor of every analogue meter.

Going further

Texts. NCERT Physics Class 12, Ch. 4 (§4.6, §4.8) · HC Verma Vol. 2, Ch. 35 · CIE 9702 Topic 20.2 · IGCSE 0625 §4.5.1 (patterns and rules only) · IB DP Physics guide, D.3.

Provenance. All results standard. A verifier should recompute: wire anchors $2\times10^{-7}\times10/0.05 = 4.0\times10^{-5}$ T and $2\times10^{-7}\times12/0.025 = 9.6\times10^{-5}$ T; loop centre $4\pi\times10^{-7}\times100\times0.5/0.2 = 3.14\times10^{-4}$ T; coil example $6.28\times10^{-4}$ T centre, 6-8-10 triangle giving $1.36\times10^{-4}$ T at $x = 8$ cm, ratio $(10/6)^3 = 4.63$ ; one-eighth landmark at $x = \sqrt{3}R$ .

Prerequisites. The Biot–Savart Law · The Magnetic Field & Field Lines. Leads to. Ampère's Circuital Law · Solenoids & Toroids (next in this chapter) · Net Fields of Wire Systems (coming in the E4 problem gym) · Magnetic Field of Steady Currents (UG deep dive).

Why does the wire's field fall as 1/d when every piece of it obeys inverse-square?

Because a wire is not a point — it is an infinite supply of pieces. Stand close and only the nearest short stretch of wire matters; back away and pieces farther along the wire, which were too oblique to matter before, swing into effective broadside range. The number of usefully contributing pieces grows in proportion to your distance, while each piece weakens by distance squared: one power cancels, leaving $1/d$ . The same reasoning run for a loop seen from far away goes the other way — opposite sides of the ring nearly cancel — and steepens the falloff to $1/x^3$ . Geometry decides the exponent; the per-piece law never changes.

How do I keep the thumb rule and the curl rule from getting mixed up?

Anchor each rule to what is straight. In a straight wire, the current is the straight thing — so the thumb (your straight digit) rides the current, and the curled fingers play the circling field. In a loop, the current is the circling thing — so the curled fingers ride the current, and the thumb plays the straight field through the hole. Straight part of the hand = straight part of the physics. One right hand, two assignments, zero ambiguity — and in both cases the field and current are perpendicular partners, each circling the other.

Is the field zero anywhere inside a current loop?

No — inside the loop (in its plane), the field is nonzero everywhere and actually grows as you approach the wire itself. The centre is the minimum of the in-plane interior field, not a zero. What is true: the field is weakest at the centre along the plane, strongest near the conductor, and outside the loop in its plane the direction reverses (the return path of the field lines). MCQs exploit the confusion with "the field vanishes at the centre" as a tempting distractor — it never does for a single loop carrying current.

The axis formula has R squared upstairs and a 3/2 power downstairs — how do I remember it?

Rebuild it from its checkpoints instead of memorising blind. At $x = 0$ it must give $\mu_0 I/2R$ — try it: $R^2/(R^2)^{3/2} = 1/R$ ✓. Far away it must look like a dipole, $1/x^3$ — try it: $R^2/(x^2)^{3/2} = R^2/x^3$ ✓. The $(R^2 + x^2)^{1/2}$ is just the slant distance from the ring to your axis point (draw the right triangle: radius, axis, slant). So the formula is "centre value, demoted by the cube of slant-over-radius" — and the ratio form $B_0/B = ((R^2+x^2)^{1/2}/R)^3$ is both the memory aid and the fastest way to compute, as the worked example shows.

Why is a loop's far field called a dipole field — what does it share with a bar magnet?

Everything measurable. From distances large compared with $R$ , the loop's field has the same shape as a small bar magnet's: field threading out of one face (the north face), around, and into the other, with the same $1/x^3$ weakening and the same angular pattern. Nature exploits the disguise in reverse: a bar magnet's field is the summed field of countless microscopic current loops (electrons circulating and spinning in its atoms) — a hypothesis Ampère made in the 1820s that turned out exactly right. The strength of the equivalence is captured by one number, the magnetic moment, which Current Loops as Magnetic Dipoles builds later in this chapter.

What happens to these formulas if the loop isn't a full circle — say a semicircular arc?

The centre-of-loop logic survives beautifully, because the derivation only used "every slice equidistant, broadside, and cooperating". A semicircle is half the slices, so half the field: $B = \mu_0 I/4R$ . A quarter-arc gives a quarter, and in general an arc subtending any fraction of the full circle contributes that fraction of $\mu_0 I/2R$ . This is a favourite JEE-Main twist — often with straight feed-in wires attached whose own contributions at the centre are zero if they point radially at it (the Biot–Savart blind spot from the previous module: a slice contributes nothing along its own line). Fraction the circle, zero the radial feeds, add.

Can I use these formulas near the middle of a real, finite wire?

Yes, with judgement. Equation (1) assumed an infinite wire, but "infinite" really means long compared with your distance: near the middle of a 1-metre wire, at 2 cm away, the missing far ends would have contributed almost nothing, and (1) is accurate to better than a percent. It fails near the wire's ends (roughly half the value right at an end) and at distances comparable to the wire's length. The rule of thumb: trust the formula when $d$ is at most a tenth of the distance to the nearest end. Exam questions signal the safe regime with the word "long"; real engineering checks it numerically.

Fields of Straight Wires & Circular Loops

Where this lives today

The physical picture

The straight wire, priced

The circular loop, earned

Limiting cases

Worked examples

Board level — the compass by the cable (CBSE/ISC pattern, 3 marks)

Competitive level — one coil, two checkpoints (JEE-Main pattern, ~3 minutes)

Exam traps

Going further

Before this

Where this leads