A cosmic ray muon with the same charge as an electron and a mass of 1.9 ×10^-28 kg is moving tow

step 1 In this question we have a stream of mumes hitting the earth at an angle of 25 degrees of the ve

A cosmic ray muon with the same charge as an electron and a...

A cosmic ray muon with the same charge as an electron and a mass of $1.9 \times 10^{-28} \mathrm{kg}$ is moving toward the ground at an angle of $25^{\circ}$ from the vertical with a speed of $7.0 \times 10^{7} \mathrm{m} / \mathrm{s}$. As it crosses point $P$, the muon is at a horizontal distance of $85.0 \mathrm{cm}$ from a high-voltage power line. At that moment, the power line has a current of 16.0 A. What is the magnitude and direction of the force on the muon at the point $P$ in the diagram?

Key Concepts

Magnetic Field from a Current?Carrying Wire

A current flowing through a long, straight conductor generates a magnetic field in the surrounding space. The magnitude of this magnetic field at a given distance is determined by Ampère’s law and can be calculated using the formula B = (??I)/(2?r), where ?? is the permeability of free space, I is the current, and r is the distance from the wire. This concept is central when analyzing forces in setups involving current?carrying conductors and nearby charged particles.

Lorentz Force on a Moving Charge

A moving charged particle in a magnetic field experiences a force given by F = q(v × B), known as the Lorentz force. This force depends on the particle’s charge, its velocity, and the magnetic field through which it moves. The magnitude and direction of the force are determined by the vector cross product, making it essential to understand both the mathematical and physical interpretation of cross products in electromagnetism.

Vector Cross Product Application in Magnetic Forces

The cross product in the Lorentz force equation determines both the magnitude and the direction of the force acting on a charged particle in a magnetic field. The operation results in a vector that is perpendicular to the plane formed by the velocity vector and the magnetic field vector. This mathematical tool is crucial for understanding how different orientations of velocity and magnetic field affect the resultant force.

Charge Sign and Force Direction Determination

The sign of the charge of a particle affects the direction of the Lorentz force. For a positive charge, the force follows the direction given by the right-hand rule applied to the cross product of the velocity and magnetic field. Conversely, a negative charge experiences a force in the opposite direction. This concept is important for correctly predicting the behavior of charged particles in electromagnetic fields.

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