The Definition of B — Question 3

Question 3

A force magnitude alone leaves a twofold ambiguity in the velocity-field angle

A nonzero charge of magnitude q moves at speed v in a field of known magnitude B. Its measured magnetic-force magnitude is $F=qvB/2$. Let theta lie between zero and 180 degrees.

1. Find all possible theta values.

2. If the sign of the velocity component along the field is independently found to be negative, choose the angle and find the parallel and perpendicular speeds.

Question 3 - Solution

A force magnitude alone leaves a twofold ambiguity in the velocity-field angle

(a) Method and calculation

$$F=qvB\sin\theta\Rightarrow\sin\theta=1/2.$$$$\theta=30.0^\circ\quad\text{or}\quad150^\circ.$$

(b) Interpretation and further calculation

$$v_\parallel=v\cos\theta<0\Rightarrow\theta=150^\circ.$$$$v_\parallel=-\tfrac{\sqrt3}{2}v,\quad v_\perp=\tfrac12v.$$

Physical check

Magnetic-force magnitude measures perpendicular motion and cannot by itself distinguish parallel from antiparallel components.