Calculating the Magnetic Field Due to a Current — Question 6

Question 6

A specified error tolerance determines how long a wire must be

A segment of total length 2L is observed a distance a from its midpoint. Its field is $B_f=B_\infty L/\sqrt{L^2+a^2}$, where $B_\infty=\mu_0I/(2\pi a)$. Define the relative approximation error using the exact finite field: $\epsilon=(B_\infty-B_f)/B_f$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the minimum L/a for an error no larger than 1.00 percent.

2. Derive the leading error for L much larger than a and compare its predicted threshold.

Question 6 - Solution

A specified error tolerance determines how long a wire must be

(a) Method and calculation

$$\epsilon=\sqrt{1+(a/L)^2}-1\le.0100.$$$$\boxed{L/a\ge[(1.0100)^2-1]^{-1/2}}=7.05346.$$

(b) Interpretation and further calculation

$$\sqrt{1+u}=1+u/2+O(u^2)\Rightarrow\epsilon\simeq a^2/(2L^2).$$$$\epsilon\le.0100\Rightarrow L/a\gtrsim\sqrt{50}=7.07107.$$

Physical check

The definition of relative error matters; normalizing by the infinite-wire estimate gives a slightly different exact threshold.