91Ó°ÊÓ

When a charged particle moves through a magnetic field, it experiences a force that affects its velocity components. The velocity vector of a particle is made up of three components: \( v_x \), \( v_y \), and \( v_z \). Each of these represents the speed in the respective directions of the coordinate axes.
In the context of a uniform magnetic field, as given in our exercise, the magnetic force equation is expressed as \( \overrightarrow{\boldsymbol{F}} = q(\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{B}}) \). This formula shows that the force depends on the cross product of velocity and the magnetic field vectors.

By solving the force equations step by step, we can find the values for specific components of the velocity. For instance:

• Using \( F_x = q(-v_y B_z) \), we determined \( v_x = -1.06 \times 10^2 \text{ m/s} \).
• From \( F_y = q(v_x (-B_z)) \), we found \( v_y = 4.86 \times 10^1 \text{ m/s} \).

The \( v_z \) component remains undetermined from force equations, indicating that movements along the \( z \)-axis do not influence the force experienced in this setup.

The vector cross product is a fundamental concept in calculating the magnetic force on a charged particle. It is expressed in the equation \( \overrightarrow{\boldsymbol{F}} = q(\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{B}}) \). This cross product between velocity \( \overrightarrow{\boldsymbol{v}} \) and magnetic field \( \overrightarrow{\boldsymbol{B}} \) is key to understanding why force is always perpendicular to the plane formed by these two vectors.
A cross product results in a third vector that is orthogonal to the original pair:

• The direction of this vector follows the right-hand rule, which helps determine the direction of the force vector \( \overrightarrow{\boldsymbol{F}} \).
• The magnitude is computed using the formula \( |\overrightarrow{\boldsymbol{v}}| |\overrightarrow{\boldsymbol{B}}| \sin(\theta) \), where \( \theta \) is the angle between the velocity and magnetic field vectors.

In our exercise, the force is calculated in such a way that proves the angle between the velocity and force is \( 90 \) degrees. This attribute of the cross product fundamentally establishes how the velocity vector components interact with the magnetic field.

A uniform magnetic field is one where the magnetic field lines are parallel and equally spaced. This implies that the strength and direction of the magnetic field remain constant across the space it's acting within.
In practical terms, a uniform magnetic field is represented mathematically as a constant vector, such as the one in our example, \( \overrightarrow{\boldsymbol{B}} = -1.25 \hat{\boldsymbol{k}} \). In this scenario, the field is directed along the negative \( z \)-axis and has a uniform magnitude of \( 1.25 \text{ T} \).

This uniformity greatly simplifies calculations as it means:

• No direction change in the force as the particle moves, making it easier to compute velocity components affected by the magnetic force.
• The effects of the magnetic field can be precisely calculated irrespective of the particle’s position in the field.

As a result, in a uniform magnetic field, the task of predicting the motion of charged particles becomes more manageable, allowing for a clear calculation of forces like in our exercise.