What does Ampere's circuital law state?

Ampere's circuital law states that the line integral of the magnetic field B around any closed loop equals μ₀ times the total current passing through the surface bounded by that loop, written ∮B·dl = μ₀I. The integral is taken over the closed loop coinciding with the boundary of the surface, and the result is independent of the size or shape of the loop.

Why is Ampere's law called the magnetic analogue of Gauss's law?

Ampere's law is to the Biot–Savart law what Gauss's law is to Coulomb's law. Both Ampere's and Gauss's laws relate a field quantity on the boundary or periphery (magnetic or electric field) to its source in the interior (enclosed current or enclosed charge). Both are derived from the inverse-square laws yet make field evaluation easy whenever the configuration has symmetry.

How does the magnetic field vary inside and outside a thick current-carrying wire?

For a long straight wire of radius a carrying current uniformly over its cross-section, the field inside (r a) only the full current I is enclosed, giving B = μ₀I/2πr, so B ∝ 1/r and falls off with distance.

What is the meaning of 'enclosed current' in Ampere's law?

Enclosed current Iₑ is the net current that actually pierces the surface bounded by the chosen Amperian loop, taken with the right-hand sign convention. Currents that lie outside the loop, or that enter and leave the surface, contribute nothing. Inside a thick wire only the fraction of current within radius r is enclosed, which is why the inside field is smaller than the surface field.

When can Ampere's law be used to find the magnetic field easily?

Ampere's law always holds for steady currents, but it gives the field by inspection only when symmetry lets you choose a loop on which B is either tangential and constant in magnitude, normal to the loop, or zero. Such symmetry exists for an infinite straight wire (cylindrical symmetry), a toroid, and a long solenoid; without it the integral cannot be reduced to BL.

What is the magnetic field inside a toroid from Ampere's law?

For a toroid of N turns carrying current I, taking a circular Amperian loop of radius r concentric with the toroid gives B(2πr) = μ₀NI, so B = μ₀NI/2πr. The field exists only within the windings, is tangential to the circle, and is the result that, in the limit of a very large radius, reduces to the long-solenoid field B = μ₀nI.

Ampere's Circuital Law — NEET Physics

Statement of the law

Ampere's circuital law provides an alternative and appealing way of expressing the field–current relationship contained in the Biot–Savart law. Consider an open surface bounded by a closed curve C, with current passing through it. Break the boundary into small line elements $d\mathbf{l}$, take the tangential component of the magnetic field $B_t$ at each element, and multiply by the element length: $B_t\,dl = \mathbf{B}\cdot d\mathbf{l}$. Summing over the boundary in the limit of vanishingly small elements gives an integral.

Ampere's law states that this integral equals $\mu_0$ times the total current passing through the surface:

$$\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I$$

where $I$ is the total current through the surface and the integral is taken over the closed loop coinciding with the boundary C. As NIOS Section 18.4 stresses, the result is independent of the size or shape of the closed loop. The relation carries a sign convention given by the right-hand rule: curl the fingers of the right hand in the sense the boundary is traversed in the loop integral, and the thumb points in the direction in which current $I$ is counted positive.

The magnetic analogue of Gauss's law

Ampere's circuital law is not new in content from the Biot–Savart law. Both relate the magnetic field and the current, and both express the same physical consequences of a steady electrical current. In NCERT's words, Ampere's law is to the Biot–Savart law what Gauss's law is to Coulomb's law. Both Ampere's and Gauss's laws relate a physical quantity on the periphery or boundary — the magnetic or electric field — to another physical quantity, the source in the interior, which is the enclosed current or the enclosed charge respectively.

One restriction must be remembered: Ampere's circuital law holds for steady currents that do not fluctuate with time. With that caveat, symmetry turns the law into a calculation tool, in exactly the way Gauss's law is wielded in electrostatics.

Aspect	Gauss's law (electric)	Ampere's law (magnetic)
Integral form	`∮E·dA = q_enc/ε₀`	`∮B·dl = μ₀I_enc`
Integral over	Closed surface	Closed loop (line)
Source in interior	Enclosed charge	Enclosed current
Underlying inverse-square law	Coulomb's law	Biot–Savart law
Made tractable by	Symmetry of charge	Symmetry of current

Choosing an Amperian loop

For many applications a much simplified version of the law suffices. We assume it is possible to choose a loop — called an Amperian loop — such that at each point of the loop one of three conditions holds: (i) $\mathbf{B}$ is tangential to the loop and is a non-zero constant $B$; or (ii) $\mathbf{B}$ is normal to the loop; or (iii) $\mathbf{B}$ vanishes. Let $L$ be the length of the loop for which $\mathbf{B}$ is tangential, and let $I_e$ be the current enclosed. The law then reduces to

$$B L = \mu_0 I_e$$

This reduction is only valid when the configuration has a symmetry — such as the cylindrical symmetry of an infinite straight wire — that guarantees one of the three conditions everywhere on the loop. Without symmetry the integral cannot be pulled apart, and one falls back on the Biot–Savart law.

Figure 1 · Amperian loop around a straight wire

Application 1 — long straight wire

For an infinite straight current-carrying wire, the natural Amperian loop is a circle of radius $r$ with the wire along its axis. By symmetry the field is tangential to the circle and has the same magnitude at every point on it, so the left-hand side of $BL = \mu_0 I_e$ becomes $B \cdot 2\pi r$. The full current $I$ threads the loop, giving

$$B \times 2\pi r = \mu_0 I \quad\Rightarrow\quad B = \frac{\mu_0 I}{2\pi r}$$

This is the same field obtained, with far more labour, by integrating the Biot–Savart law over the whole wire. NCERT highlights four features of this result that are favourite NEET hooks: the field has cylindrical symmetry (it depends only on $r$); its lines form closed concentric circles rather than originating and terminating on charges; even for an infinite wire the field at a finite distance is finite and blows up only as $r \to 0$; and its direction follows the right-hand grip rule — grasp the wire with the thumb along the current and the fingers curl in the direction of $\mathbf{B}$.

