To become familiar with the order of magnitude of magnetic...

To become familiar with the order of magnitude of magnetic effects, consider the situation in a television cathode ray tube (not a flat-panel TV). In order to bend the electron trajectory so that the electron hits the top of the screen rather than going straight through to the center of the screen, you need a radius of curvature in the magnetic field of about $20 \mathrm{cm}$ (Figure 20.95 ). If the electrons are accelerated through a 15,000 $\mathrm{V}$ potential difference, they have a speed of $0.7 \times 10^{8} \mathrm{m} / \mathrm{s}$ Calculate the magnitude of the magnetic field required to make the electrons hit the top of the screen. Is the magnetic field into or out of the page?

Key Concepts

Magnetic Field Direction and Charge Sign

The direction of the force on a charged particle in a magnetic field is determined by the right-hand rule. However, for negatively charged particles such as electrons, the force direction is opposite to that predicted by the right-hand rule used for positive charges. This concept is essential for understanding how the magnetic field influences the trajectory of charged particles.

Energy Conversion from Potential to Kinetic Energy

Charged particles accelerated through a potential difference gain kinetic energy, as described by the relation eV = (1/2)mv^2, where V is the voltage and e is the charge of the particle. This conversion from electrical potential energy to kinetic energy is fundamental in determining the speed of the particle, which is a key step in analyzing its subsequent motion in a magnetic field.

Centripetal Force in Circular Motion

When a charged particle moves in a circular path under a magnetic force, the magnetic force acts as the centripetal force, which is necessary to sustain circular motion. The centripetal force is expressed as F = m(v^2/r), with m representing the mass, v the speed of the particle, and r the radius of curvature. Equating the magnetic force and centripetal force allows one to solve for unknown quantities such as the magnetic field strength.

Lorentz Force

The Lorentz force describes the force experienced by a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This force causes the particle to move in a curved path if the velocity is not parallel to the magnetic field direction.

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