A particle with charge `7.00 mu C` is moving with velocity `vec v =- (4xx10^3 ms^-1) hat j.` The magnetic force on the particle is measured to be `vec F = +(8.4 xx 10^-2 N) hat i - (5.60 xx 10^-2 N) hat k.`
a. Calculate all the components of the magnetic field you can from this information.
b. Are there components of the magnetic field that cannot be determined by measurement of the force? Explain.
c. Calculate the scalar product `vec B. vec F.` What is the angle between `vec B and vec F`?

Explanation

(a) Let `vecB=B_x hati+B_yhatj+B_zhatk, then vecF=qvecvxxvecB`
`implies 8.40xx10^-2hati-5.60xx10^-2hatk`
`=7xx10^-6(-4xx10^3)hatjxx(B_xhati+B_yhatj+B_zhatk)`
`implies 84hati-56hatk=28(-B_xhatk+B_zhati)`
`implies B_x=-56/28=-2T, B_z=-84/28=-3T`
(b) `B_y` is indeterminate.
(c) Because `vecFbot vecB, so vecB.vecF=0`.
Let us check it
`vecB.vecF=(-2hati+B_yhatj-3hatk).(8.40xx10^-2hati-5.70xx10^-2hatk)=0`