Ampère's Law
Circulation of B counts the current inside.
Lesson at a glanceShow the key points
- ∮B⃗·dl⃗ = μ₀ i_through (35.10).
- Right-hand rule decides which currents are positive.
- Outside currents don't contribute; useful with symmetry.
- Long wire: B·2πd = μ₀i.
Gauss’s law relates the flux of E through a closed surface to the charge inside. Ampère’s law relates the circulation of B round a closed loop to the current through it.
Circulation and sign conventions
Draw any closed plane curve and put an arrow on it. Curl the fingers of your right hand along the arrow: the thumb marks the positive side of the enclosed area. (Looking at the loop, an anticlockwise arrow means the positive side faces you.)
At each small step dl⃗ along the curve take B⃗·dl⃗, using the total field there, and add them round the whole loop. This line integral ∮B⃗·dl⃗ is the circulation of B⃗. Currents crossing the area toward the positive side count as positive, the others as negative.
- B⃗ on the left is the field of all currents everywhere.
- i on the right counts only currents passing through the loop; outside currents contribute zero circulation.
A loop is traversed so that its positive side points into the page. Wires through it carry 3 A into the page, 5 A into the page and 2 A out of the page; another wire outside carries 7 A. Find ∮B⃗·dl⃗.
A long straight wire, again
Take a circle of radius d round a long straight wire, in a plane perpendicular to it. By symmetry B⃗ is tangent to the circle and has the same size all round it, so ∮B⃗·dl⃗ = B · 2πd. The current through the circle is i:
- 1∮B⃗·dl⃗ = μ₀ × (net current through the loop) (35.10).
- 2Right-hand rule fixes which currents are positive.
- 3Outside currents shape B⃗ but add nothing to the circulation.
- 4Symmetry + Ampère: long wire gives μ₀i/2πd immediately.
Check yourself
Keep exploring
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