91Ó°ÊÓ

Understanding electron velocity is crucial when dealing with problems involving electric and magnetic fields. An electron, which is a charged particle, will move through space at a certain velocity, often measured in meters per second (m/s). In the context of our exercise, the electron moves with a velocity of \(1.20 \times 10^{4} \mathrm{m/s}\) in the positive x-direction. Knowing the direction of velocity helps us analyze how the electron will interact with electric and magnetic fields around it.

When an electron moves through a magnetic field, the direction and magnitude of its velocity play a significant role in determining the force it experiences due to the magnetic field. In this exercise, since the electron is not accelerating in the x-direction, it tells us that the velocity contributes no additional forces in that direction. Therefore, understanding the electron velocity aids in setting up the vector nature of the problem, which is fundamental in analyzing the coupled effects of electric and magnetic fields.

The magnetic field has unique characteristics that make it essential to understand when analyzing the motion of charged particles like electrons. A magnetic field affects moving charges by exerting a force perpendicular to both the velocity of the charge and the magnetic field direction. This force is given by the formula: \[ F = q(v \times B) \] where \( F \) is the magnetic force, \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field.

In our exercise, because the electron does not accelerate in the x-direction, we deduced that there cannot be a component of the magnetic field in the x-direction. Therefore, the force equation simplifies, implying a relationship between the magnetic and electric fields. The absence of acceleration in the y and z directions also indicates that the magnetic force cannot have components in these directions either if the resultant magnetic force needs to remain zero.

• This leads to the solution stating that the magnetic field \( B = 0 \).
• The electron’s lack of directional change underlines the electric field as the predominant force, nullifying the magnetic component based on the existing vectors.

Acceleration occurs when a force acts upon a mass, as described by Newton’s second law \( F = ma \). In an electric field, the force acting on a charged particle like an electron is defined by \( F = Eq \), where \( E \) is the electric field and \( q \) is the charge of the particle.

In the scenario provided by the exercise, the electron is experiencing an acceleration of \(2.00 \times 10^{12} \mathrm{m/s^2}\) in the positive z-direction. This acceleration is entirely due to the electric field present in that direction, which has a magnitude of \(20.0 \mathrm{N/C}\). By breaking down this information, we use the equation:

\[ a = \frac{F}{m} = \frac{Eq}{m} \] We observe that this relationship confirms that the acceleration is due to the electric field alone as there is no magnetic field affecting it in this case (\( B = 0 \)).

• This allows us to understand that the charged particle is solely influenced by the electric field, which provides the force necessary for the observed acceleration in the z-direction.

This isolated behavior emphasizes the dominant role of the electric field when magnetic influence is null or orthogonal to the direction of motion.