91Ó°ÊÓ

In physics, the cross product is a mathematical operation that helps us find a vector that is perpendicular to two other vectors. This is particularly useful when dealing with forces in electromagnetic fields. The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) is denoted as \( \vec{A} \times \vec{B} \). It is important to note that the result of a cross product is always a vector.

To perform a cross product, consider two vectors in three-dimensional space represented as \( \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \). The cross product is given by:

• \( \vec{A} \times \vec{B} = (A_y B_z - A_z B_y) \hat{i} + (A_z B_x - A_x B_z) \hat{j} + (A_x B_y - A_y B_x) \hat{k} \)

This formula calculates a new vector that is orthogonal (perpendicular) to both \( \vec{A} \) and \( \vec{B} \).

In the context of this physics problem, the velocity vector \( \vec{v} \) and the magnetic field vector \( \vec{B} \) are considered. By taking their cross product, we can find the direction of the magnetic force acting on the particle.

When a charged particle moves through electric \( \vec{E} \) and magnetic \( \vec{B} \) fields, it experiences forces that can change its path. For the particle in question to remain undeflected, the net force acting on it must be zero.

The forces involved are the electric force \( \vec{F}_e = q \vec{E} \) and the magnetic force \( \vec{F}_m = q(\vec{v} \times \vec{B}) \), where \( q \) is the charge of the particle. For the condition of no deflection, these forces must counterbalance each other:

• \( \vec{F}_e + \vec{F}_m = 0 \)

This implies that the electric field \( \vec{E} \) must be such that \( \vec{E} = \vec{v} \times \vec{B} \), thereby perfectly negating the effects of the magnetic force on the particle's motion.

Understanding this principle is key for applications involving electromagnetic fields, such as in particle accelerators and other scientific equipment that manipulate charged particle paths.

Electric and magnetic forces are fundamental in understanding how charged particles interact with fields.

• Electric Force: This force depends on the electric field \( \vec{E} \) and is given by \( \vec{F}_e = q \vec{E} \). It acts in the direction of the electric field for a positive charge and opposite for a negative charge.
• Magnetic Force: This is influenced by the velocity of the particle \( \vec{v} \) and the magnetic field \( \vec{B} \). The force is expressed as \( \vec{F}_m = q( \vec{v} \times \vec{B}) \) and acts perpendicular to both the velocity and magnetic field vectors.

The particle's charge also determines the direction of these forces, as illustrated in the exercise. For \( +0.640 \ ext{nC} \), the electric force points in the same direction as the cross product result \( \hat{i} \), while for \( -0.320 \ ext{nC} \), it points in the opposite direction. These principles highlight the delicate balance and interactions between electric and magnetic components when managing particle trajectories.