Problem 44 - Fundamentals of Physics

Question

In the product \(\vec{F}=q \vec{v} \times \vec{B}\), take \(q=2\), $$\vec{v}=2.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}} \text { and } \vec{F}=4.0 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}$$ What then is \(\vec{B}\) in unit-vector notation if \(B_{x}=B_{y} ?\)

Step-by-Step Solution

Answer

\(\vec{B} = -3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\)

Step 1: Understanding the Problem

We have the equation \( \vec{F} = q \vec{v} \times \vec{B} \), where \( q = 2 \), \( \vec{v} = 2.0\hat{\mathrm{i}} + 4.0\hat{\mathrm{j}} + 6.0\hat{\mathrm{k}} \), and \( \vec{F} = 4.0\hat{\mathrm{i}} - 20\hat{\mathrm{j}} + 12\hat{\mathrm{k}} \). We need to find \( \vec{B} \) given that \( B_{x} = B_{y} \).

Step 2: Express Cross Product Components

The cross product \( \vec{v} \times \vec{B} \) can be expressed by the determinant: \[ \begin{vmatrix} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \ 2 & 4 & 6 \ B_{x} & B_{y} & B_{z} \end{vmatrix} \] Expanding this determinant, we get:\[ \vec{v} \times \vec{B} = (4B_{z} - 6B_{y})\hat{\mathrm{i}} - (2B_{z} - 6B_{x})\hat{\mathrm{j}} + (2B_{y} - 4B_{x})\hat{\mathrm{k}} \]

Step 3: Substitute into Force Equation

Substitute the expanded cross product into the force equation: \[ \vec{F} = 2((4B_{z} - 6B_{y})\hat{\mathrm{i}} - (2B_{z} - 6B_{x})\hat{\mathrm{j}} + (2B_{y} - 4B_{x})\hat{\mathrm{k}}) \] Simplifying, we have:\[ \vec{F} = (8B_{z} - 12B_{y})\hat{\mathrm{i}} - (4B_{z} - 12B_{x})\hat{\mathrm{j}} + (4B_{y} - 8B_{x})\hat{\mathrm{k}} \]

Step 4: Equate Components with Given Force

Since \( \vec{F} = 4.0\hat{\mathrm{i}} - 20\hat{\mathrm{j}} + 12\hat{\mathrm{k}} \), equate the components:1. \( 8B_{z} - 12B_{y} = 4 \)2. \( 4B_{z} - 12B_{x} = 20 \)3. \( 4B_{y} - 8B_{x} = 12 \)

Step 5: Simplify Equations with Condition \( B_x = B_y \)

Given \( B_x = B_y \), substitute into the equations:- Equation 1 simplifies to \( 8B_{z} - 12B_{x} = 4 \) - Equation 2 becomes \( 4B_{z} - 12B_{x} = 20 \)- Equation 3 changes to \( 4B_{x} - 8B_{x} = 12 \rightarrow -4B_x = 12 \rightarrow B_x = -3 \) Therefore, \( B_y = B_x = -3 \).

Step 6: Solve for \( B_z \)

Substitute \( B_x = -3 \) and \( B_y = -3 \) into Equation 1: \( 8B_{z} - 12(-3) = 4 \rightarrow 8B_{z} + 36 = 4 \rightarrow 8B_{z} = -32 \rightarrow B_{z} = -4 \).

Step 7: Final Unit-Vector Notation of \( \vec{B} \)

Combine all components: \( \vec{B} = -3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}} \).

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that focuses on vector fields and differentiation and integration of vectors. It is widely used in physics to describe and analyze physical phenomena that have both magnitude and direction.
In physics, **vector quantities** can represent various physical entities such as velocity, force, and acceleration. The calculation involving vector calculus enables us to understand and solve problems related to fields like electromagnetism and fluid dynamics.

• **Vectors** are represented as arrows with direction and magnitude.
• The **calculus** part involves operations like differentiating and integrating these vectors.

Understanding vector calculus is crucial as it helps in performing complex calculations related to force, velocity, and other vector quantities in physics.

Force Equation

The force equation in physics describes how various aspects like mass and acceleration interact to influence the motion of an object. In the provided problem, the force equation is given as \(\vec{F} = q \vec{v} \times \vec{B}\), where:

• \(\vec{F}\) is the force vector.
• \(q\) symbolizes the charge of a particle.
• \(\vec{v}\) is the velocity vector of the particle.
• \(\vec{B}\) represents the magnetic field vector.

This specific equation describes the force experienced by a charged particle moving through a magnetic field, known as the Lorentz force. The concept underscores how magnetic fields interact with moving charges, producing a force. It is essential to understand the force equation to predict an object's movement when it is under the influence of various forces.

Cross Product

The cross product is a vector operation used in physics to determine a vector that is perpendicular to two given vectors. Mathematically, given vectors \(\vec{A}\) and \(\vec{B}\), the cross product \(\vec{A} \times \vec{B}\) is calculated using determinants.
For example, in the given problem, the cross product \( \vec{v} \times \vec{B} \) is expressed by the determinant:
\[ \begin{vmatrix} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \ 2 & 4 & 6 \ B_{x} & B_{y} & B_{z} \end{vmatrix} \]
This method gives us a new vector that is perpendicular to both \(\vec{v}\) and \(\vec{B}\). The cross product is specifically useful in physics for determining torques and the magnetic forces on charged particles. Understanding how to compute a cross product is instrumental for solving problems involving multidimensional vectors.

Magnetic Fields

Magnetic fields are a vector field that exerts forces on moving charges, as expressed by the force equation mentioned earlier. They have a direction and magnitude that can influence the behavior of charged particles. In the exercise, vector \(\vec{B}\) represents the magnetic field.
The magnetic force can alter the trajectory of moving charged particles, and is utilized in applications such as electric motors and generators. Key points to understand about magnetic fields include:

• They are produced by moving electric charges or intrinsic magnetic moments of elementary particles.
• Their direction is defined from the north to the south pole of a magnet.
• Magnetic fields can be visually represented using field lines that show the direction and strength of the field.

The understanding of magnetic fields and their interaction with charges is essential for comprehending various technologies and natural phenomena that leverage magnetism.