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Builds toward

Stacking many such loops into a tightly wound coil gives the uniform interior field of a solenoid. See Solenoid Magnetic Field for the rectangular-loop derivation that gives $B = \mu_0 n I$.

Application 2 — inside a thick wire

NCERT Example 4.7 takes a long straight wire of circular cross-section (radius $a$) carrying steady current $I$ distributed uniformly across the cross-section, and asks for the field in the regions $r < a$ and $r > a$. This is the single most-tested consequence of Ampere's law for NEET.

Outside ($r > a$). The Amperian loop is a circle concentric with the cross-section, $L = 2\pi r$, and the current enclosed is the full $I$. The law gives the familiar straight-wire result:

$$B (2\pi r) = \mu_0 I \quad\Rightarrow\quad B = \frac{\mu_0 I}{2\pi r}, \qquad B \propto \frac{1}{r}\;(r>a)$$

Inside ($r < a$). Now the current enclosed $I_e$ is not $I$ but is less than this value. Since the distribution is uniform, $I_e = I\left(\dfrac{\pi r^2}{\pi a^2}\right) = I\dfrac{r^2}{a^2}$. Applying Ampere's law,

$$B (2\pi r) = \mu_0 I\frac{r^2}{a^2} \quad\Rightarrow\quad B = \left(\frac{\mu_0 I}{2\pi a^2}\right) r, \qquad B \propto r\;(r

So the field rises linearly from zero at the axis to a maximum $B = \mu_0 I / 2\pi a$ at the surface, then falls off as $1/r$ outside. The kink at $r = a$ is where the two regimes meet.

Figure 2 · B versus r for a thick wire

NEET Trap

Only the ENCLOSED current counts

The right side of Ampere's law is $\mu_0 I_{enclosed}$, not $\mu_0$ times the total current of the system. Inside a thick wire, only the fraction $I r^2/a^2$ pierces the loop, which is exactly why the inside field is smaller and grows linearly. Currents lying outside the loop, however large, add nothing.

Inside a uniform wire $B \propto r$; outside $B \propto 1/r$. The maximum field is at the surface, $r = a$.

Application 3 — the toroid

A toroid, as NIOS Section 18.4.2 describes it, is essentially an endless solenoid formed by bending a straight solenoid into a circular shape. To find the field at a point P inside the toroid at distance $r$ from the centre O, draw a circular Amperian loop through P concentric with the toroid. The field is everywhere tangential to this circle and equal in magnitude at all of its points, so $\oint \mathbf{B}\cdot d\mathbf{l} = B\,(2\pi r)$, since $2\pi r$ is the circumference.

If the toroid has $N$ total turns each carrying current $I$, the total current threaded by the circular path is $NI$. Ampere's law then gives

$$B (2\pi r) = \mu_0 N I \quad\Rightarrow\quad B = \frac{\mu_0 N I}{2\pi r}$$

The field exists only within the windings; outside the toroid it is effectively zero. In the limit of a very large radius, a section of the toroid behaves like a long straight solenoid, and this expression reduces to $B = \mu_0 n I$ with $n$ the turns per unit length — the forward link picked up in the solenoid note.

Figure 3 · Amperian loop inside a toroid

NEET Trap

No symmetry → no shortcut

Ampere's law is always true for steady currents, but it yields the field by inspection only when a symmetric loop can be found on which $\mathbf{B}$ is tangential and constant, normal, or zero. For a finite wire, a square loop, or a point near the end of a short solenoid, no such loop exists — the integral still equals $\mu_0 I_{enc}$, but you cannot factor out $B$, so use Biot–Savart instead.

A loop with the right symmetry is what converts $\oint\mathbf{B}\cdot d\mathbf{l}$ into $B \cdot L$.

Applications at a glance

The three standard configurations differ only in the Amperian loop chosen and the current it encloses. The structure of the calculation is identical each time.

Geometry	Amperian loop	Enclosed current	Result
Long straight wire	Circle, radius r, wire on axis	`I`	`B = μ₀I / 2πr` (B ∝ 1/r)
Inside a thick wire (uniform current, radius a)	Circle, radius r < a	`I·r²/a²`	`B = μ₀Ir / 2πa²` (B ∝ r)
Toroid, N turns	Circle, radius r, concentric	`NI`	`B = μ₀NI / 2πr`
Long solenoid (limit of toroid)	Rectangle straddling the wall	`nLI`	`B = μ₀nI`

Quick Recap

Ampere's circuital law in one screen

$\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{enclosed}$ — line integral of B around a closed loop equals μ₀ times the current threading it; valid for steady currents and independent of loop shape.
It is the magnetic analogue of Gauss's law: boundary field related to interior source; Ampere's law is to Biot–Savart what Gauss's law is to Coulomb's law.
Symmetry lets the law reduce to $BL = \mu_0 I_e$, where B is tangential and constant on length L.
Straight wire: $B = \mu_0 I / 2\pi r$. Inside a thick wire: $B = \mu_0 I r / 2\pi a^2$ (B ∝ r), peaking at the surface, then $B \propto 1/r$ outside.
Toroid: $B = \mu_0 N I / 2\pi r$, field confined within the windings; reduces to the solenoid result $B = \mu_0 nI$ for large radius.

Ampere's Circuital Law