Write the expression for the force `vecF` acting on a charged particle of charge q moving with a velocity `vecv` in the presence of both electric field `vecE` and magnetic field `vecB` . Obtain the condition under which the particle moves underflected through the fields.

Explanation

A charged particle having charge q and velocity `vecv` experiences a total force `vecF` in the presence of an electric field `vecE` and a magnetic field `vecB`, which is expressed as :
`vecF = vecF_(E) + vecF_(B) = q [ vec(e) + (vec(v) + vec(B))]`
If `vecv , vecE and vecB` are in mutually perpendicular directions as shown in fig.
`vecv = v hat i, vecE = E hat j and vecB = B hatk`, then
`vecF_(E) = q vec(E) = q E hatj` along y-axis, and
`vecF_(B) = q (vecv xx vecB) = q ( v hati xx B hatk) = q v B (hati xx hatk) = -q v B hatj`
If values of `vecE and vecB` are adjusted such that magnitude of `F_E` and `F_B` are equal then naturally being euqal and opposite they nullify each other and net force on the charged particle is zero. As a result particle continues to travel along a straight line without suffering any deviation. For such an eventuality
`q E = q v B implies v = E/B`.