Beam of Electrons A beam of electrons whose kinetic energy...

Beam of Electrons A beam of electrons whose kinetic energy is $K$ emerges from a thin-foil "window" at the end of an accelerator tube. There is a metal plate a distance $d$ from this window and perpendicular to the direction of the emerging beam (Fig. $29-28$ ). Show that we can prevent the beam from hitting the plate if we apply a uniform magnetic field $\vec{B}$ such that its magnitude is

Beam of Electrons A beam of electrons whose kinetic energy is $K$ emerges from a thin-foil "window" at the end of an accelerator tube.
There is a metal plate a distance $d$ from this window and perpendicular to the direction of the emerging beam (Fig. $29-28$ ). Show that we can prevent the beam from hitting the plate if we apply a uniform magnetic field $\vec{B}$ such that its magnitude is

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field. This force, given by F = q(v × B), is always perpendicular to the velocity of the particle and the magnetic field. This perpendicular nature of the force leads to a change in the direction of the particle’s velocity, rather than its speed, resulting in circular or helical motion. Understanding the Lorentz force is crucial for predicting how charged particles, such as electrons, move in the presence of magnetic fields.

Circular Motion of Charged Particles

When a charged particle enters a uniform magnetic field perpendicularly, it experiences a force that causes it to move in a circular path. The radius of this circular motion, often referred to as the Larmor radius, depends on the particle’s mass, velocity, charge, and the strength of the magnetic field. This concept is essential in applications like cyclotrons and beam steering in accelerators, where controlling the trajectory of charged particles is important.

Kinetic Energy and Momentum Relationship

The relationship between a particle’s kinetic energy and its momentum is critical for determining its speed and thus the radius of its circular motion in a magnetic field. For non-relativistic speeds, the kinetic energy (K) is related to the momentum (p) by the equation p = ?(2mK). This relationship allows one to express the radius of the trajectory in terms of kinetic energy and the magnetic field, linking energy considerations with motion dynamics.

Magnetic Field Control of Particle Trajectories

A uniform magnetic field can be used to manipulate the path of a charged particle to avoid obstacles or to guide it along a desired path. By carefully selecting the magnitude of the magnetic field, one can ensure that the centripetal force induced by the field is sufficient to bend the particle’s trajectory away from a target object or region. This principle is commonly applied in designing beamlines and other systems where precise control over charged particle motion is required.

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