Ampère's Law — Question 8

Question 8

An off-center circular integration path does not justify pulling B outside the integral

An infinite wire carrying I toward +z crosses the xy plane at the origin. A circular path of radius R is centered at x=d, with $0<d<R$, and is traversed counterclockwise. The wire is inside the path, but not at its center. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the circulation and the field magnitudes at the nearest and farthest points on the path.

2. Explain why setting the local field equal to mu0 I divided by 2 pi R is invalid; give a bound using the path-average tangential component.

Question 8 - Solution

An off-center circular integration path does not justify pulling B outside the integral

(a) Method and calculation

$$\boxed{\oint_C\vec B\cdot d\vec\ell=\mu_0I},\quad B_{\mathrm{near}}=\frac{\mu_0I}{2\pi(R-d)}.$$$$B_{\mathrm{far}}=\frac{\mu_0I}{2\pi(R+d)},\quad B_{\mathrm{near}}\ne B_{\mathrm{far}}.$$

(b) Interpretation and further calculation

$$\langle B_t\rangle_C=\frac1{2\pi R}\oint_C\vec B\cdot d\vec\ell=\frac{\mu_0I}{2\pi R}.$$$$\boxed{B_{\mathrm{far}}<\langle B_t\rangle_C<B_{\mathrm{near}}}.$$The law determines an average tangential component; neither constant magnitude nor tangency everywhere follows for this path.

Physical check

Ampère law remains exact for the displaced path; what fails is the symmetry argument needed to turn its integral into B times circumference.