What Produces a Magnetic Field? — Question 3

Question 3

Current reversal separates a fixed background from a reversible source

A vector sensor measures $(8\hat x+6\hat y)$ microtesla with current +I and $(2\hat x-2\hat y)$ microtesla with current -I. Assume the background is constant and the source field is odd in I.

1. Extract the background vector and the source vector at +I.

2. Find their magnitudes and predict the reading at zero current.

Question 3 - Solution

Current reversal separates a fixed background from a reversible source

(a) Method and calculation

$$\vec B_+=\vec B_0+\vec B_s,\quad\vec B_-=\vec B_0-\vec B_s.$$$$\vec B_0=\tfrac12(\vec B_++\vec B_-)=(5\hat x+2\hat y)\,\mathrm{\mu T},\quad\vec B_s=(3\hat x+4\hat y)\,\mathrm{\mu T}.$$

(b) Interpretation and further calculation

$$B_0=\sqrt{29}\,\mathrm{\mu T},\quad B_s=5.00\,\mathrm{\mu T}.$$$$\vec B(0)=\vec B_0=(5\hat x+2\hat y)\,\mathrm{\mu T}.$$

Physical check

This extraction fails if the background drifts between readings or the source has an even-in-current contribution.