Television Tube Each of the electrons in the beam of a...

Television Tube Each of the electrons in the beam of a television tube has a kinetic energy of $12.0 \mathrm{keV}$. The tube is oriented so that the electrons move horizontally from geomagnetic south to geomagnetic north. The vertical component of Earth's magnetic field points down and has a magnitude of $55.0 \mu \mathrm{T}$. (a) In what direction will the beam deflect? (b) What is the magnitude of the acceleration of a single electron due to the magnetic field? (c) How far will the beam deflect in moving $20.0 \mathrm{~cm}$ through the television tube?

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through a magnetic field. This force is given by the equation F = q(v × B), where q is the charge, v is the velocity of the particle, and B is the magnetic field vector. The force is always perpendicular to both the velocity and the magnetic field, which leads to circular or curved trajectories of the particle.

Kinetic Energy and Velocity

The kinetic energy of a particle is directly related to its velocity through the equation K = ½ m v², where m is the mass of the particle. This relationship allows us to determine the speed of a charged particle when its kinetic energy is known, which is essential for analyzing its motion in a magnetic field.

Centripetal Force and Circular Motion

When a charged particle moves perpendicularly to a uniform magnetic field, the magnetic force acts as the centripetal force that bends its path into a circular trajectory. The radius of this circular path can be found using the balance between the Lorentz force and the centripetal force, with the formula r = m v / (q B). This concept allows for the calculation of deflection distances over part of a circular path.

Right-Hand Rule for Charged Particles

The right-hand rule is a useful mnemonic for determining the direction of the force on a moving charge in a magnetic field. For a positive charge, pointing the fingers in the direction of velocity and curling them toward the magnetic field shows the direction of the force. For a negative charge like an electron, the force direction is opposite to that indicated by the right-hand rule, which is crucial for understanding the actual motion of the particle.

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