Explain velocity of charge in combined electric and magnetic field in reference to velocity selector.

Explanation

1. A charge q moving with velocity `vecv` in presence of both electric and magnetic fields experiences a force given by,
`vecF=vecF_(E)+vecF_(B)`
= `vecE_(q)+q(vecv+vecB)" "...(1)`

2. Electric and magnetic field are perpendicular to each other and also perpendicular to the velocity of the particle as shown in figure.
`vecF_(E)=qvecE=qEhatj" "...(2)`
and `vecF_(B)=q(vecvxxvecB)=q(vhatixxBhatk)`
`vecF_(B)=-qvB(hatj)" "...(3)" "(becausehatixxhatk=-hatj)`
`thereforevecF=vecF_(E)+vecF_(B)=q[E-VB]hatj`
3. Thus, `vecF_(E)andvecF_(B)` both are opposite to each other
4. Suppose, we adjust the value of `vecEandvecB` such that magnitudes of the two forces are equal. Then, total force on the charge is zero and the charge will move in the fields undeflected.
5. Thus, Eq = qvB
`thereforev=E/B" "...(4)`
6. This condition can be used to select charged particles of a particular velocity out of a beam containing charges moving with different speeds (irrespective of their charge and mass). The crossed `vecEandvecB` fields therefore, serve as a velocity selector. Only particles with speed `E/B` pass undeflected through the region of crossed fields.
7. This method was employed by J.J. Thomson in 1897 to measure the charge to mass ratio `e/m` of an electron.
8. The principle is also employed in mass spectrometer, a device that separates charged particles, usually ions according to their charge to mass ratio.