A source injects an electron of speed v=1.5 ×10^7 m / s into a uniform magnetic field of magnit

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A source injects an electron of speed v=1.5 ×10^7 m / s...

A source injects an electron of speed $v=1.5 \times 10^{7} \mathrm{~m} / \mathrm{s}$ into a uniform magnetic field of magnitude $B=1.0 \times 10^{-3} \mathrm{~T}$. The velocity of the electron makes an angle $\theta=10^{\circ}$ with the direction of the magnetic field. Find the distance $d$ from the point of injection at which the electron next crosses the field line that passes through the injection point.

Key Concepts

Velocity Component Decomposition

When a charged particle enters a magnetic field at an angle, its velocity can be decomposed into two orthogonal components: one parallel to the magnetic field and one perpendicular to it. The parallel component remains unchanged since the magnetic force does not act along the field direction, while the perpendicular component is responsible for the circular motion of the particle.

Cyclotron Period and Pitch

The time taken for one complete circle (the cyclotron period) is dependent on the mass of the particle, the charge of the particle, and the strength of the magnetic field. The pitch of the helical path is the product of the parallel velocity component and the cyclotron period, indicating the distance along the direction of the magnetic field between successive crossings of the same field line.

Helical Motion

When the perpendicular circular motion is combined with the uniform motion along the magnetic field direction, the resulting trajectory of the charged particle is a helix. This helical motion is characterized by a constant pitch, which is the distance traveled parallel to the magnetic field in one complete revolution of the circular motion.

Lorentz Force

The Lorentz force describes the force on a charged particle moving in a magnetic field, given by F = q(v × B). This force acts perpendicular to both the velocity of the particle and the magnetic field, and it is responsible for the curved trajectories of charged particles in magnetic fields.

Circular Motion in a Magnetic Field

The perpendicular component of the charged particle's velocity, under the influence of the Lorentz force, results in uniform circular motion. The radius of this circular path, often called the cyclotron radius, is determined by the balance of the magnetic force and the centripetal force required to maintain the circular motion.

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