A velocity selector uses a 60 -mT magnetic field perpendicular to a 24 -kN/C electric field. At what speed will charged particles pass through the selector undeflected?

Electric force plays a crucial role in the concept of a velocity selector, a device used to filter charged particles based on their speeds. The electric force, denoted as (\( F_e \)), acts on a charged particle within an electric field and is calculated by the product of the charge (\( q \)) and the electric field (\( E \)). That is, (\( F_e = qE \)). In the context of our exercise, this force is counterbalanced by the magnetic force to ensure that only particles with a certain velocity pass through undeflected.

The concept can be better understood by imagining a particle with a specific charge being exposed to an electric field. As a result of this exposure, it experiences a force pushing it in the direction of the field if the charge is positive, or in the opposite direction if the charge is negative. This force is directly proportional to both the magnitude of the charge and the strength of the electric field, implying that larger charges or stronger fields result in a greater force.

The magnetic force, represented by (\( F_m \)), is another fundamental concept in this problem. It affects the behavior of a charged particle when it moves perpendicular to a magnetic field. The magnitude of this force is contingent upon the charge of the particle (\( q \)), its velocity (\( v \)), and the strength of the magnetic field (\( B \)), and can be calculated by (\( F_m = qvB \)). Notably, this force does not act along the direction of motion but perpendicularly to both the velocity and the magnetic field, a right-hand rule usually illustrates this direction.

If we consider a charged particle traveling through a magnetic field, it experiences a force that tends to curve its path. The higher the velocity or the stronger the magnetic field, the more significant the deflection. This also means that, unlike electric forces, magnetic forces do not do work on the particle, as the force remains perpendicular to its motion, preventing a change in the particle's kinetic energy.

Charged particle deflection is a phenomenon that occurs when a charged particle passes through simultaneous electric and magnetic fields. When the electric force (\( F_e \)) is exactly balanced by the magnetic force (\( F_m \)), the particle continues its path undeflected. This balance implies that (\( qE = qvB \)), leading to the formula (\( v = \frac{E}{B} \)) used to calculate the velocity at which a particle will not be deflected by the fields in the velocity selector.

In the practical application of a velocity selector, we find that by adjusting the strengths of the electric and magnetic fields, we can 'dial in' the speed of the particles that will be transmitted straight through, while particles with other speeds will be deflected and filtered out. This is pivotal in experimental and analytical techniques, such as mass spectrometry, which relies on this exact principle to separate particles based on their mass-to-charge ratio.