An alpha particle travels at a velocity v⃗ of magnitude 550 m / s through a uniform magnetic fi

step 1 In this question, we have an upper particle. So it's moving at 550 per second, assuming it moves

An alpha particle travels at a velocity v⃗ of magnitude 550...

An alpha particle travels at a velocity $\vec{v}$ of magnitude $550 \mathrm{~m} / \mathrm{s}$ through a uniform magnetic field $\vec{B}$ of magnitude $0.045 \mathrm{~T}$. (An alpha particle has a charge of $+3.2 \times 10^{-19} \mathrm{C}$ and a mass of $6.6 \times$ $\left.10^{-27} \mathrm{~kg} .\right)$ The angle between $\vec{v}$ and $\vec{B}$ is $52^{\circ} .$ What is the magnitude of (a) the force $\vec{F}_{B}$ acting on the particle due to the field and (b) the acceleration of the particle due to $\vec{F}_{B}$ ? (c) Does the speed of the particle increase, decrease, or remain the same?

An alpha particle travels at a velocity $\vec{v}$ of magnitude $550 \mathrm{~m} / \mathrm{s}$ through a uniform magnetic field $\vec{B}$ of magnitude $0.045 \mathrm{~T}$. (An alpha particle has a charge of $+3.2 \times 10^{-19} \mathrm{C}$ and a mass of $6.6 \times$ $\left.10^{-27} \mathrm{~kg} .\right)$ The angle between $\vec{v}$ and $\vec{B}$ is $52^{\circ} .$ What is the magnitude of (a) the force $\vec{F}_{B}$ acting on the particle due to the field and
(b) the acceleration of the particle due to $\vec{F}_{B}$ ? (c) Does the speed of the particle increase, decrease, or remain the same?

Key Concepts

Magnetic Lorentz Force

This concept describes the force experienced by a charged particle moving through a magnetic field. The magnitude of the force is given by F = q v B sin(?), where q is the charge of the particle, v is its velocity, B is the magnetic field strength, and ? is the angle between the velocity and the field. This force is always perpendicular to both the magnetic field and the particle’s velocity, which has significant implications in determining the trajectory of the particle.

Uniform Magnetic Field

A uniform magnetic field is one in which the magnetic field strength and direction are constant in space. In such a field, charged particles experience a constant magnitude magnetic force as long as their velocity and the field remain fixed in magnitude and direction relative to each other. This simplifies the analysis of the motion of the particle and allows the use of standard formulas such as F = qvB sin(?).

Newton's Second Law

Newton’s Second Law states that the net force on an object is equal to its mass multiplied by its acceleration (F = ma). In the context of a charged particle in a magnetic field, this law is used to relate the magnetic Lorentz force to the acceleration of the particle, thereby allowing the calculation of the acceleration given the force calculated from the Lorentz force formula.

Circular Motion and Centripetal Acceleration

Because the magnetic force is always perpendicular to the particle’s velocity, it does not do work on the particle and instead changes only the direction of the velocity. This results in circular (or helical if there is a component of velocity parallel to the magnetic field) motion, where the acceleration directed towards the center of the circle is called centripetal acceleration, defined as a = v^2/r. In a purely circular motion scenario, the speed remains constant though the direction of motion changes.

Work Done by Magnetic Force

The work done by a force is defined as the force component in the direction of displacement multiplied by the displacement. Since the magnetic force on a moving charged particle is always perpendicular to its motion, it does no work on the particle. Consequently, the kinetic energy and speed of the charged particle remain constant even though its direction of motion is continuously changing.

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