Find the condition under which the charged particles moving with different speeds in the presence of electric and magnetic field vectors can be used to select charged particles of a particular speed.

Explanation

To find the condition under which charged particles moving with different speeds can be selected in the presence of electric and magnetic fields, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces on Charged Particles**:
- When charged particles move through an electric field (\( \vec{E} \)) and a magnetic field (\( \vec{B} \)), they experience forces due to both fields. The electric force (\( \vec{F_E} \)) is given by:
\[
\vec{F_E} = q \vec{E}
...