Problem 57 - University Physics with Modern Physics

Question

A negative point charge \(q=-7.20 \mathrm{mC}\) is moving in a reference frame. When the point charge is at the origin, the magnetic field it produces at the point \(x=25.0 \mathrm{cm}, y=0, z=0\) is \(\vec{B}=(6.00 \mu \mathrm{T}) \hat{j},\) and its speed is 800 \(\mathrm{km} / \mathrm{s}\) . (a) What are the \(x\) - \(y\) .and \(z\) -components of the velocity \(\vec{v}_{0}\) of the charge? (b) At this same instant, what is the magnitude of the magnetic field that the charge produces at the point \(x=0, y=25.0 \mathrm{cm}, z=0 ?\)

Step-by-Step Solution

Answer

The velocity components are in the x-z plane, with the y-component of the magnetic field being 6.00 μT at (25.0 cm, 0, 0). The magnetic field at (0, 25.0 cm, 0) depends on the velocity component perpendicular to the radial vector at that point.

Step 1: Understanding the Problem

We have a negative point charge, a magnetic field it produces, and its velocity. We need to find the components of the charge's velocity and the magnetic field at a specific point.

Step 2: Apply Biot-Savart Law

The Biot-Savart Law relates the magnetic field produced by a moving charge to its velocity and the distance from the charge. It's given by: \[ \vec{B} = \frac{\mu_0}{4\pi} \frac{q \vec{v} \times \hat{r}}{r^2} \] where \( \vec{v} \) is the velocity, \( \hat{r} \) is the unit vector from the charge to the point, and \( r \) is the distance.

Step 3: Find Velocity Direction

The problem tells us \( \vec{B} = (6.00 \mu \text{T}) \hat{j} \) at \((x, y, z) = (0.25, 0, 0)\). This suggests the cross-product \( \vec{v} \times \hat{r} \) gives a vector in the y-direction. Therefore, \( \vec{v} \) must be in the x-z plane.

Step 4: Write Cross Product Equation

By examining the cross-product \( \vec{v} \times \hat{r} \) in the given setup, we see that \( \vec{v} = v_x\hat{i} + v_z\hat{k} \) and \( \hat{r} = \hat{i} \), leading to \( \vec{v} \times \hat{r} = v_z\hat{j} \). Thus, \( 6.00 \mu \text{T} = \frac{\mu_0}{4\pi} \frac{-7.20 \cdot 10^{-3} \cdot v_z}{0.25^2} \).

Step 5: Solve for Velocity Components

Using the cross-product equation, set \( v_x^2 + v_z^2 = (800000)^2 \) (since the speed is 800 km/s) and solve for \( v_z \) using the magnetic field equation to find precise values of \( v_x \) and \( v_z \).

Step 6: Calculate the Magnetic Field at New Point

Repeat the previous analysis for the position \((0, 0.25, 0)\). Here, \( \hat{r} = \hat{j} \), calculate a new \( \vec{v} \times \hat{r} = -v_x\hat{k} \) and find that magnitude of the resulting \( \vec{B} \).

Step 7: Compute Magnitude from Perpendicular Velocity Component

Knowing \( \vec{v} \times \hat{r} = -v_x\hat{k} \) from Step 6, the magnitude of \( \vec{B} \) produced is: \[ |\vec{B}| = \frac{\mu_0}{4\pi} \frac{|q| \cdot |v_x|}{r^2} \]. Solve for \(|\vec{B}|\) using the known values and constants.

Key Concepts

Point Charge

A "point charge" is simply a model used in physics to represent a charged particle that has no size, it is just a tiny point with a certain amount of charge. This concept is essential in understanding electrostatics and electromagnetism. In our exercise, we are dealing with a negative point charge, which means it has an excess of electrons giving it a negative charge value (for example, as given as \(q=-7.20 \mathrm{mC}\) in the problem). Point charges are fundamental in physics problems because they simplify calculations by focusing on the effects of the charge rather than the physical dimensions of the particle.
The charge being negative indicates the direction of the electric and magnetic fields relative to the charge's motion, which is crucial when applying the Biot-Savart Law. This model allows us to easily calculate the influence of the charge at any location in space, making it a convenient way to analyze forces and fields in classical electromagnetism.

Magnetic Field

The "magnetic field" is a field produced in space around charged particles when they are in motion. It can be thought of as the way in which magnetic forces are transmitted. The Earth's own magnetic field is an example of such a field. In the problem presented, the magnetic field is produced by the moving point charge, and is given at a specific point as \(\vec{B}=(6.00 \mu \mathrm{T}) \hat{j}\).
When dealing with moving charges, the Biot-Savart Law is utilized to find the amount and direction of the magnetic field produced. The magnetic field itself is a vector field, which means it has both a magnitude and a direction. In this exercise, understanding the direction of the magnetic field is particularly important to solve the problem.
By using the given magnetic field vector, we can infer information about the components of the velocity of the point charge, and other properties, with the aid of formulas and vector analyses.

Velocity Components

"Velocity components" refer to the breaking down of the velocity vector into parts that represent movement in the x, y, and z directions. In our exercise, a key task is determining these components for the point charge. Given that the speed is constant (800 \(\mathrm{km/s}\)), we can denote the velocity vector as \(\vec{v} = v_x\hat{i} + v_z\hat{k}\), because the resulting magnetic field tells us that the velocity is in the x-z plane.
Breaking velocity into components helps us analyze physical situations, particularly when dealing with force and field calculations in varying directions. Here, we use the condition that the speed or magnitude of velocity is the square root of the sum of squares of these components, \(v_x^2 + v_z^2 = (800000)^2\). This gives us a system to solve for individual components after equating variables through the Biot-Savart Law.
Determining these components is vital for calculating magnetic fields at different points and understanding the behavior of charges in magnetic fields.

Cross-Product

A "cross-product" is a mathematical operation that takes two vectors and produces another vector perpendicular to the plane of the input vectors. In physics, it is often used to determine vectors such as torque and magnetic force. For vector \(\vec{A}\) and \(\vec{B}\), the cross-product is represented by \(\vec{A} \times \vec{B}\). In the context of the Biot-Savart Law used in this exercise, the cross-product helps find the direction and magnitude of magnetic fields generated by moving charges.
Specifically, the vector \(\vec{v} \times \hat{r}\) is crucial to find out which direction the magnetic field is oriented. In our problem, knowing \(\vec{B} = (6.00 \mu \text{T}) \hat{j}\) and the position vector \(\hat{r}\), we deduce that \(\vec{v} \times \hat{r} = v_z \hat{j}\), from which further analysis is derived.
The cross-product also confirms that the orientation of the vectors resulting from the motion of the charge generates specific directions for the magnetic field, guiding us in step-by-step solutions to understand complex vector systems involving electromagnetism.