Magnetic Field & Charged Particle

Q: A proton moves at 2.0 \times 10^6 m/s perpendicular to a 0.5 T magnetic field. What is the magnitude of the force on the proton?

1.6 \times 10^{-13} N F_B = qvB = (1.6 \times 10^{-19}) \times (2.0 \times 10^6) \times 0.5 = 1.6 \times 10^{-13} N.

Q: If a particle enters a magnetic field and its path is a straight line, what can you conclude about its velocity relative to the field?

The velocity is parallel or anti-parallel to the field. The force is F_B = qvB \sin \ heta. If \\theta = 0° or 180°, \\sin \ heta = 0, and the particle experiences no magnetic force.

Visualize the motion of charged particles in magnetic fields using the Lorentz force F = qvB sin θ. Understand that magnetic force is perpendicular to both velocity and field, causing circular or helical motion. Calculate the radius of circular paths r = mv/(qB) for particles in uniform fields. Explore applications in cyclotrons, mass spectrometers, and the Aurora Borealis where Earth's magnetic field deflects charged solar wind particles toward the poles.

MAGNETIC FORCES ON MOVING CHARGES

When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field. This is the **Lorentz Force**: . Because the force is always perpendicular to the direction of motion, it does no work and cannot change the particle's speed—it only changes the direction, causing the particle to move in a circular or helical path.

THE RIGHT-HAND RULE

To determine the direction of the magnetic force on a **positive** charge, use the Right-Hand Rule: Point your fingers in the direction of the velocity (), curl them toward the magnetic field (), and your thumb will point in the direction of the force (). For a **negative** charge (like an electron), the force points in the exact opposite direction.

HOW TO USE THIS VISUALIZATION

1. **Launch a Particle**: Select between a Proton (positive) and an Electron (negative). 2. **Adjust the Field**: Change the strength and direction of the magnetic field (In/Out of page vs. Up/Down). 3. **Set Initial Velocity**: Change the speed and entry angle of the particle. Observe how faster particles move in larger circles. **Try This**: Launch a proton into a field pointing "Into Page." Which way does it curve? Now launch an electron with the same settings. Compare the radius of their paths—why is the electron's radius so much smaller?

CORE FORMULAS

Magnetic Force on a moving charge

Radius of circular path in a uniform field

Cyclotron frequency

AP EXAM CONNECTION

Unit: Unit 5: Magnetism and Electromagnetic Induction (Topic 5.1)
Learning Objective: LO 2.D.1

COMMON MISCONCEPTIONS

Thinking the magnetic force does work on the particle (it doesn't, speed remains constant).
Forgetting to reverse the direction for negative charges.
Assuming the force is in the direction of the field (it's always perpendicular).

KEY TAKEAWAYS

Force is perpendicular to and .
No work is done by -fields.
Circular motion results from .
Radius depends on momentum ().
Right-Hand Rule determines direction.

PRACTICE QUESTIONS

Q1 (QUANTITATIVE): A proton moves at m/s perpendicular to a 0.5 T magnetic field. What is the magnitude of the force on the proton?

Show Answer & Explanation

Answer: 1.6 \times 10^{-13} N

Explanation: N.

Q2 (CONCEPTUAL): If a particle enters a magnetic field and its path is a straight line, what can you conclude about its velocity relative to the field?

Show Answer & Explanation

Answer: The velocity is parallel or anti-parallel to the field.

Explanation: The force is . If ° or 180°, , and the particle experiences no magnetic force.