13. Magnetic deflection of electron beam. Deflection of an...

13. Magnetic deflection of electron beam. Deflection of an electron beam in a cathode-ray tube may be accomplished by magnetic as well as by electrostatic means. A beam of electrons of energy $W$ enters a region of transverse uniform magnetic field of strength $B$. (Neglect fringe effects. See Fig. 4.22.) (a) If $x$ is the distance from the point at which the electron entered the field to the point at which it leaves it, show that $$ y=r\left[1-\sqrt{1-\left(\frac{x}{r}\right)^{2}}\right] $$ where $r$ is the radius of curvature of the electron in the transverse magnetic field. The radius of curvature is the radius of the circle that will match (coincide with) the curved portion of the path. (b) If $R$ is the radius of the magnet poles, then $x \approx 2 R$ when $r \gg R$. Use the binomial expansion to show $y \approx 2 R^{2} / r$.

13. Magnetic deflection of electron beam. Deflection of an electron beam in a cathode-ray tube may be accomplished by magnetic as well as by electrostatic means. A beam of electrons of energy $W$ enters a region of transverse uniform magnetic field of strength $B$. (Neglect fringe effects. See Fig. 4.22.)
(a) If $x$ is the distance from the point at which the electron entered the field to the point at which it leaves it, show that
$$
y=r\left[1-\sqrt{1-\left(\frac{x}{r}\right)^{2}}\right]
$$
where $r$ is the radius of curvature of the electron in the transverse magnetic field. The radius of curvature is the radius of the circle that will match (coincide with) the curved portion of the path.
(b) If $R$ is the radius of the magnet poles, then $x \approx 2 R$ when $r \gg R$. Use the binomial expansion to show $y \approx 2 R^{2} / r$.

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in an electromagnetic field. In the context of a magnetic deflector, the force is perpendicular to both the velocity of the particle and the magnetic field, causing the electron to travel in a curved path rather than a straight line.

Uniform Magnetic Field

A uniform magnetic field is one in which the magnetic field vector has the same magnitude and direction at every point in space. This consistency ensures that a charged particle entering the field will experience a constant magnitude of magnetic force, which is essential for analyzing the particle’s curved trajectory.

Circular Motion of Charged Particle

When a charged particle moves perpendicular to a uniform magnetic field, the constant perpendicular force causes it to describe a circular path. This motion is characterized by a constant radius known as the radius of curvature, which depends on the particle's speed, charge, mass, and the magnetic field's strength.

Radius of Curvature

The radius of curvature in the context of a charged particle in a magnetic field is the radius of the circle that best approximates the particle's curved path. It is determined by balancing the centripetal force required for circular motion with the magnetic force acting on the particle.

Binomial Approximation

The binomial approximation is a mathematical method used to simplify expressions that involve powers of binomials, particularly when one of the terms is small compared to the other. It is especially useful in physics for approximating expressions where higher order terms can be neglected, thereby simplifying the analysis of systems where exact solutions are complex.

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