Problem 66 - University Physics with Modern Physics

Question

A particle of charge \(q > 0\) is moving at speed \(v\) in the \(+ z\) -direction through a region of uniform magnetic field \(\vec { \boldsymbol { B } }\) . The magnetic force on the particle is \(\vec { \boldsymbol { F } } = F _ { 0 } ( 3 \hat { \boldsymbol { \imath } } + 4 \hat { \boldsymbol { J } } ) ,\) where \(F _ { 0 }\) is a positive constant. (a) Determine the components \(B _ { x } , B _ { y } ,\) and \(B _ { z }\) , or at least as many of the three components as is possible from the information given. (b) If it is given in addition that the magnetic field has magnitude \(6 F _ { 0 } / q v ,\) determine as much as you can about the remaining components of \(\vec { B } .\)

Step-by-Step Solution

Answer

\(B_x = -\frac{4F_0}{qv}\), \(B_y = \frac{3F_0}{qv}\), \(B_z = \pm \frac{\sqrt{11}F_0}{qv}\)."

Step 1: Understand the Magnetic Force Equation

The magnetic force on a moving charge is given by the equation \( \vec{F} = q (\vec{v} \times \vec{B}) \), where \( \vec{v} \) is the velocity vector in the \( +z \)-direction and \( \vec{B} \) represents the magnetic field vector. Since the particle is moving in the \( z \)-direction, we write: \( \vec{v} = v \hat{k} \). The given force is \( \vec { oldsymbol { F } } = F _ { 0 } ( 3 \hat { oldsymbol { \i } } + 4 \hat { oldsymbol { J } } ) \).

Step 2: Apply the Cross Product

Calculate \( \vec{v} \times \vec{B} \). With \( \vec{v} = v \hat{k} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \), the cross product is: \( \vec{v} \times \vec{B} = v (B_x \hat{k} \times \hat{i} + B_y \hat{k} \times \hat{j} + B_z \hat{k} \times \hat{k}) \). Using the right-hand rule, \( \hat{k} \times \hat{i} = -\hat{j} \), \( \hat{k} \times \hat{j} = \hat{i} \), and \( \hat{k} \times \hat{k} = \vec{0} \). Thus, \( \vec{v} \times \vec{B} = v (-B_x \hat{j} + B_y \hat{i}) \).

Step 3: Equate to the Given Force

Equating the calculated cross product to the given force, we have: \( qv(-B_x \hat{j} + B_y \hat{i}) = F_0 (3 \hat{i} + 4 \hat{j}) \). Comparing components, we get \(-qvB_x = 4F_0 \) and \( qvB_y = 3F_0 \).

Step 4: Solve for \(B_x \) and \(B_y \)

From the equations derived:1. \( B_x = -\frac{4F_0}{qv} \)2. \( B_y = \frac{3F_0}{qv} \)

Step 5: Analyzing the Magnitude of \( \vec{B} \)

Given the magnitude of \( \vec{B} = \frac{6F_0}{qv} \), we use the formula for the magnitude of a vector: \( |\vec{B}| = \sqrt{B_x^2 + B_y^2 + B_z^2} \). Substitute the known components:\[\sqrt{\left(-\frac{4F_0}{qv}\right)^2 + \left(\frac{3F_0}{qv}\right)^2 + B_z^2} = \frac{6F_0}{qv}.\]

Step 6: Solve for \( B_z \)

Squaring both sides of the magnitude equation: \[\left(\frac{4F_0}{qv}\right)^2 + \left(\frac{3F_0}{qv}\right)^2 + B_z^2 = \left(\frac{6F_0}{qv}\right)^2.\] Calculate and solve: \( \frac{16F_0^2}{q^2v^2} + \frac{9F_0^2}{q^2v^2} + B_z^2 = \frac{36F_0^2}{q^2v^2} \) \( B_z^2 = \frac{36F_0^2}{q^2v^2} - \frac{25F_0^2}{q^2v^2} \) \( B_z^2 = \frac{11F_0^2}{q^2v^2} \) \( B_z = \pm \frac{\sqrt{11}F_0}{qv} \)

Key Concepts

Particle Charge

In the world of electromagnetism, the concept of a particle charge is crucial. A particle, such as an electron or proton, carries a charge, often denoted as \( q \). The charge can be either positive or negative, determining how the particle interacts with electric and magnetic fields. In the exercise, we consider a particle with a positive charge, i.e., \( q > 0 \).

Understanding the charge of a particle helps in predicting the direction and magnitude of forces that act on it when subjected to magnetic or electric fields.
- Positive charges are repelled by other positive charges and attracted to negative ones.- The magnetic force on a charged particle also depends on its velocity, as it experiences a force perpendicular to both its motion and the magnetic field lines.

The relationship between the charge, velocity, and magnetic field is vital in producing a magnetic force that follows the equation: \[ \vec{F} = q(\vec{v} \times \vec{B}) \]This equation underlines how the charge interacts with the magnetic field through the velocity vector, making understanding particle charge foundational in electromagnetism.

Magnetic Field Components

Magnetic fields can be broken down into components along the coordinate axes: \( B_x \), \( B_y \), and \( B_z \). These components describe the magnetic field's strength and direction in space.

In the current scenario, we have a magnetic field expressed in the format: \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \)The task is to find these components using the information given.

When the particle moves solely in the \(+z\) direction (with velocity \(v\hat{k}\)), and a magnetic force expressed as \( \vec{F} = F_0(3 \hat{i} + 4 \hat{j}) \) acts on it:

• The component \(B_x\) impacts the force in the \(\hat{j}\) direction.
• Similarly, \(B_y\) affects the force in the \(\hat{i}\) direction.
• Interestingly, \(B_z\) does not contribute since the cross product with the same direction \(\hat{k}\) results in zero.

Understanding magnetic field components is key to determining how fields interact with charged particles and can help explain phenomena like the Lorentz force, crucial in many technological applications ranging from MRI machines to particle accelerators.

Cross Product Calculation

One of the cornerstones in dealing with forces in electromagnetic fields is the cross product, which involves the interaction between vectors. Here, the force that a magnetic field exerts on a moving charged particle is dictated by this operation.

The cross product \( \vec{v} \times \vec{B} \), involves the velocity vector \( \vec{v} = v \hat{k} \) and the magnetic field vector \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \). Let's break this down step-by-step:

- The term \( \hat{k} \times \hat{i} \) simplifies to \(-\hat{j} \).- The term \( \hat{k} \times \hat{j} \) results in \(\hat{i}\).- Finally, \( \hat{k} \times \hat{k} \) yields \(\vec{0}\).
Thus, the cross product is:\[ \vec{v} \times \vec{B} = v(-B_x \hat{j} + B_y \hat{i}) \]Evaluating this product allows us to solve for the unknown components of the magnetic field, as by equating it to the given magnetic force \( \vec{F} = F_0(3 \hat{i} + 4 \hat{j}) \), we derive the necessary expressions to find \(B_x\) and \(B_y\).

The cross product ultimately gives a vector perpendicular to the plane formed by the interacting vectors, providing insight into the direction and magnitude of forces applied during the interaction. This concept is broadly applicable in physics and engineering, making it an essential tool in understanding complex systems.