Ampere’s circuital law

What is Ampere’s circuital law?

In plain words: walk round any closed path in a magnetic field and keep a running total of how much of the field points along your path. The total depends only on the current threading through your path, not on the shape of the path. When the path is chosen to match the symmetry of the currents, that total gives the field almost at once.

Statement of the law

• Take an open surface with a boundary C, with a current passing through the surface. Split the boundary into small elements dl, and at each take the tangential component of the field, B_t. Add the products B_t dl = B·dl all round the boundary (NCERT pages 117–118, Figure 4.12).
• In the limit of many small elements the sum becomes a line integral, and Ampere’s law states (NCERT Eq. [4.13(a)], page 118):

• I is the total current through the surface. Sign: curl the fingers of the right hand in the sense in which the boundary is traversed; the thumb gives the direction in which the current counts as positive (page 118).
• Simplified form. Often the loop, called an amperian loop, can be chosen so that at each point B is either tangential to it with a constant non-zero size B, or normal to it, or zero. If L is the length on which B is tangential in the chosen traversal direction (opposite-directed portions need minus signs) and I_e the current enclosed, then (NCERT Eq. [4.13(b)], page 118):

• Ampere’s law is not new in content: it follows from the Biot–Savart law and expresses the same physics, just as Gauss’s law follows from Coulomb’s law (NCERT page 119; Points to Ponder 4, page 134). Like Gauss’s law, it relates a field on a boundary to the source inside it.
• In this vacuum form, I includes all enclosed current, and the magnetostatic version holds for steady currents that do not change with time (page 119).
• It holds for any loop, but it gives the field easily only when there is enough symmetry. For example, it cannot be used to get B = μ₀I/(2R) at the centre of a circular loop (NCERT page 120).
• Currents outside the loop do not appear on the right-hand side, although they do change B at points on the loop; their contributions to the integral round the loop add up to zero (reasoned from the law).

Field of an infinitely long straight wire

• For a straight infinite wire carrying a current I, take the amperian loop to be a circle of radius r centred on the wire, at right angles to it. The field is tangential to the circle and the same all round it, so the left side of the law is B × 2πr (NCERT page 118):

• Cylindrical symmetry: the field has the same size at every point of the circle, so it depends on only one coordinate, r (NCERT page 119).
• Field lines are circles round the wire, the same circles traced by iron filings in NCERT Figure 4.1(c). Magnetic field lines form closed loops, unlike electric field lines, which start on positive charges and end on negative ones. This explains Oersted’s observations (page 119).
• Finite field: although the wire is infinite, the field at a non-zero distance is finite; it is proportional to I and inversely proportional to r, and tends to blow up only when we come very close to the wire (page 119). Strictly, that happens only in the ideal zero-radius filament model; for a wire of finite radius the field stays finite, as the thick-wire result below shows (reasoned).
• Right-hand rule for a straight wire (NCERT page 119):

Grasp the wire in your right hand with your extended thumb pointing in the direction of the current. Your fingers will curl around in the direction of the magnetic field.

NCERT Physics Part I (Class 12), Section 4.6, page 119

• There are two right-hand rules: this one, and the one for the axis of a current loop. Fingers and thumb swap roles between them (NCERT footnote, page 119).
• Inside a thick wire of radius a carrying a uniform current I, an amperian circle of radius r < a encloses only I_e = Ir²/a². Then B(2πr) = μ₀Ir²/a², so B = μ₀Ir/(2πa²), rising in proportion to r. Outside, r > a, the field is μ₀I/(2πr), as for a thin wire (NCERT Example 4.7, Eqs. [4.15(a)]–[4.15(b)], pages 119–120). The field is largest at the surface, r = a (NCERT Figure 4.14).
• NCERT slip: page 118 says the straight-wire case “is exhibited in the Example 4.8 below”; the straight-wire example that follows is Example 4.7 (page 119); Example 4.8 (page 122) is about a solenoid.

Solved examples

Common mistakes to avoid

Quick revision summary

• Ampere’s law: the line integral of B round a closed loop equals μ_₀ times the current through it, with the sign set by the right-hand rule.
• With a well-chosen amperian loop it reduces to BL = μ_₀I_e; it holds for steady currents and follows from the Biot–Savart law.
• Long straight wire: B × 2πr = μ_₀I, so B = μ_₀I/(2πr), circles round the wire, cylindrical symmetry; direction by grasping the wire with the thumb along the current.
• Thick wire of radius a with a uniform current: B ∝ r inside, B ∝ 1/r outside, largest at the surface.
• The law holds for every loop but gives B easily only for symmetric currents (not, for example, at the centre of a loop).

Questions students ask

What is the difference between the Biot–Savart law and Ampere’s circuital law?

