A proton of velocity 1.5 ×10^6 m s^-1 enters a region of uniform magnetic field B=0.50 T The mag

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A proton of velocity 1.5 ×10^6 m s^-1 enters a region of...

A proton of velocity $1.5 \times 10^{6} \mathrm{m} \mathrm{s}^{-1}$ enters a region of uniform magnetic field $B=0.50 \mathrm{T}$ The magnetic field is directed vertically up (along the positive $z$ direction) and the proton's velocity is initially on the $z$ -x plane making an angle of $30^{\circ}$ with the positive $x$ axis. (See Figure 6.37 .) (a) Show that the proton will follow a helical path around the magnetic field lines. (b) What is the radius of the helix? (c) How many revolutions per second does the proton make? (d) How fast is the proton moving along the field lines? (e) What is the vertical separation of the coils of the helix?

A proton of velocity $1.5 \times 10^{6} \mathrm{m} \mathrm{s}^{-1}$ enters a region of uniform magnetic field $B=0.50 \mathrm{T}$ The magnetic field is directed vertically up (along the positive $z$ direction) and the proton's velocity is initially on the $z$ -x plane making an angle of $30^{\circ}$ with the positive $x$ axis. (See Figure 6.37 .)
(a) Show that the proton will follow a helical path around the magnetic field lines.
(b) What is the radius of the helix?
(c) How many revolutions per second does the proton make?
(d) How fast is the proton moving along the field lines?
(e) What is the vertical separation of the coils of the helix?

Key Concepts

Radius of Curvature (Cyclotron Radius)

The radius of the circle traced by the perpendicular motion of a charged particle in a magnetic field is known as the cyclotron radius. It is determined by the balance between the Lorentz force and the centripetal force required for circular motion, and it depends on the particle's mass, charge, the magnetic field strength, and the perpendicular component of the velocity.

Lorentz Force

The Lorentz force is the fundamental force on a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This force is always perpendicular to both the velocity and the magnetic field, which leads to circular or curved paths when the particle's velocity has a component perpendicular to the field.

Velocity Component Decomposition

Decomposing the velocity of a charged particle into components parallel and perpendicular to the magnetic field is crucial. The perpendicular component contributes to circular motion due to the Lorentz force, while the parallel component remains unaffected by the magnetic field. This decomposition explains the resulting helical trajectory of the particle.

Circular Motion in a Magnetic Field

When only the perpendicular component of a charged particle's velocity is considered, the particle undergoes circular motion. This motion is characterized by a constant radius and a constant angular frequency, which depend on the magnitude of the magnetic field, the particle's charge, mass, and the speed associated with the perpendicular component.

Helical Motion

A helical motion results from the combination of circular motion (due to the perpendicular velocity component) and linear motion (due to the parallel velocity component). The particle spirals along the direction of the magnetic field, forming a helix, with the axis of the helix aligned with the field lines.

Cyclotron Frequency

Cyclotron frequency is the rate at which a charged particle revolves in a circle under the influence of a magnetic field. It depends only on the charge, the magnetic field strength, and the mass of the particle, and is independent of the particle's speed. This frequency is key in determining the number of revolutions per second in circular or helical motion.

Pitch of the Helix

The pitch of the helix is the vertical distance between consecutive turns of the helical path. It is determined by the parallel component of the velocity and the period of the circular motion. This concept explains how quickly the particle moves along the direction of the magnetic field while simultaneously undergoing circular motion.

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