Problem related to deflection of electrons under magnetic and electric fields

Question

A beam of electrons passes first through a magnetic field and then through an electric field such that the beam remains undeflected.

What are the possible ways of arranging the two fields ?

My approach Keeping magnets at the same plane as the paper , the beam would deflect outwards perpendicular to plane of paper if the magnetic field is in the plane of the paper downwards.

Now the beam can pass through the electric field acting in direction perpendicular to plane of paper outwards such that positive plate is kept below the paper and negative plate is kept above it, thus causing deflection inwards into the plane of the paper.

Is my approach/solution correct ?

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Are there any other possible combinations or approaches possible?

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Answer 1

The exact geometry of your proposed solution is unclear. However, the simple fact is that the electric and magnetic fields must be perpendicular to each other and to the velocity of the electron beam to achieve the desired zero deflection.

This diagram illustrates the forces exerted by a magnetic field on a beam of charged particles, using the right hand to create a set of orthogonal axes:

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Unfortunately, the diagram is for a beam of positive charges. To apply this method to beam of electrons, simply use your left-hand, or use your right hand and remember to reverse the direction of the magnetic field

In any event, it is clear that a sideways magnetic field, to the right, will create an upward force on the beam of electrons. So you would need an upward electric field (acting on electrons, remember) to exert an opposite and equal downward force on the electrons.

It's also relevant to mention that the magnetic field force is dependent on the velocity of the electrons, while the electric field force is not. So the condition of zero deflection for a beam of electrons will serve to sort the electrons by velocity.