The Biot-Savart Law and Ampere's Law
You know that a current makes a magnetic field — hold a compass near a wire carrying a heavy current
and the needle swings. But how much field, and pointing which way, at a given point
in space? That is the question this page answers. It is the magnetic twin of a story you have already
met for electricity: to find the electric field of a charge you had two tools — brute-force summation
(Coulomb's law) and clever symmetry (Gauss's law). Magnetism has exactly the same pair.
The brute-force tool is the Biot–Savart law: chop the current into tiny pieces, work
out the little field each piece makes, and add them all up. It always works, but the sum can be
painful. The elegant tool is Ampère's law: when the field has enough symmetry, one
line integral hands you the answer in a single line of algebra. Learn to reach for the right tool and
the magnetic field of almost any everyday arrangement — a wire, a loop, a solenoid, the coil inside an
MRI scanner — falls out cleanly.
Biot–Savart: the field of a tiny piece of current
Start small. Take a current I flowing along a wire and zoom in on an
infinitesimal length d\vec{\ell} of it, pointing in the direction the
current flows. Jean-Baptiste Biot and Félix Savart measured, in 1820, exactly how much magnetic field
this little segment contributes at a point a displacement \vec{r} away:
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The field contributed by a current element is
d\vec{B} = \frac{\mu_0}{4\pi}\,\frac{I\,d\vec{\ell}\times\hat{r}}{r^{2}},
where \hat{r} is the unit vector from the element toward the field
point and r is the distance to it.
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It is a vector product. The d\vec{\ell}\times\hat{r}
means the field wraps around the current: point your right thumb along the current and
your fingers curl the way \vec{B} points.
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It is an inverse-square law, just like Coulomb's — the
1/r^{2} makes distant current elements contribute far less.
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The constant \mu_0 = 4\pi\times 10^{-7}\ \text{T·m/A} is the
permeability of free space; the tidy 4\pi is bundled
into it so the answers for wires and loops come out clean.
To find the total field you integrate this over the whole current path:
\vec{B} = \frac{\mu_0 I}{4\pi}\int \frac{d\vec{\ell}\times\hat{r}}{r^{2}}.
The cross product is what makes this fiddly — d\vec{\ell} and
\hat{r} change direction as you march along the wire. Let's turn the crank
on the two cases that matter most.
Biot–Savart in action
A long straight wire. Add up the contributions of every element of an infinitely long
straight wire carrying current I, at a perpendicular distance
r. Every element pushes the field in the same circulating direction (around
the wire), and doing the integral gives a famously simple result:
B = \frac{\mu_0 I}{2\pi r}.
The field circles the wire, falls off as 1/r (not
1/r^2 — the extra factor of r comes from summing
an infinite line of elements), and its direction is set by the right-hand rule: thumb along the
current, fingers curl the way \vec{B} points.
On the axis of a circular loop. Now bend the wire into a single circular loop of
radius R carrying current I, and ask for the
field at a point on its axis, a distance x from the centre. By symmetry the
off-axis parts of every element's contribution cancel and only the axial part survives, leaving
B = \frac{\mu_0 I R^{2}}{2\,(R^{2}+x^{2})^{3/2}}.
Set x = 0 to get the field right at the centre of the loop,
a result worth memorising:
B_{\text{centre}} = \frac{\mu_0 I}{2R}.
Notice how much work the cross product and the integral did here. That is Biot–Savart's honest cost:
it always works, but you pay for it. When the geometry is symmetric enough, there is a shortcut.
Ampère's law: the shortcut through symmetry
Gauss's law let you skip the Coulomb integral by relating the flux of \vec{E}
through a closed surface to the charge inside. Ampère's law does the analogous magic for
magnetism, but with a closed loop instead of a closed surface:
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Around any closed loop, the line integral of the magnetic field equals
\mu_0 times the current threading the loop:
\oint \vec{B}\cdot d\vec{\ell} = \mu_0\, I_{\text{enc}}.
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I_{\text{enc}} is the net current passing through any
surface bounded by the loop; its sign follows a right-hand rule (curl your fingers the way you
walk the loop, your thumb points the "positive" current direction).
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It is one of Maxwell's four equations (in this static form), the magnetic partner
of Gauss's law — and, like Gauss's law, it is always true but only useful when
symmetry lets you pull B out of the integral.
Watch it recover the straight wire in one line. Around a wire the field must, by
symmetry, be constant in size on a circle of radius r and point along it.
So \oint \vec{B}\cdot d\vec{\ell} = B\,(2\pi r), the loop encloses the full
current I, and Ampère's law gives
B\,(2\pi r) = \mu_0 I, i.e. B = \dfrac{\mu_0 I}{2\pi r}.
The same answer Biot–Savart earned through a whole integral drops out in a single step.
Four workhorse fields from Ampère's law
Pick the Ampèrian loop that matches the symmetry and the field falls out.
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Straight wire (outside): circular loop of radius r →
B = \dfrac{\mu_0 I}{2\pi r} (as above).
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Inside a fat wire of radius R carrying uniform current:
a loop of radius r < R encloses only the fraction of current inside it,
I_{\text{enc}} = I\,(r^2/R^2), so
B = \frac{\mu_0 I\, r}{2\pi R^{2}} \quad (r < R).
