EM-14 · Magnetostatics field

Ampère's Law

The circulation, the magnetic field added up once around a closed loop, is proportional only to the current passing through that loop. Growing the Ampère loop or moving the wire outside it, you confirm for yourself that the circulation does not depend on the size of the loop.

Grow the loop and the circulation stays the same

The gold wire carries current upward, and the circle around it is the Ampère loop. The small arrows on the loop are the magnetic field: grow the loop with the slider and the field weakens as 1/R, but the perimeter grows as 2πR. So the circulation ∮B·dl stays put on the indicator. Move the wire outside the loop with the button, and the circulation disappears.

Amperian loop radius RR = 1.50

Drag to rotate; the slider sets the loop radius R, and the buttons put the wire inside or outside.

The circulation around the loop

∮B·dl = μ₀ I_enc · only the current threading the loop counts

B shrinks as 1/R while the perimeter grows as 2πR, so their product, the circulation, stays the same.

B 67% · 2πR 150% · ∮ 100% (invariant)

Inside · μ₀I ·

What the Ampère law says

In EM-13 you saw that the magnetic field around a straight current draws concentric circles. Ampère's law sums this up in one line. The magnetic field added up once around a closed loop, the circulation ∮B·dl, is proportional only to the current passing through the loop. ∮B·dl = μ₀ I_enc. The constant μ₀ is the permeability of free space. Just as Gauss's law linked flux and charge, Ampère's law links circulation and current. This form, however, is the law for currents that do not change in time; when the current changes, the displacement current you will meet later must be added.

Observe∮B·dl = μ₀ I_enc

The circulation is the enclosed current times μ₀.

Independent of loop size

Doubling a loop around a wire halves the field on the loop (B = μ₀I/(2πR)) but doubles the perimeter (2πR). Their product, the circulation, does not change. The same holds for a dented loop or a square one: as long as it encloses the same current, the circulation is always μ₀I. That is because only the enclosed current sets the circulation.

Choose∮B·dl = ?

Only the current passing through the loop counts.

Only the threading current

When the wire is outside the loop, going once around the loop you meet stretches where the field points along the loop and stretches where it points against it. The two contributions cancel exactly, so the circulation is 0. That is why Ampère's law counts only the current that actually passes through the loop. However large a current runs outside, it adds nothing to the circulation. The field B at each point on the loop, however, is affected by outside currents, so without symmetry the law still holds but cannot give the field directly.

With symmetry, B without integrating

With good symmetry, Ampère's law becomes a shortcut past the Biot-Savart integral. Take a circular loop concentric with a straight current, and the field has constant magnitude on the loop and runs along it, so the circulation reduces at once to B·(2πr). This gives B = μ₀I/(2πr), exactly the straight-current result found by integration in EM-13. A solenoid yields B = μ₀nI with a rectangular loop, and a toroid is solved the same way. The key is choosing a path that fits the symmetry, as practiced in EM-01.

Fill inB · 2πr = ?

With symmetry, the field is constant on the loop and the circulation turns into a product.

On your ownB = ?

Ampère's law brings back the field of a straight current.

Back to the first screen

On the first screen, growing the Ampère loop did not budge the circulation. As the field on the loop weakened as 1/R, the perimeter grew as 2πR, and the two canceled exactly. Moving the wire outside dropped the circulation to 0, which means that what was added and what was subtracted while going around the loop balanced out. In the end, what sets the circulation is not the size or shape of the loop but only the current passing through it. ∮B·dl = μ₀I_enc.

Ampère's law states that the circulation ∮B·dl, the magnetic field added up around a closed loop, is proportional only to the current passing through the loop. ∮B·dl = μ₀ I_enc. It does not depend on the shape and size of the loop or on currents outside the loop. With good symmetry, B can be pulled out of the integral and the field found without integrating (straight wire B = μ₀I/2πr).

What comes next

So far we have looked at currents making magnetic fields. Biot-Savart (EM-13) and Ampère (EM-14) are both laws of current → magnetic field. From the next unit we look the other way, at magnetic fields exerting forces on moving charges: the magnetic force and the Lorentz force F = qv × B (EM-15). The force on a current in a magnetic field also comes from here.