A positive charge q=3.20 ×10^-19 C moves with a velocity...

A positive charge $q=3.20 \times 10^{-19} \mathrm{C}$ moves with a velocity $\mathbf{v}=(2 \mathbf{i}+3 \mathbf{j}-1 \mathbf{k}) \mathrm{m} / \mathrm{s}$ through a region where both a uniform magnetic field and a uniform electric field exist. (a) What is the total force on the moving charge (in unit-vector notation) if $\mathbf{B}=$ $(2 \mathbf{i}+4 \mathbf{j}+1 \mathbf{k}) \mathbf{T}$ and $\mathbf{E}=(4 \mathbf{i}-1 \mathbf{j}-2 \mathbf{k}) \mathrm{V} / \mathrm{m}^{2}$ (b) What angle does the force vector make with the positive $x$ axis?

A positive charge $q=3.20 \times 10^{-19} \mathrm{C}$ moves with a velocity $\mathbf{v}=(2 \mathbf{i}+3 \mathbf{j}-1 \mathbf{k}) \mathrm{m} / \mathrm{s}$ through a region
where both a uniform magnetic field and a uniform electric field exist. (a) What is the total force on the moving charge (in unit-vector notation) if $\mathbf{B}=$ $(2 \mathbf{i}+4 \mathbf{j}+1 \mathbf{k}) \mathbf{T}$ and $\mathbf{E}=(4 \mathbf{i}-1 \mathbf{j}-2 \mathbf{k}) \mathrm{V} / \mathrm{m}^{2}$
(b) What angle does the force vector make with the positive $x$ axis?

Key Concepts

Lorentz Force

The Lorentz force law describes the total force acting on a charged particle moving in the presence of electric and magnetic fields. It is given by the equation F = q(E + v × B), where q is the charge, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This concept is foundational in electromagnetism as it combines both the electric force (which acts in the direction of the field) and the magnetic force (which acts perpendicular to both the velocity and the magnetic field).

Vector Cross Product

The cross product is a mathematical operation used to determine a vector that is perpendicular to two given vectors. In the context of the Lorentz force, it is used to calculate the magnetic force component (v × B), which points in a direction perpendicular to both the velocity of the particle and the magnetic field vector. Understanding the properties of the cross product is crucial when evaluating the directional effects of magnetic forces on moving charges.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. In this scenario, the electric force (qE) and the magnetic force (q(v × B)) are added together component-wise to obtain the total force acting on the charge. This concept is important for accurately determining both the magnitude and the direction of the resultant force in electromagnetism problems.

Angle Calculation

Determining the angle that a vector makes with a particular axis involves using inverse trigonometric functions, typically the arctan function. This process requires the identification of relevant vector components, usually in a two-dimensional plane if the problem is simplified, and then computing the angle relative to the chosen axis. Correctly computing this angle is essential for understanding the directional aspects of forces in vector analysis.

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