The force on a charged particle moving in a magnetic field can be computed as the vector sum of the force due to each separate component of the magnetic field. As an example, a particle with charge q is moving with speed v in the -y direction. It is moving in a uniform magnetic field `vec B = B_x hat i + B_y hat j + B_z hat k.`
a. What are the components of the force `vec F` exerted on the particle by the magnetic field?
b. If `qgt0`, what must the signs of the components of `vec B` be if the components of `vec F` are all non-negative?
c. If `qlt0 and B_x = B_y = B_z gt 0,` find the direction and magnitude of `vec F` in terms of `|q|,` v and `B_x.`

Explanation

(a) `vec F=qvecvxxvecB=-qv[B_x(hatjxxhati)+B_z(hatjxxhatk)]`
`=qvB_xhatk-qvB_zhati`
(b) `B_xgt0, B_Zlt0, sign of B_y` does not matter.
(c) `vecF=|q|VB_xhati-|q|VB_xhatk implies |vecF|=sqrt2|q|vB_x`