Acceleration and magnetic deflection of electrons. (This...

Acceleration and magnetic deflection of electrons. (This problem and Probs. 12 to 14 are reviews of material in Chap. 3.) Suppose electrons are liberated at rest at point 0 on a metallic plane (see Fig. $4.21$ ) and are accelerated toward a parallel plane $0.25 \mathrm{~cm}$ away by an electric field. A tiny hole at $P$ permits a beam of electrons to escape into a region free of electric fields (all in a high vacuum, of course). The electric field is produced by applying voltages of $-300$ and $0 \mathrm{~V}$ to the metallic planes, as indicated. It is desired to bend the beam through a $90^{\circ}$ angle in a circular path of radius $0.5 \mathrm{~cm}$ by a magnetic field $B$ of circular outline as shown. Calculate the field strength required; also state its direction. (Note: An electric field intensity of 1 statvolt $/ \mathrm{cm}$ is equal to $300 \mathrm{~V} / \mathrm{cm}$.)

Acceleration and magnetic deflection of electrons. (This problem and Probs. 12 to 14 are reviews of material in Chap.
3.) Suppose electrons are liberated at rest at point 0 on a metallic plane (see Fig. $4.21$ ) and are accelerated toward a parallel plane $0.25 \mathrm{~cm}$ away by an electric field. A tiny hole at $P$ permits a beam of electrons to escape into a region free of electric fields (all in a high vacuum, of course). The electric field is produced by applying voltages of $-300$ and $0 \mathrm{~V}$ to the metallic planes, as indicated. It is desired to bend the beam through a $90^{\circ}$ angle in a circular path of radius $0.5 \mathrm{~cm}$ by a magnetic field $B$ of circular outline as shown. Calculate the field strength required; also state its direction. (Note: An electric field intensity of 1 statvolt $/ \mathrm{cm}$ is equal to $300 \mathrm{~V} / \mathrm{cm}$.)

Key Concepts

Electric Field Acceleration

When charged particles such as electrons are subjected to an electric field, they experience a force that accelerates them. This force is given by F = qE, where q is the charge and E is the electric field strength. The work done by the field increases the kinetic energy of the particles as they are accelerated from rest, which is crucial in determining their resulting speed before entering other regions of force.

Kinetic Energy and Electron Velocity

The kinetic energy acquired by an electron accelerated from rest by an electric field is directly related to the electric potential difference it experiences. The energy gained, expressed as qV, is converted into kinetic energy (½mv²) of the electron, allowing one to calculate the final speed of the electrons after acceleration. This conversion is key to understanding subsequent motion in the presence of magnetic fields.

Magnetic Force on Charged Particles

When a charged particle moves in a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field, known as the Lorentz force (F = q(v × B)). This force is responsible for changing the direction of the particle's motion without altering its kinetic energy, leading to deflected trajectories such as circular or helical paths.

Circular Motion in a Magnetic Field

A particle moving in a magnetic field under the influence of the Lorentz force undergoes circular motion if the magnetic field is uniform. The balance between the centripetal force required for circular motion and the magnetic force gives rise to the relationship r = mv/(qB), where r is the radius of the circular path. This concept is essential for determining the necessary magnetic field strength to achieve a specific curvature in the particle's trajectory.

Right-Hand Rule and Field Direction Determination

The direction of the magnetic force acting on a moving charge is determined using the right-hand rule, which involves pointing the fingers in the direction of the particle's velocity and curling them toward the magnetic field direction; the thumb then points in the direction of the force. For electrons, which have negative charge, the resultant magnetic force is opposite to that predicted by the right-hand rule for positive charges, a detail that is critical in correctly orienting magnetic fields in practical applications.

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