The field grows linearly from zero at the centre, peaks at the surface, then falls off as
1/r outside.
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Solenoid (a long tightly-wound coil, n turns per unit
length): a rectangular loop straddling the wall gives a uniform field inside and essentially zero
outside,
B = \mu_0 n I.
Remarkably, the field inside does not depend on how fat the solenoid is or where you stand
inside it.
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Toroid (a solenoid bent into a doughnut, N total turns):
a circular loop of radius r inside the doughnut encloses
N I, giving
B = \frac{\mu_0 N I}{2\pi r}.
The chart below plots the wire field across the whole range — the linear rise
B\propto r inside the conductor and the 1/r
decay outside — meeting at the surface. Drag the sliders for the wire radius
R and current I and watch the peak move.
Worked examples
Example 1 — field beside a wire. A power cable carries
I = 100\ \text{A}. What is B at
r = 5\ \text{cm} = 0.05\ \text{m}?
B = \frac{\mu_0 I}{2\pi r} = \frac{(4\pi\times 10^{-7})(100)}{2\pi (0.05)} = \frac{2\times 10^{-7}\times 100}{0.05} = 4\times 10^{-4}\ \text{T}.
About 0.4\ \text{mT} — roughly ten times the Earth's field. (Handy shortcut:
\mu_0/2\pi = 2\times 10^{-7}.)
Example 2 — centre of a loop. A flat coil of radius
R = 0.10\ \text{m} carries I = 3\ \text{A}. The
field at its centre is
B_{\text{centre}} = \frac{\mu_0 I}{2R} = \frac{(4\pi\times 10^{-7})(3)}{2(0.10)} = \frac{(1.257\times 10^{-6})(3)}{0.20} \approx 1.9\times 10^{-5}\ \text{T}.
Example 3 — inside a solenoid. A solenoid has
n = 500\ \text{turns/m} and carries I = 2\ \text{A}.
Its interior field is
B = \mu_0 n I = (4\pi\times 10^{-7})(500)(2) \approx 1.26\times 10^{-3}\ \text{T}.
A uniform 1.3\ \text{mT} everywhere inside — that uniformity is exactly why
solenoids are the field-makers of choice.
Example 4 — the Ampèrian derivation in full (solenoid). Take a rectangular loop with
one long side of length L running inside the solenoid parallel to its
axis, the opposite long side well outside, and the two short sides crossing the wall. Walk the
loop and split the integral into four pieces:
- Inside leg: \vec{B}\parallel d\vec{\ell}, contributes
B L.
- Outside leg: B \approx 0, contributes nothing.
- The two short legs: \vec{B}\perp d\vec{\ell} inside and zero outside, so
\vec{B}\cdot d\vec{\ell} = 0.
Thus \oint\vec{B}\cdot d\vec{\ell} = B L. The loop encloses the length
L of solenoid, which carries nL turns, each with
current I, so I_{\text{enc}} = n L I. Ampère's law
gives B L = \mu_0 n L I, and the L cancels:
B = \mu_0 n I.
Ampère's law is always true, but only useful with symmetry — exactly like
Gauss's law. The equation \oint\vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}}
holds for every closed loop around any current, tangled or tidy. But it only lets you
solve for B when the symmetry is high enough to pull the (constant)
field out of the integral, so that \oint\vec{B}\cdot d\vec{\ell} collapses
to B\times(\text{loop length}). With a lopsided current the left side is a
genuine, un-simplifiable integral and Ampère's law tells you almost nothing on its own — you fall back
to Biot–Savart.
Two more traps. First, I_{\text{enc}} is the current the loop
bounds — a wire threading the loop counts, a wire outside it does not, and its sign follows
the right-hand rule relative to the direction you walk the loop. Second, don't expect the solenoid
field to depend on radius: inside a long solenoid it is \mu_0 n I
everywhere, independent of how far you are from the axis. Uniformity is the whole point.
The integral form is one face of a deeper local law. Apply
Stokes' theorem,
which turns a line integral around a loop into a surface integral of the curl over any surface the loop
bounds: \oint\vec{B}\cdot d\vec{\ell} = \int(\nabla\times\vec{B})\cdot d\vec{A}.
Writing the enclosed current as the flux of the current density,
I_{\text{enc}} = \int\vec{J}\cdot d\vec{A}, and demanding the two match for
every loop forces the integrands to be equal point by point:
\nabla\times\vec{B} = \mu_0\,\vec{J}.
That is Ampère's law in differential form — a statement about the field at a single
point rather than around a loop. Maxwell later spotted that this version is incomplete: a
changing electric field (as in a charging capacitor, where no real current crosses the gap) also makes
magnetic field. He patched it with a "displacement current" term,
\mu_0\varepsilon_0\,\partial\vec{E}/\partial t, and that single fix — the
realisation that changing fields feed each other — is what predicted electromagnetic waves and revealed
that light itself is one. The humble coil law you just learned is a corner of Maxwell's equations, and
the same physics is what makes the giant superconducting solenoid of an MRI scanner hum.