Ampere's circuital law: long straight wire and solenoid

Ampere's circuital law

Imagine any closed path in space. We call it an Amperian loop; we choose it, just as we choose a Gaussian surface in electrostatics. Break the loop into small pieces dl and add up B·dl all the way round. This sum is the line integral ∮ B·dl. Ampere's circuital law says that the line integral of B round any closed loop equals μ₀ times the net current threading the loop: ∮ B·dl = μ₀I. Currents outside the loop do not count. The law holds for steady currents. It is most useful when symmetry lets us pick a loop on which B is tangential and of constant size. Then the integral is simply B × (length of the loop).

Derivation: field of an infinitely long straight wire

An infinitely long straight wire carries a steady current I. Find B at a distance r from it.

1. 1
   Choose an Amperian loop: a circle of radius r, centred on the wire, in a plane perpendicular to the wire.
2. 2
   By symmetry, B has the same size at every point of this circle and is tangential to it.
3. 3
   So B·dl = B dl at every point, and ∮ B·dl = B ∮ dl = B × 2πr.
4. 4
   The current threading the loop is I.
5. 5
   Ampere's law: B × 2πr = μ₀I.
6. 6★
   So, B = μ₀I/2πr. The field falls as 1/r, and its direction is given by the right-hand thumb rule.

★ key step

The straight solenoid (qualitative treatment)

A solenoid is a long insulated wire wound in a close helix, a spiral like a spring. Each turn is nearly a circular loop. Inside, the fields of all the turns point the same way along the axis and add up. So the field inside a long solenoid is strong, uniform and parallel to the axis. Outside, the fields of neighbouring turns tend to cancel, so the field is very weak, nearly zero for a long solenoid. Ampere's law gives the result B = μ₀nI, where n is the number of turns per unit length. You need this result, not its derivation. B does not depend on the length or the radius, as long as the solenoid is long compared with its radius. One end behaves like a north pole and the other like a south pole, just like a bar magnet. A soft iron core inside makes the field many times stronger.

How to draw the field of a solenoid

• Draw two parallel rows of small circles, top and bottom, for the turns cut along their length.
• Put ⊙ in every circle of the top row and ⊗ in every circle of the bottom row, to show the current.
• Inside, draw straight, parallel, equally spaced lines along the axis, with arrows.
• Outside, draw a few lines that leave one end, spread out and curve round to enter the other end.
• Label the end where the lines come out N and the other end S. Check with the rule: looking at an end, anticlockwise current means N.

Worked example: wire and solenoid

Long straight wire vs long solenoid