An electron moving along the +x -axis at 5.0 ×10^6 m / s enters a magnetic field that makes a

step 1 We have period of the electron given by 2 pi m over QB and that will come out as 1 .78 times 10

An electron moving along the +x -axis at 5.0 ×10^6 m / s...

An electron moving along the $+x \quad$ -axis at $5.0 \times 10^{6} \mathrm{m} / \mathrm{s}$ enters a magnetic field that makes a $75^{\circ}$ angle with the $x$ -axis of magnitude 0.20 T. Calculate the (a) pitch and (b) radius of the trajectory.

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field, given by the cross product of its velocity and the magnetic field. This force causes the particle to move in a circular path when the velocity is perpendicular to the field, and it is the underlying principle behind the motion of charged particles in magnetic fields.

Velocity Decomposition

When a charged particle enters a magnetic field at an angle, its velocity can be decomposed into two components: one parallel to the magnetic field and one perpendicular to it. The parallel component remains unaffected by the magnetic force, leading to linear motion along the field direction, while the perpendicular component undergoes circular motion.

Circular Motion in a Magnetic Field

The perpendicular component of the particle's velocity results in circular motion due to the Lorentz force providing the necessary centripetal force. The radius of this circle can be computed by equating the magnetic force to the centripetal force, linking charge, magnetic field strength, and mass of the particle.

Helical Trajectory

The overall motion of the particle, combining the circular motion from the perpendicular velocity component and the linear motion from the parallel component, results in a helical trajectory. The helix is characterized by parameters such as the radius of the circular motion and the pitch, which is the distance along the magnetic field between successive turns.

Period of Circular Motion and Pitch

The period of the circular motion, determined by the charge and mass of the particle as well as the magnetic field strength, dictates the time taken to complete one full circle. Multiplying the period by the parallel component of the velocity gives the pitch of the helix, which is the distance the particle travels along the magnetic field during one complete revolution.

Please give Ace some feedback