The Biot–Savart law gives the field of each small element of a current, to be added up over the whole circuit, while Ampere’s circuital law relates the line integral of B round a closed loop to the current through it (NCERT pages 113 and 118). For steady currents they hold the same physics: Ampere’s law can be derived from the Biot–Savart law, as Gauss’s law from Coulomb’s law (NCERT Points to Ponder 4, page 134). In use, the Biot–Savart law works for any shape, such as a loop or a short straight piece, but needs an integration, while Ampere’s law gives B quickly only where symmetry makes it constant along a loop: a long straight wire, a thick wire, a long solenoid (NCERT page 120).

What is the difference between Gauss’s law and Ampere’s circuital law?

Gauss’s law relates the electric flux through a closed surface to the charge inside, ϕ = q/ε₀ (NCERT Chapter 1, page 30), while Ampere’s circuital law relates the line integral of B round a closed loop to the current through it, ∮ B·dl = μ₀I (NCERT page 118): one sums over a surface, the other along a curve. Both link a field on a boundary to the source inside, and both give the field easily only for a symmetric choice of surface or loop (NCERT pages 118–119). Gauss’s law for magnetism is different again: zero magnetic flux through any closed surface (NCERT Chapter 5, page 143).

Why can Ampere’s circuital law not be used to find the field of a finite straight wire?

Ampere’s circuital law cannot give the field of a finite straight wire because the law holds only for steady currents round complete circuits, and a finite piece is not a whole circuit (beyond the NCERT text, reasoned from NCERT pages 118–120). For the whole circuit the law still holds on a circle round the piece, but the rest of the circuit makes B vary round it, so it cannot be taken out as B × 2πr. The piece’s own field, from the Biot–Savart law, is the same all round but does not obey the law alone: opposite the middle of a piece twice as long as r, it is about 71% of μ₀I/(2πr).

Why did Maxwell modify Ampere’s circuital law?

Maxwell modified Ampere’s circuital law because, as ∮ B·dl = μ₀I, it holds only for steady currents and contradicts itself when a current changes, as while a capacitor charges (NCERT Chapter 8, Section 8.2, pages 202–203). A loop round the wire bounds both a flat surface the current crosses and a pot-shaped surface between the plates that it does not, so the law gives two different fields at one point. He added a displacement current, ε₀ dΦ_E/dt, to the conduction current, giving the Ampere–Maxwell law (NCERT pages 203–204), which reduces to this law for steady currents.

Are Ampere’s law, Ampere’s circuital law and Ampere’s force law the same thing?

Ampere’s law and Ampere’s circuital law are two names for the same law, ∮ B·dl = μ₀I: NCERT uses both (NCERT pages 117–118), and the JEE Main and NEET (UG) 2026 syllabuses write “Ampere’s law” (Unit 13). Ampere’s force law is a different result: the law of the force between current-carrying wires, which Ampere studied from 1820 to 1825 and published in 1827 (NCERT pages 118 and 122). For two long parallel wires it makes parallel currents attract and antiparallel currents repel.

Which applications of Ampere’s circuital law are in the JEE Main and NEET syllabus?

Two applications of Ampere’s circuital law are named in both syllabuses: Unit 13 of JEE Main and of NEET (UG) 2026 lists “Ampere’s law and its applications to infinitely long current carrying straight wire and solenoid”, which NCERT Class 12 Physics Part I covers in Sections 4.6 and 4.7 (pages 117–122). The field inside and outside a thick wire, NCERT Example 4.7, uses the same method on a straight wire. The toroid, done with Ampere’s law in older NCERT editions, has no section in the current book, which names it only in passing (NCERT Examples 5.1 and 5.3, pages 140 and 144), and neither syllabus lists it.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Find the field 5.0 cm from a long straight wire carrying 15 A.

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   B = μ_₀I/(2πr) = 2 × 10⁻⁷ × 15/0.050 = 6.0 × 10⁻⁵ T (NCERT Eq. (4.14), page 118).

2. Q2. An amperian circle surrounds two long wires carrying 3 A and 5 A in opposite directions. What is the line integral of B round it, in terms of μ_₀?

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   Count the 5 A direction as positive, that is, traverse the circle the way the right-hand fingers curl when the thumb points along the 5 A current. The enclosed current is then 5 − 3 = 2 A, so the integral is μ_₀ × (2 A) = 8π × 10⁻⁷ T m ≈ 2.5 × 10⁻⁶ T m. Traversed the other way it is −μ_₀ × (2 A) (NCERT Eq. [4.13(a)] and its sign rule, page 118).

3. Q3. For a long straight wire, how does the field change if you go three times as far from it?

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   It falls to one-third, since B ∝ 1/r (NCERT page 119).