Calculating the Magnetic Field Due to a Current — Question 10

Question 10

An asymmetric finite wire can be inferred from two endpoint angles

A straight segment lies on the x axis from x=-a to x=+3a and carries I toward +x. The field point is $P=(0,a,0)$. Only this segment contribution is required. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the field at P without treating P as a midpoint.

2. What fraction of the total is contributed by the portion x=0 to x=3a? Explain why tripling that portion length does not triple its field.

Question 10 - Solution

An asymmetric finite wire can be inferred from two endpoint angles

(a) Method and calculation

$$B_z=\frac{\mu_0 I}{4\pi a}\left[\frac{x}{\sqrt{x^2+a^2}}\right]_{-a}^{3a}.$$$$\boxed{\vec B=\frac{\mu_0 I}{4\pi a}\left(\frac1{\sqrt2}+\frac3{\sqrt{10}}\right)\hat z}.$$

(b) Interpretation and further calculation

$$f_+=\frac{3/\sqrt{10}}{1/\sqrt2+3/\sqrt{10}}=\boxed{\frac3{3+\sqrt5}}=.572949.$$$$dB_z/dx=\frac{\mu_0Ia}{4\pi(x^2+a^2)^{3/2}}\to0\quad(|x|\to\infty).$$More distant elements contribute less because of both their distance and cross-product angle.

Physical check

The two endpoint contributions have the same sign because the perpendicular from P falls within the segment; their angles are unequal.