Calculating the Magnetic Field Due to a Current — Question 2

Question 2

Oppositely traversed arcs form an annular sector with a nonzero central field

A closed wire follows an inner circular arc of radius a counterclockwise through angle phi, a radial connector, an outer arc of radius b clockwise through the same angle, and a second radial connector. Here $b>a$; all paths lie in the xy plane. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Derive the field at the common center, including the radial segments.

2. Find its magnitude and direction for a=.0400 m, b=.100 m, phi=120 degrees and I=2.00 A.

Question 2 - Solution

Oppositely traversed arcs form an annular sector with a nonzero central field

(a) Method and calculation

$$d\vec\ell\parallel\vec r\Rightarrow\vec B_{\mathrm{radial}}=0,\quad B_{\mathrm{arc}}=\frac{\mu_0 I}{4\pi R}\int_0^\phi d\theta.$$$$\vec B=\boxed{\frac{\mu_0 I\phi}{4\pi}\left(\frac1a-\frac1b\right)\hat z}.$$

(b) Interpretation and further calculation

$$\phi=2\pi/3,\quad a^{-1}-b^{-1}=25-10=15\,\mathrm{m^{-1}}.$$$$\vec B=\boxed{2\pi\,\mu\mathrm T\,\hat z}\simeq6.28319\,\mu\mathrm T\,\hat z.$$

Physical check

The smaller-radius arc dominates because each arc field scales as inverse radius; the field vanishes when the radii coincide.