A proton having an initial velocity of 20.0 i Mm / s enters...

A proton having an initial velocity of 20.0 i $\mathrm{Mm} / \mathrm{s}$ enters a uniform magnetic field of magnitude $0.300 \mathrm{~T}$ with a direction perpendicular to the proton's velocity. It leaves the field-filled region with velocity $-20.0 \mathrm{j} \mathrm{Mm} / \mathrm{s}$. Determine (a) the direction of the magnetic field, (b) the radius of curvature of the proton's path while in the field, (c) the distance the proton traveled in the field, and (d) the time interval during which the proton is in the field.

A proton having an initial velocity of 20.0 i $\mathrm{Mm} / \mathrm{s}$ enters a uniform magnetic field of magnitude $0.300 \mathrm{~T}$ with a direction perpendicular to the proton's velocity. It leaves the field-filled region with velocity $-20.0 \mathrm{j} \mathrm{Mm} / \mathrm{s}$. Determine
(a) the direction of the magnetic field, (b) the radius of curvature of the proton's path while in the field, (c) the distance the proton traveled in the field, and (d) the time interval during which the proton is in the field.

Key Concepts

Time of Flight in Circular Motion

The time a charged particle spends in a magnetic field while undergoing circular motion can be calculated from the arc length of its motion and its constant speed. In the context of circular motion, the time of flight is directly related to the angle swept by the trajectory and the particle's constant speed, illustrating the time it takes to traverse a part of the circular path.

Radius of Curvature in Magnetic Fields

The radius of curvature of a charged particle's circular path in a magnetic field is determined by balancing the magnetic Lorentz force and the centripetal force required for circular motion. The relation r = (mv)/(qB) connects the particle’s mass, velocity, charge, and the magnetic field strength to the radius of the trajectory.

Circular Motion of Charged Particles

If the velocity of a charged particle is perpendicular to a uniform magnetic field, it experiences a constant magnitude force that always points towards the center of a circular path. This centripetal force causes the particle to execute uniform circular motion, with the speed remaining constant while the direction changes continuously.

Uniform Magnetic Fields

A uniform magnetic field is one that has the same magnitude and direction at every point in space. When a charged particle moves in such a field, the consistent force leads to predictable motion, often resulting in circular or helical paths, depending on the particle's velocity components relative to the field.

Magnetic Lorentz Force

The magnetic Lorentz force is the force experienced by a moving charged particle in a magnetic field. This force, given by F = q(v × B), acts perpendicular to both the velocity of the particle and the magnetic field direction. It is fundamental in determining the trajectory of charged particles in magnetic fields.

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