The Definition of B — Question 5

Question 5

One force measurement determines only the magnetic field perpendicular to the velocity

A particle of known nonzero q and nonzero velocity v has measured magnetic force F. No electric or mechanical force contributes.

1. Derive the most general B compatible with the measurement and state its consistency condition.

2. For $\vec v=v\hat x$ and $\vec F=F\hat z$ with q positive, give B explicitly and identify the field of minimum magnitude.

Question 5 - Solution

One force measurement determines only the magnetic field perpendicular to the velocity

(a) Method and calculation

$$\vec v\cdot\vec F=0,\quad\vec F\times\vec v=q[v^2\vec B-(\vec v\cdot\vec B)\vec v].$$$$\vec B=\frac{\vec F\times\vec v}{qv^2}+\lambda\hat v,\quad\lambda\text{ arbitrary}.$$

(b) Interpretation and further calculation

$$\vec B=\frac F{qv}\hat y+\lambda\hat x.$$$$B^2=\left(\frac F{qv}\right)^2+\lambda^2\Rightarrow B_{\min}=\frac F{qv}\quad(\lambda=0).$$

Physical check

Adding a component parallel to the probe velocity leaves the magnetic force unchanged.