The Definition of B — Question 10

Question 10

Two orthogonal near-null force tests place a finite upper bound on the entire field

A charge of magnitude q is tested with velocities $v\hat x$ and $v\hat y$. In each trial the magnetic-force magnitude is no greater than Fmax. Assume q and v are nonzero and no other forces act. Set $b=F_{\max}/(qv)$.

1. Derive inequalities for the field components and the sharp upper bound on B.

2. Give a field that attains the bound and explain the exact-null limit.

Question 10 - Solution

Two orthogonal near-null force tests place a finite upper bound on the entire field

(a) Method and calculation

$$B_y^2+B_z^2\le b^2,\quad B_x^2+B_z^2\le b^2.$$$$B^2+B_z^2\le2b^2\Rightarrow B\le\sqrt2\,b.$$

(b) Interpretation and further calculation

$$\vec B=b\hat x+b\hat y:\quad|\vec F_x|=|\vec F_y|=qvb=F_{\max}.$$$$F_{\max}=0\Rightarrow B=0.$$

Physical check

One null direction permits a parallel field; two nonparallel exact-null directions exclude every nonzero field.