Calculating the Magnetic Field Due to a Current — Question 8

Question 8

A semicircular detour and tangent leads reinforce at their common center

A wire travels from x=-infinity to x=0 along y=-R, then counterclockwise around the right semicircle centered at the origin to (0,+R), then toward x=-infinity along y=+R. Current I follows this path. Find the field at the origin. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Calculate the contributions of the semicircle and the two semi-infinite straight leads.

2. Find the total and the fraction supplied by the curved part.

Question 8 - Solution

A semicircular detour and tangent leads reinforce at their common center

(a) Method and calculation

$$B_{\mathrm{arc}}=\frac{\mu_0 I\pi}{4\pi R}=\frac{\mu_0 I}{4R},\quad B_{\mathrm{lead}}=\frac{\mu_0 I}{4\pi R}.$$$$\vec B_{\mathrm{arc}},\ \vec B_{\mathrm{lower}},\ \vec B_{\mathrm{upper}}\text{ all point along }+z.$$

(b) Interpretation and further calculation

$$\vec B=\boxed{\frac{\mu_0 I}{4\pi R}(\pi+2)\hat z}.$$$$\frac{B_{\mathrm{arc}}}{B_{\mathrm{total}}}=\boxed{\frac{\pi}{\pi+2}}=.611015.$$

Physical check

Tangent leads have a finite perpendicular distance from P; unlike radial leads, their fields do not vanish.