Question

The force F experienced by the charge Q=2c moving with the velocity V with a magnitude B is given by F=qv×B

Given

V= 2i+4j+6k

F=4i-20j+12k

What then is B

If Bx=By

Answer 1

Given:

${\bf v}= 2\hat i+4\hat j+6\hat k$

${\bf F}= 4\hat i-20\hat j+12\hat k$

$q=2e$

$B_x=B_y=B$

The Lorentz force

$${\bf F}=q{\bf v}\times {\bf B}=2e\begin{vmatrix} \hat i & \hat j &\hat k \\ 2 & 4 &6\\ B& B& B_z \end{vmatrix}$$

$${\bf F}=2e[(4B_z-6B)\hat i-(2B_z-6B)\hat j-2B\hat k]$$

Comparing with given force, one has

$$B=\frac{12}{-4e}=\frac{3}{|e|}$$

$$B_z=\frac{10}{e}+\frac{3B}{e}=-\frac{19}{|e|}$$

Finally,

$${\bf B}=\frac{3}{|e|}(1,1,-19/3)$$