D.3.3 Motion in combined electric and magnetic fields

Velocity Selector: Balancing Forces in Crossed Fields

Imagine you’re tasked with selecting particles moving at a specific speed from a stream of charged particles. How can you achieve this?

A velocity selector is the solution, using crossed electric and magnetic fields to filter particles by speed.

How Does a Velocity Selector Work?

• Crossed Fields:
  • An electric field ($E$) exerts a force ($F_E=qE$) on a charged particle.
  • A magnetic field ($B$) exerts a force ($F_B=qvB$) on the same particle.
• Balancing Forces:
  • To ensure the particle travels in a straight line, these forces must be equal and opposite: $qE=qvB$.
• Velocity Condition:
  • Simplifying the equation gives the condition for the particle to remain undeflected: $E=vB$, or $v=\frac{E}{B}$.

Why Does This Work?

• Electric Force:
  • The electric field exerts a constant force on the charged particle, pushing it upward.
• Magnetic Force:
  • The magnetic field exerts a force that depends on the particle’s velocity.
  • In this setup, it pushes the particle downward (for positive charges moving right).
• Balance:
  • When the particle moves at just the right speed, the electric and magnetic forces are equal and opposite.
  • They cancel each other out, and the particle travels in a straight line.

Deflection in Crossed Fields: Trajectory of a Charged Particle

What happens if the forces aren’t balanced?

• Speed Greater than $\frac{E}{B}$:
  • The magnetic force dominates, and the particle curves upward.
• Speed Less than $\frac{E}{B}$:
  • The electric force dominates, and the particle curves downward.

Charge Separation: The Basis of Electromagnetic Filters

Velocity selectors are used in electromagnetic filters to separate particles based on speed.

How Does Charge Separation Work?

• Crossed Fields:
  • Particles enter a region with perpendicular electric and magnetic fields.
• Speed Filtering:
  • Only particles with speed $v=\frac{E}{B}$ remain undeflected.
• Separation:
  • Particles with different speeds are deflected in different directions, allowing for separation.

Applications: Thomson’s Experiment and Electron Velocity Measurement

The velocity selector played a crucial role in J.J. Thomson’s experiment to measure the charge-to-mass ratio of the electron.

How Did Thomson Use a Velocity Selector?

• Setup:
  • Electrons were accelerated through a known potential difference to achieve a specific speed.
• Crossed Fields:
  • The electrons entered a region with perpendicular electric and magnetic fields.
• Balancing Forces:
  • By adjusting the fields to satisfy $E=vB$, Thomson ensured the electrons traveled undeflected.
• Measuring Velocity:
  • The velocity of the electrons was determined using the relationship $v=\frac{E}{B}$.

Calculating the Charge-to-Mass Ratio

• Circular Motion:
  • After passing through the velocity selector, the electrons entered a magnetic field, moving in a circular path.
• Radius of Path:
  • The radius $R$ of the path is given by $R=\frac{mv}{qB}$.
• Combining Equations:
  • Substituting $v=\frac{E}{B}$ into the equation for $R$, Thomson derived the charge-to-mass ratio:

$$\frac{q}{m}=\frac{E}{B^{2}R}$$

Reflection

The interplay of electric and magnetic fields in velocity selectors and electromagnetic filters is foundational to modern physics and technology.