Question 7: An electron beam passes through a region of crossed electric and magnetic fields of intensity E and B respectively. For what value of the electron speed the beam will remain unaffected?
When a charged particle is moving in a region where there are both electric and magnetic fields, it experiences a force due to both of them. The combined effect of the forces is equal to their vector sum. Now, if the electric field E and magnetic field B are crossed, the fields due to electric and magnetic fields are opposite in directions (evB and eE in the figure). Consider the figure. Let an electron of charge e moves in a crossed electric and magnetic fields.
Charge e moves to the right. The magnetic force on the electron is
Electric force on the charge is
Since both the fields exert forces in the opposite direction on the electron, therefore, if both forces are equal in magnitude they will cancel each other. The particle will pass the fields without being affected (change in direction).
This equation gives the condition when the electron will move in the field unaffected.