91Ó°ÊÓ

When electrons or any charges move through a magnetic field, they experience a magnetic force. This force is described by the formula \( \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \), where \( \mathbf{F} \) is the force, \( q \) is the charge, \( \mathbf{v} \) is the velocity vector, and \( \mathbf{B} \) is the magnetic field vector. In our case, the charge \( q \) is that of an electron, which is \(-1.6 \times 10^{-19} \, \text{C}\).
As electrons move from west to east under a downward magnetic field (the Earth's), the force acts perpendicular to both the velocity of the electrons and the magnetic field. This is due to the cross-product in the formula. The right-hand rule usually helps determine the direction, but remember: electrons are negative, so the force direction is the opposite. In this scenario, the electrons are deflected toward the north.
The perpendicular force doesn't work simply because it pushes north; it's due to how magnetic fields influence moving charges at right angles, resulting in circular or helical paths if left unobstructed. With only one component involved (vertical field), only north-south deflection happens.

Electrons in the beam have kinetic energy, which is the energy associated with their motion. This is expressed as \( KE = \frac{1}{2}mv^2 \), where \( KE \) is kinetic energy, \( m \) is mass, and \( v \) is velocity. For electrons in the television tube, kinetic energy is given as \( 2.40 \times 10^{-15} \, \text{J} \).
Knowing the kinetic energy and mass of an electron, we can solve for the velocity \( v \) of electrons. Plug in the electron mass \( 9.11 \times 10^{-31} \, \text{kg} \) to find the velocity:
\[ v = \sqrt{\frac{2 \times 2.40 \times 10^{-15}}{9.11 \times 10^{-31}}} \approx 7.28 \times 10^7 \, \text{m/s} \]
This velocity is crucial because it's part of the magnetic force equation. As velocity increases, so does the force experienced by the electrons in a magnetic field, meaning speed impacts deflection significantly.

To find the acceleration that electrons undergo due to the deflection by a magnetic field, we use the fundamental relation seen in Newton's second law \( F = ma \). Given the earlier calculated magnetic force \( 2.33 \times 10^{-16} \, \text{N} \), acceleration \( a \) is derived by dividing this force by the electron's mass.
Substituting the known values yields:
\[ a = \frac{2.33 \times 10^{-16}}{9.11 \times 10^{-31}} \approx 2.56 \times 10^{14} \, \text{m/s}^2 \]
Such enormous acceleration happens because electrons have very little mass, and even a small force can lead to a significant change in their velocity. This demonstrates how sensitive lighter particles are to magnetic fields, an important factor in designing devices like cathode ray tubes and understanding particle behavior in physics.