A positive charge q=3.20 ×10^-19 C moves with a velocity 𝐯=(2 𝐢+3 𝐣-𝐤) m / s through a region wh

step 1 Hello friends here it is given a positive charge Q 3 .2 into 10 to the power minus 19 coulum mov

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 $\mathbf{v}=(2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}) \mathrm{m} / \mathrm{s}$ through a region where both a uni- form magnetic field and a uniform electric field exist. (a) Calculate the total force on the moving charge (in unit-vector notation), taking $\mathbf{B}=(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\mathbf{k})$ T and $\mathbf{E}=(4 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}})$ V/m. (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 $\mathbf{v}=(2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}) \mathrm{m} / \mathrm{s}$ through a region where both a uni-
form magnetic field and a uniform electric field exist.
(a) Calculate the total force on the moving charge (in unit-vector notation), taking $\mathbf{B}=(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\mathbf{k})$ T and $\mathbf{E}=(4 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}})$ V/m. (b) What angle does the force
vector make with the positive $x$ axis?

Key Concepts

Lorentz Force

The Lorentz force is the total force experienced by 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 principle is fundamental in electromagnetism as it combines the contributions from both fields into one net force acting on the particle.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a third vector and is crucial for calculating the magnetic part of the Lorentz force, v × B. The direction of the resulting vector is determined by the right-hand rule, and its magnitude depends on the sine of the angle between the original vectors, making it essential for understanding how the magnetic force is oriented relative to the particle’s velocity and the magnetic field.

Vector Addition

Vector addition is the process of combining two or more vectors to obtain a resultant vector. In the context of the Lorentz force, the net force on a charged particle is found by adding the electric force vector, qE, and the magnetic force vector, q(v × B). This concept is key for determining the overall effect of multiple forces acting simultaneously on an object.

Unit Vector Notation

Unit vector notation is a standardized way to represent vectors in terms of their components along the coordinate axes, typically using i, j, and k for the x, y, and z axes respectively. This format provides clarity and precision when performing vector operations such as addition, subtraction, and cross products, which are essential for solving problems in electromagnetism and mechanics.

Angle Calculation

Calculating the angle between a vector and an axis involves using the dot product formula, which relates the cosine of the angle between two vectors to their components and magnitudes. This method is vital when determining the orientation of the resultant force relative to a specific reference direction, such as the positive x-axis, allowing one to understand the directional behavior of forces in a physical system.

Transcript

Please give Ace some feedback