A particle with charge `-5.60 nC` is moving in a uniform magnetic field `vec B = -(1.25T) hat k.` The magnetic force on the particle is measured to be `vec F = -(3.36 xx 10^-7 N) hat i + (7.42 xx 10^-7 N) hat j.`
a. Calculate all components of the velocity of the particle from this information.
b. Are there components of the velocity that cannot be determined by the measurement of the force? Explain.
c. Calculate the scalar product `vec v . vec F.` What is the angle between v and F?

Explanation

(a) Let velocity of the particle is `vecv=v_xhati+v_y hatj+v_zhatk`
then `vecF=qvecvxxvecB`
`implies -3.36xx10^-7hati+7.42xx10^-7hatj`
`=-5.6xx10^-9(v_xhati+v_y hatj+v_zhatk)xx(-1.25)hatk`
`implies -3.36hati+7.42hatj=7xx10^-2(-v_xhatj+y_yhati)`
`implies v_x=-(7.42xx10^2)/7=-106ms^-1`
and `v_y=(-3.36xx10^2)/7=-48ms^-1`
(b) The value of `v_z` is indeterminate.
(c) Because `vecFbotvecv, so vecv.vecF=0`. Let us check it
`vecv.vecF=(-106hati-48hatj+v_zhatk).(-3.36xx10^-7hati+7.42xx10^-7hatj)`
`=106xx3.36xx10^-7-48xx7.42xx10^-7=0`