EM-14 · Magnetostatics field

Ampère's Law

The circulation of the magnetic field around a closed loop is proportional only to the current threading the loop. Grow the Amperian loop or move the wire outside, and see for yourself that the circulation is independent of the loop size.

Grow the loop, the circulation stays

Grow the Amperian loop around the current-carrying wire. On the loop the field weakens as 1/R while the perimeter grows as 2πR. What happens to the circulation ∮B·dl? And what if you move the wire outside the loop?

Amperian loop radius RR = 1.50

Drag to orbit. Slider for loop radius R, buttons for wire inside/outside.

The circulation around the loop

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

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

B 67% · 2πR 150% · ∮ 100% (invariant)

Inside · μ₀I ·

What the Ampère law says

In EM-13 we saw the field around a straight current form concentric circles. The Ampère law compresses this into one line: the circulation ∮B·dl, the field summed once around a closed loop, is proportional only to the current I threading that loop. ∮B·dl = μ₀ I_enc, with the vacuum permeability μ₀ as the constant. Just as Gauss’s law tied flux to charge, Ampère’s law ties circulation to current.

Observe∮B·dl = μ₀ I_enc

The circulation is the enclosed current times μ₀.

Independent of loop size

Double the loop around the wire and the field on it halves (B = μ₀I/2πR) while the perimeter doubles (2πR). Their product, the circulation, is unchanged. A dented loop or a square one — same thing. As long as it wraps the same current, the circulation is always μ₀I, because the circulation is fixed by the current it encircles alone.

Choose∮B·dl = ?

Only the current threading the loop counts.

Only the threading current

When the wire is outside the loop, its field enters the loop on one side and leaves on the other. Summed once around, the incoming and outgoing contributions cancel exactly, so the circulation is zero. That is why the Ampère law counts only the current that actually threads the loop. However large a current sits outside, it contributes nothing to the circulation — though the field B at each point on the loop is still affected by it.

With symmetry, B without integrating

When symmetry is good, the Ampère law becomes a shortcut around the Biot–Savart integral. Around a straight current, take a circular loop concentric with it: B is constant over the loop and runs along it, so the circulation collapses to B·(2πr). That gives B = μ₀I/(2πr) — exactly the straight-wire result we found by integration in EM-13. A solenoid, with a rectangular loop, gives B = μ₀nI; a toroid follows the same way. Choosing the path that matches the symmetry, learned back in EM-01, is the key.

Fill inB · 2πr = ?

With symmetry, B is constant on the loop and the circulation becomes a product.

On your ownB = ?

The Ampère law gives back the field of a straight current.

Back to the first screen

On the first screen, growing the Amperian loop left the circulation untouched: as the field on it weakened by 1/R, the perimeter grew by 2πR, cancelling exactly. Move the wire outside and the circulation dropped to zero — every contribution that entered also left. In the end, what sets the circulation is not the loop's size or shape, but only the current threading it. ∮B·dl = μ₀I_enc.

The Ampère law states that the circulation of the magnetic field around a closed loop, ∮B·dl, is proportional only to the current threading the loop: ∮B·dl = μ₀ I_enc. It is independent of the loop's shape and size and of currents outside it. With good symmetry, you pull B out of the loop integral and find the field without integrating (a straight wire gives B = μ₀I/2πr).

What comes next

So far we have looked at current making a magnetic field. Biot–Savart (EM-13) and Ampère (EM-14) are both laws of "current → field." From the next unit we turn it around: how a magnetic field exerts force on moving charge — the magnetic force and the Lorentz force F = qv × B (EM-15). The force on a current-carrying wire in a field comes from here too.