The Definition of B — Question 4

Question 4

Two perpendicular velocity probes determine all three magnetic-field components

A positive test charge q=1.00 microcoulomb is sent through the same field with velocity $10^4\hat x$ m/s, giving force $(0,-.0300,.0200)$ N. With velocity $10^4\hat y$ m/s, its force is $(.0300,0,-.0100)$ N. Only magnetic forces are present.

1. Reconstruct B from both measurements.

2. Predict the force for velocity $10^4\hat z$ m/s and verify it is perpendicular to that velocity.

Question 4 - Solution

Two perpendicular velocity probes determine all three magnetic-field components

(a) Method and calculation

$$\vec F_x=qu(0,-B_z,B_y),\quad\vec F_y=qu(B_z,0,-B_x),\quad qu=.0100.$$$$\vec B=(1.00\hat x+2.00\hat y+3.00\hat z)\,\mathrm T.$$

(b) Interpretation and further calculation

$$\vec F_z=qu(-B_y,B_x,0)=(-.0200\hat x+.0100\hat y)\,\mathrm N.$$$$\vec F_z\cdot(u\hat z)=0.$$

Physical check

A field component invisible to one velocity direction is accessible to a nonparallel probe.