Amperian Loop

Also called Ampere loop

An Amperian loop is a closed imaginary path drawn around a current-carrying conductor, chosen so that Ampere's law can be evaluated on it. It has no physical existence and is selected purely for the symmetry it exposes.

AP Physics C: Electricity and Magnetism defines it at EK 12.4.A.2: an Amperian loop is a closed path around a current-carrying conductor. The CED describes it elsewhere as a closed imaginary path, which is the part worth holding on to. You draw it; nature does not.

It exists to make the printed law usable:

∮B⃗⋅dℓ⃗=μ0Ienc\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}

The integral is around your chosen loop and IencI_{\text{enc}} is the current passing through any surface bounded by it.

Choosing well is the whole skill. Ampere's law is always true and only useful when you can pull ∣B⃗∣\lvert \vec{B} \rvert out of the integral, which needs a loop on which the field magnitude is constant and the angle to dℓ⃗d\vec{\ell} never changes. A circle centred on a long straight wire does that. An arbitrary wobbly loop around the same wire satisfies the law and tells you nothing.

The magnetic counterpart of a Gaussian surface, and the same warning applies: current outside the loop still contributes to the field at points on it, but contributes nothing to the integral.

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