A thin, vertical solenoid carrying anti-clockwise current I with ηturns per unit length, length

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A thin, vertical solenoid carrying anti-clockwise current I...

A thin, vertical solenoid carrying anti-clockwise current $I$ with $\eta$ turns per unit length, length $l$ and radius $R$ can be used to focus off-axis charges. Define the origin at the bottom of the solenoid and the z-axis to be positive upwards (towards the top of the solenoid). (a) Determine the longitudinal and radial magnetic field at a small perpendicular distance $r \ll R$ from the cylindrical axis as a function of z-coordinate $z$. Hint: we have calculated one of these fields in a previous problem. (b) Now, suppose that particles with charge $q$ and mass $m$ are placed at radial distances $r \ll R$ at the bottom of the solenoid. They are given a large velocity $v_0$ in the positive $z$-direction such that their radial coordinates are approximately constant throughout the solenoid. If $l \gg R$, approximately how large should $v_0$ be as compared to the other parameters for this to occur? Now, determine the instantaneous velocities of these particles at the instance they exit from the top of the solenoid. (c) Determine the time $t$ after this instance, at which they coincide with the z-axis, assuming that the magnetic field outside the solenoid is zero. Would this set-up function as a good lens to focus charges (of the same magnitude) with different initial radial distances?

A thin, vertical solenoid carrying anti-clockwise current $I$ with $\eta$ turns per unit length, length $l$ and radius $R$ can be used to focus off-axis charges. Define the origin at the bottom of the solenoid and the z-axis to be positive upwards (towards the top of the solenoid).
(a) Determine the longitudinal and radial magnetic field at a small perpendicular distance $r \ll R$ from the cylindrical axis as a function of z-coordinate $z$. Hint: we have calculated one of these fields in a previous problem.
(b) Now, suppose that particles with charge $q$ and mass $m$ are placed at radial distances $r \ll R$ at the bottom of the solenoid. They are given a large velocity $v_0$ in the positive $z$-direction such that their radial coordinates are approximately constant throughout the solenoid. If $l \gg R$, approximately how large should $v_0$ be as compared to the other parameters for this to occur? Now, determine the instantaneous velocities of these particles at the instance they exit from the top of the solenoid.
(c) Determine the time $t$ after this instance, at which they coincide with the z-axis, assuming that the magnetic field outside the solenoid is zero. Would this set-up function as a good lens to focus charges (of the same magnitude) with different initial radial distances?

Key Concepts

Lorentz Force and Charged Particle Dynamics

Charged particle dynamics in the presence of a magnetic field are governed by the Lorentz force, which acts perpendicular to both the velocity of the particle and the magnetic field. This results in circular or helical motion characterized by the cyclotron frequency and Larmor radius. In the context of a solenoid, understanding how the magnetic field—both uniform and varying components—affects the trajectory is critical for designing devices intended to control or focus charged particles.

Magnetic Focusing and Lens Aberration

Magnetic focusing utilizes the spatial variation of the magnetic field to converge charged particle trajectories towards a focal point, making the magnetic field act like a lens. In practice, the focusing quality depends on how sensitively the focal properties depend on the initial conditions of the particles, such as their radial position. Aberrations occur when particles with different initial radial distances focus at different times or positions, potentially preventing a precise convergence. This concept is key in applications ranging from electron optics to accelerators, where maintaining beam quality is essential.

Solenoid Magnetic Field

This concept involves understanding the magnetic field produced by a solenoid. In an ideal long solenoid, the field is nearly uniform along the axis with magnitude proportional to the product of the current and the turn density (typically given by ???I). However, near the ends of the solenoid or when considering regions outside the core, the field is non?uniform, and fringe fields appear. Analyzing a solenoid’s field is fundamental for problems in electromagnetism where the spatial distribution of the magnetic field influences charged particle behavior.

Field Expansion in Cylindrical Coordinates

When considering points at a small perpendicular distance from the axis of symmetry (r ? R), it is useful to perform a series expansion of the magnetic field components in cylindrical coordinates. This technique allows one to approximate both the longitudinal and radial components of the magnetic field as functions of the axial coordinate z and the small radial distance r. Such expansions are essential in predicting the behavior of particles slightly off the symmetry axis in devices like solenoids.

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