3. Motion of Charged Particles in a Uniform Magnetic Field

Concept Explanation:
A charged particle moving in a uniform magnetic field follows a circular or helical path. This occurs because the Lorentz force acts as a centripetal force, constantly changing the direction of the particle's velocity without changing its speed.

Radius of Circular Path:
The radius r of the circular path can be determined by balancing the centripetal force with the Lorentz force:
r = (m × v) / (q × B)
where:
m is the mass of the particle,
v is the velocity of the particle,
q is the charge of the particle,
B is the magnetic field strength.

Example Problem:
An electron with charge 1.6 × 10^-19 C and mass 9.1 × 10^-31 kg is moving at 1 × 10^7 m/s perpendicular to a magnetic field of strength 0.01 T. Calculate the radius of the electron’s circular path.

Solution:
Using the formula:
r = (m × v) / (q × B)
r = (9.1 × 10^-31 × 1 × 10^7) / (1.6 × 10^-19 × 0.01)
r = 5.69 × 10^-2 m
So, the radius of the electron’s circular path is approximately 5.69 cm.