Calculating the Magnetic Field Due to a Current — Question 1

Question 1

A finite straight segment approaches the long-wire field in a controlled limit

A straight segment on the x axis extends from $-L$ to $+L$ and carries steady current I toward +x. Find only this segment contribution at $P=(0,a,0)$; the rest of the closed circuit is distant. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Integrate the Biot–Savart law to obtain the field vector.

2. Evaluate for I=3.00 A, L=.120 m and a=.0500 m, and compare with an infinite wire.

Question 1 - Solution

A finite straight segment approaches the long-wire field in a controlled limit

(a) Method and calculation

$$d\vec\ell=dx\hat x,\quad\vec r=-x\hat x+a\hat y,\quad d\vec\ell\times\vec r=a\,dx\hat z.$$$$\vec B=\frac{\mu_0 I}{4\pi}\int_{-L}^L\frac{a\,dx}{(x^2+a^2)^{3/2}}\hat z=\boxed{\frac{\mu_0 I L}{2\pi a\sqrt{L^2+a^2}}\hat z}.$$

(b) Interpretation and further calculation

$$B=(12.0\,\mu\mathrm T)\frac{.120}{.130}=\boxed{11.0769\,\mu\mathrm T}.$$$$B_\infty=\frac{\mu_0 I}{2\pi a}=12.0\,\mu\mathrm T,\quad B/B_\infty=12/13.$$

Physical check

The field points out of the xy plane. The finite segment contributes less than the infinite wire, with equality approached as L/a grows.