A proton moves perpendicular to a uniform magnetic field 𝐁 at a speed of 1.00 ×10^7 m / s and un

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A proton moves perpendicular to a uniform magnetic field 𝐁...

A proton moves perpendicular to a uniform magnetic field $\overrightarrow{\mathbf{B}}$ at a speed of $1.00 \times 10^{7} \mathrm{m} / \mathrm{s}$ and undergoes an acceleration of $2.00 \times 10^{13} \mathrm{m} / \mathrm{s}^{2}$ in the positive $x$ -direction when its velocity is in the positive z - direction. Determine the magnitude and direction of the field.

Key Concepts

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the force exerted on a charged particle moving in a magnetic field. By pointing the fingers in the direction of the velocity and curling them towards the direction of the magnetic field, the thumb points in the direction of the magnetic force for a positive charge. This tool is essential for visualizing and determining vector directions in electromagnetic problems.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic field strength and direction are constant in space. In such a field, a charged particle moving through the field experiences a constant force (in magnitude) as long as its speed remains constant and its velocity is perpendicular to the field. This uniformity is key to simplifying the analysis of the particle’s motion, making it possible to apply the Lorentz force law directly.

Lorentz Force

The Lorentz force describes the force experienced by a charged particle moving in electric and magnetic fields. In the case of a magnetic field, the force is given by F = q(v × B), where q is the charge, v is the velocity vector, and B is the magnetic field vector. The force acts perpendicular to both the velocity and the magnetic field, causing the particle to change direction without doing work, which is a fundamental concept in electromagnetism.

Circular Motion of Charged Particles

When a charged particle moves perpendicular to a uniform magnetic field, the Lorentz force acts as a centripetal force, causing the particle to follow a circular path. The balance between the magnetic force (qvB) and the required centripetal force (mv²/r) for circular motion allows one to determine parameters such as the radius of the circle or the magnitude of the magnetic field. This concept is crucial in understanding phenomena like cyclotron motion in particle accelerators.

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