A charged particle is accelerated from rest through a...

A charged particle is accelerated from rest through a potential difference $\Delta V .$ The particle then passes straight through a velocity selector (field magnitudes $E$ and $B$ ). Derive an expression for the charge-to-mass ratio $(q / m)$ of the particle in terms of $\Delta V, E$, and $B$.

Key Concepts

Energy Conservation in Particle Acceleration

When a charged particle is accelerated through a potential difference, the work done by the electric field is converted into kinetic energy. This is encapsulated by the equation (1/2)mv² = q?V, which shows that the kinetic energy gained by the particle is directly proportional to the product of its charge and the potential difference, offering a direct relationship between the velocity achieved and the provided accelerating voltage.

Velocity Selector Principle

A velocity selector employs perpendicular electric and magnetic fields to filter particles based on their speed. In such a setup, the electric force (qE) acting on the particle is balanced by the magnetic force (qvB) when the particle's velocity equals E/B. This precise balance ensures that only particles with the selected velocity pass through undeflected, providing a method to control and measure particle speeds.

Derivation of Charge-to-Mass Ratio

By combining the energy conservation relation with the velocity selector condition, one can derive an expression for the charge-to-mass ratio of the particle. Substituting the velocity obtained from the balance of forces (v = E/B) into the kinetic energy equation leads to an equation that expresses q/m in terms of the potential difference, electric field, and magnetic field, which is a fundamental technique in determining the properties of charged particles in electromagnetic fields.

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