What Produces a Magnetic Field? — Question 10

Question 10

A linear source model can be tested with a two-coil closure experiment

At a fixed sensor, coil A alone adds $\vec a=(2\hat x+\hat y)$ microtesla and coil B alone adds $\vec b=(-\hat x+3\hat y)$ microtesla. With neither energized the reading is $\vec c=(4\hat x-2\hat y)$ microtesla. Assume all measurements use the same source currents.

1. Predict the both-on reading under linear superposition and construct a combination of four readings that should vanish.

2. The measured both-on reading is $(5.2\hat x+2\hat y)$ microtesla. Find the closure residual and give two possible experimental explanations without concluding which is responsible.

Question 10 - Solution

A linear source model can be tested with a two-coil closure experiment

(a) Method and calculation

$$\vec B_A=\vec c+\vec a,\quad\vec B_B=\vec c+\vec b,\quad\vec B_{AB}^{\mathrm{pred}}=\vec c+\vec a+\vec b=(5\hat x+2\hat y)\,\mathrm{\mu T}.$$$$\vec D=\vec B_{AB}-\vec B_A-\vec B_B+\vec B_0=0\quad\text{under the model}.$$

(b) Interpretation and further calculation

$$\vec D=(.200\hat x+0\hat y)\,\mathrm{\mu T}.$$$$|\vec D|=.200\,\mathrm{\mu T}.$$Possible causes include sensor drift or changed coil currents when both are connected; nonlinear magnetic material response is another possibility.

Physical check

A failed closure test rejects the complete experimental model, not necessarily superposition for fixed microscopic sources.