91Ó°ÊÓ

The Lorentz force equation is an essential tool for understanding how charged particles move in electric and magnetic fields. Given by the formula \( \vec{F} = q(\vec{v} \times \vec{B}) \), where:

• \( \vec{F} \) is the force experienced by the charge.
• \( q \) is the charge of the particle.
• \( \vec{v} \) represents the particle's velocity.
• \( \vec{B} \) stands for the magnetic field.
• \( \times \) denotes the cross product.

This equation reveals that the force is perpendicular to both the velocity and the magnetic field, making it a perfect fit for the problem at hand. We know the charge, velocity, and force; our goal is to uncover the magnetic field components using this relationship.

Understanding the cross product is crucial for solving this type of problem. The cross product of two vectors, \( \vec{a} \times \vec{b} \), results in a third vector that is perpendicular to both \( \vec{a} \) and \( \vec{b} \). Its magnitude is calculated as:\[|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}| \sin(\theta)\]where \( \theta \) is the angle between the two vectors. In our problem, the velocity vector \( \vec{v} \) is along the \( \hat{J} \) direction. When crossed with the magnetic field \( \vec{B} \) (with an unknown direction), the cross product yields the components of force \( \vec{F} \) parallel to the \( \hat{I} \) and \( \hat{K} \) axes. Thus, we can connect these components back to the field strengths along those directions.

The magnetic force experienced by a charged particle is defined by how it splits along the coordinate axes when broken into components. When calculating these components using the Lorentz force, we find that:

• The force component \( F_x \) is due to the velocity and the magnetic field component \( B_z \).
• The force component \( F_z \) is influenced by the velocity and the magnetic field component \( B_x \).
• The component \( F_y \), associated with \( B_y \), does not appear as it results in no force (because the cross product with a parallel component is zero).

This step is a fantastic application of breaking complex interactions into understandable parts. By computing the field components, \( B_x = 0.176 \ \mathrm{T} \) and \( B_z = 0.257 \ \mathrm{T} \), we understand how each field component influences the magnetic force.

The angle between vectors is calculated using their scalar or dot product, which helps understand their geometric orientation. When two vectors are perpendicular, the scalar product is zero. Mathematically, the scalar product of two vectors \( \vec{A} \cdot \vec{B} \) is given by:\[\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)\]This formula aligns perfectly with our exercise, where the angle \( \theta \) between the magnetic field \( \vec{B} \) and force \( \vec{F} \) vectors leads to a zero scalar product. As a result, the angle between \( \vec{B} \) and \( \vec{F} \) is found to be \( 90^\circ \), indicating they are perpendicular. This perpendicular nature simplifies the process of calculation and aids in visualizing the alignment of vectors.