In a long, straight, vertical lightning stroke, electrons...

In a long, straight, vertical lightning stroke, electrons move downward and positive ions move upward and constitute a current of magnitude 20.0 kA. At a location 50.0 m east of the middle of the stroke, a free electron drifts through the air toward the west with a speed of 300 m/s. (a) Make a sketch showing the various vectors involved. Ignore the effect of the Earth’s magnetic field. (b) Find the vector force the lightning stroke exerts on the electron. (c) Find the radius of the electron’s path. (d) Is it a good approximation to model the electron as moving in a uniform field? Explain your answer. (e) If it does not collide with any obstacles, how many revolutions will the electron complete during the $60.0-\mu \mathrm{s}$ duration of the lightning stroke?

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in an electromagnetic field. It is given by the vector sum of the electric force (if any) and the magnetic force, where the magnetic force is proportional to the charge, its velocity, and the magnetic field, and acts perpendicular to both the velocity and the magnetic field. This force is responsible for the deflection and curving of charged particle trajectories in magnetic fields.

Uniform Magnetic Field Approximation

The uniform magnetic field approximation assumes that the magnetic field strength and direction remain constant over the region of interest. This simplifies the analysis of charged particle motion, such as predicting circular or helical trajectories. However, if the magnetic field varies significantly over the particle’s path—as it might near the current-carrying source—this approximation may not hold and can lead to errors in predicted motion.

Cyclotron Frequency and Period

When a charged particle moves in a circular orbit under the influence of a constant magnetic field, its motion is characterized by a specific angular frequency known as the cyclotron frequency. This frequency, and the associated period of revolution, are determined by the charge and mass of the particle and the strength of the magnetic field. These parameters allow one to predict how many complete cycles a charged particle will execute in a given time interval.

Magnetic Field from a Current-Carrying Wire

This concept describes how an electric current generates a magnetic field in the space surrounding it, as quantified by the Biot–Savart law and Ampère’s law. In the context of a long, straight wire, the magnetic field forms concentric circles around the wire, with its magnitude inversely proportional to the distance from the wire. This is a fundamental idea in electromagnetism and is essential for analyzing the interaction between current-carrying conductors and moving charged particles.

Circular Motion of Charged Particles in a Magnetic Field

A charged particle moving perpendicular to a uniform magnetic field experiences a magnetic force that acts as a centripetal force, causing the particle to move in a circular path. The balance between the magnetic force and the centripetal force determines the radius of the orbit, making this concept crucial for understanding particle dynamics in magnetic environments such as cyclotrons or in regions near current-carrying conductors.

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