Problem 62 - University Physics with Modern Physics

Question

A particle with charge \(q\) is moving with speed \(v\) in the \(- y\) -direction. It is moving in a uniform magnetic field \(\vec { B } =\) \(B _ { x } \hat { i } + B _ { y } \hat { J } + B _ { z } \hat { k } .\) (a) What are the components of the force \(\vec { \boldsymbol { F } }\) exerted on the particle by the magnetic field? (b) If \(q > 0 ,\) what must the signs of the components of \(\vec { B }\) be if the components of \(\vec { \boldsymbol { F } }\) are all nonnegative? (c) If \(q < 0\) and \(B _ { x } = B _ { y } = B _ { z } > 0 ,\) find the direction of \(\vec { \boldsymbol { F } }\) and find the magnitude of \(\vec { \boldsymbol { F } }\) in terms of \(| q | , v ,\) and \(B _ { x }\)

Step-by-Step Solution

Answer

(a) \( F_x = qvB_z \), \( F_y = 0 \), \( F_z = qvB_x \). (b) \( B_x, B_z > 0 \). (c) Direction is negative \( xz \)-diagonal; magnitude is \(|q|vB_x\sqrt{2}\).

Step 1: Determine Force Components Using Lorentz Force Law

The Lorentz force exerted on a charged particle moving in a magnetic field is given by \( \vec{F} = q(\vec{v} \times \vec{B}) \). The particle, with charge \( q \), moves with speed \( v \) in the \(-y\)-direction, so its velocity vector is \( \vec{v} = -v \hat{j} \). The magnetic field vector is \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \). To find the force \( \vec{F} \), compute the cross product \( \vec{v} \times \vec{B} \):\[ \vec{v} \times \vec{B} = (-v \hat{j}) \times (B_x \hat{i} + B_y \hat{j} + B_z \hat{k}) \].Calculate each component:- \((-v \hat{j}) \times (B_x \hat{i}) = vB_x \hat{k}\),- \((-v \hat{j}) \times (B_y \hat{j}) = 0\),- \((-v \hat{j}) \times (B_z \hat{k}) = vB_z \hat{i}\).Thus, \( \vec{v} \times \vec{B} = vB_z \hat{i} + vB_x \hat{k} \), and the force is:\[ \vec{F} = q(vB_z \hat{i} + vB_x \hat{k}) = qvB_z \hat{i} + qvB_x \hat{k} \].Therefore, the force components are \( F_x = qvB_z \), \( F_y = 0 \), and \( F_z = qvB_x \).

Step 2: Determine Signs of Magnetic Field Components for Positive Force (q>0)

Given that \( q > 0 \), all components of the force \( \vec{F} \) need to be nonnegative. The force components derived are \( F_x = qvB_z \) and \( F_z = qvB_x \). Since \( q > 0 \) and \( v > 0 \), to ensure \( F_x \geq 0 \) and \( F_z \geq 0 \), both \( B_x \) and \( B_z \) must be positive. Thus, the required signs are:- \( B_x > 0 \)- \( B_z > 0 \)Since \( F_y = 0 \), \( B_y \) does not affect the conditions.

Step 3: Determine Direction and Magnitude of Force for q 0

Assuming \( q < 0 \) and \( B_x = B_y = B_z > 0 \), the force components are \( F_x = qvB_z \) and \( F_z = qvB_x \). Since both components depend on the direction of charge, we get:- \( F_x = - |q| v B_x \)- \( F_z = - |q| v B_x \)The force vector \( \vec{F} = - |q| v B_x (\hat{i} + \hat{k}) \). The force direction is along the vector that makes equal angles with \( \hat{i} \) and \( \hat{k} \). This is a diagonal in the \( xz \)-plane, pointing in the negative \( x \text{-} z \) direction. The magnitude is given by:\[ |\vec{F}| = \sqrt{ (-|q| v B_x)^2 + 0^2 + (-|q| v B_x)^2 } = |q|vB_x \sqrt{2} \].

Key Concepts

Charged Particle

A charged particle, such as an electron, proton, or ion, carries an electric charge denoted by \( q \). This charge can be positive or negative, influencing how the particle interacts with electric and magnetic fields. A charged particle moves through fields, feeling forces that depend on both its charge and its motion.

• When \( q > 0 \), the particle is positively charged (like a proton).
• When \( q < 0 \), it's negatively charged (such as an electron).

Understanding the charge is vital because it dictates the direction of forces exerted on the particle. In the context of the Lorentz force, the sign of \( q \) determines whether the force direction aligns or opposes the expected vectors derived from the cross product operation with the magnetic field.

Magnetic Field

A magnetic field is a vector field represented as \( \vec{B} \), influencing the motion of charged particles. It is described by three components: \( B_x \), \( B_y \), and \( B_z \). These components correspond to the field's strength and direction in the \( x \), \( y \), and \( z \) axes, respectively.
The magnetic field impacts charged particles through a force that does not perform work but changes their trajectory.

• Uniform magnetic field: The field is consistent throughout a region, meaning \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \).
• The field direction is crucial for determining the components of the force via vector cross products.

In this exercise, understanding magnetic field components helps determine how each axis influences the particle's path via the Lorentz force.

Vector Cross Product

The vector cross product is an operation on two vectors, resulting in a third vector perpendicular to the plane containing the initial vectors. In the context of the Lorentz force, the cross product is used to find the magnetic part of the force on a charged particle.
For a particle with velocity \( \vec{v} = -v \hat{j} \) moving in a magnetic field \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \), the cross product \( \vec{v} \times \vec{B} \) is calculated as:

• \((-v \hat{j}) \times (B_x \hat{i}) = vB_x \hat{k}\)
• \((-v \hat{j}) \times (B_y \hat{j}) = 0\)
• \((-v \hat{j}) \times (B_z \hat{k}) = vB_z \hat{i}\)

This results in a vector \( \vec{v} \times \vec{B} = vB_z \hat{i} + vB_x \hat{k} \), indicating that the force exerted on the particle by the magnetic field is perpendicular to its velocity.

Force Components

Force components describe the individual effects of a force along the different axes of a coordinate system. Utilizing the Lorentz force formula, \( \vec{F} = q(\vec{v} \times \vec{B}) \), we find how the product of charge and velocity interacts with a magnetic field to produce force.
The exercise showcases the decomposition of the force into parts:

• \( F_x = qvB_z \): Component along the \( x \)-axis.
• \( F_y = 0 \): No force along the \( y \)-axis as derived from the cross product result.
• \( F_z = qvB_x \): Component along the \( z \)-axis.

If \( q > 0 \), the components are directly proportional to the same axis components of the magnetic field. If \( q < 0 \), the components are inverted, indicating a reverse direction for the force. Analyzing these helps in predicting the trajectory and behavior of charged particles in various field configurations.