What Produces a Magnetic Field? — Question 9

Question 9

One zero sensor projection does not establish a zero magnetic field

A directional sensor reports $b=\vec B\cdot\hat n$. It first reads zero along +z. After rotation, it reads +3.00 microtesla along +x and -4.00 microtesla along +y. The field is unchanged during the measurements.

1. Reconstruct the field vector and magnitude.

2. Find a unit direction in the xy plane that also gives zero reading, and explain the inference error in using only the first reading.

Question 9 - Solution

One zero sensor projection does not establish a zero magnetic field

(a) Method and calculation

$$B_z=0,\quad B_x=3.00\,\mathrm{\mu T},\quad B_y=-4.00\,\mathrm{\mu T}.$$$$\vec B=(3\hat x-4\hat y)\,\mathrm{\mu T},\quad B=5.00\,\mathrm{\mu T}.$$

(b) Interpretation and further calculation

$$\hat n=(4\hat x+3\hat y)/5,\quad\vec B\cdot\hat n=(12-12)/5=0.$$$$b=0\Rightarrow\vec B\perp\hat n\quad\text{or}\quad\vec B=0.$$

Physical check

A single scalar projection cannot determine three vector components.