Obtain an expression for magnetic Lorentz force?

Explanation

When an electric charge q is moving with velocity `vecv` in the magnetic field `vecB`, it experiences a force , called magnetic force `vecF_(m) `.
After careful experiments , Lorentz deduced the force experienced by a moving charge in the magnetic field `vecF_(m) `.
`vecF_(m) = q (vecv xx vecB)` ....(1)
In magnitude , `F_(m) = q v B sin theta` .....(2)
The equations (1) and equation (2) imply
1. `vecF_(m) ` is directly proportional to the magnetic field `vecB`
2. `vecF_(m) ` is directly proportional to the velocity `vecv`
3. `vecF_(m) ` is directly proportional to sine of the angle between the velocity and magnetic field
4. `vecF_(m) ` is directly proportional to the magnitude of the charge q
5. The direction of `vecF_(m) ` is always perpendicular to `vecv " and " vecB " as " vecF_(m) ` in the cross product of `vecv " and " vecB`

7. The direction of `vecF_(m) ` on negative charge is opposite to the direction of `vecF_(m) ` on positive charge provided other factors are identical as shown Figure.
8. If velocity `vecv ` of the charge q is along magnetic field `vecB " then, " vecF_(m) ` is zero.