A proton moves in a helical path at speed v=4.0 ×10^7 m / s high in the atmosphere, where the E

step 1 All right, so I have a proton moving in a magnetic field, which is causing it to go in a helical

A proton moves in a helical path at speed v=4.0 ×10^7 m / s...

A proton moves in a helical path at speed $v=4.0 \times$ $10^{7} \mathrm{~m} / \mathrm{s}$ high in the atmosphere, where the Earth's magnetic field has magnitude $B=1.0 \times 10^{-6} \mathrm{~T}$. The proton's velocity makes an angle of $25^{\circ}$ with the magnetic field. (a) Find the radius of the helix. [Hint: Use the perpendicular component of the velocity.] (b) Find the pitch of the helix-the distance between adjacent "coils" [Hint: Find the time for one revolution; then find how far the proton moves along a field line during that time interval.]

Key Concepts

Helical Motion

Helical motion occurs when a charged particle moves in a uniform magnetic field with an initial velocity that has both parallel and perpendicular components relative to the field. The perpendicular component undergoes circular motion under the Lorentz force, while the parallel component remains unaffected, resulting in a spiral (helical) path along the direction of the magnetic field.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field, given by F = q(v × B). It acts perpendicular to both the particle's velocity and the magnetic field, ensuring that the force changes only the direction of the velocity without altering its magnitude, thereby causing the circular component of the motion.

Decomposition of Velocity Components

To analyze motion in a magnetic field, the particle's velocity must be decomposed into two components: one parallel to the magnetic field, which remains constant, and one perpendicular, which is responsible for the circular motion. This separation is crucial for determining both the radius of the circular path and the pitch of the helical trajectory.

Radius of the Circular Motion

The radius of the circular motion in the plane perpendicular to the magnetic field can be determined by balancing the magnetic Lorentz force with the centripetal force required for circular motion. This leads to a formula where the radius depends on the particle’s mass, its perpendicular velocity component, charge, and the magnetic field strength.

Cyclotron Period and Pitch

The cyclotron period is the time it takes for the charged particle to complete one full circular orbit in the magnetic field. The pitch of the helical path—the distance between adjacent turns—is calculated by multiplying the particle’s velocity component parallel to the magnetic field by the cyclotron period, linking the time of one revolution to the longitudinal displacement.

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