Ampère's Law — Question 1

Question 1

Signed circulation counts enclosed currents rather than nearby currents

A closed path is traversed counterclockwise as viewed from +z. Its spanning surface is pierced by 4.00 A and 1.50 A toward +z and 3.00 A toward -z. A further 20.0 A wire passes outside the path. All currents are steady. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Calculate the circulation of B around the path.

2. Find the circulation for the reversed traversal, and explain whether the outside wire can change B on the path.

Question 1 - Solution

Signed circulation counts enclosed currents rather than nearby currents

(a) Method and calculation

$$\oint_C\vec B\cdot d\vec\ell=\mu_0I_{\mathrm{enc}},\quad I_{\mathrm{enc}}=4.00+1.50-3.00=2.50\,\mathrm A.$$$$\boxed{\oint_C\vec B\cdot d\vec\ell=\pi\times10^{-6}\,\mathrm{T\,m}}.$$

(b) Interpretation and further calculation

$$d\vec\ell\to-d\vec\ell\Rightarrow\boxed{\oint_{-C}\vec B\cdot d\vec\ell=-\pi\times10^{-6}\,\mathrm{T\,m}}.$$$$\oint_C\vec B_{\mathrm{outside}}\cdot d\vec\ell=0\quad\text{although }\vec B_{\mathrm{outside}}\ne0\text{ locally}.$$

Physical check

Ampère circulation is a signed sum through the surface; it is not a sum of field magnitudes around the boundary.