Problem 5 - University Physics with Modern Physics

Question

A -4.80-\(\mu\)C charge is moving at a constant speed of 6.80 \(\times\) 10\(^5\) m/s in the +\(x\) direction relative to a reference frame. At the instant when the point charge is at the origin, what is the magnetic-field vector it produces at the following points: (a) \(x =\) 0.500 m, \(y =\) 0, \(z =\) 0; (b) \(x =\) 0, \(y =\) 0.500 m, \(z =\) 0; (c) \(x =\) 0.500 m, \(y =\) 0.500 m, \(z =\) 0; (d) \(x =\) 0, \(y =\) 0, \(z =\) 0.500 m?

Step-by-Step Solution

Answer

(a) Zero; (b) \(-1.31 \times 10^{-6} \; \hat{k} \; \text{T}\); (c) \(-6.57 \times 10^{-7} \; \hat{k} \; \text{T}\); (d) \(1.31 \times 10^{-6} \; \hat{j} \; \text{T}\).

Step 1: Understand the Biot-Savart Law

The Biot-Savart law describes the magnetic field produced by a moving charge. For a point charge, the magnetic field at a point in space is given by:\[\mathbf{B} = \frac{\mu_0}{4\pi} \frac{q \mathbf{v} \times \mathbf{r}}{r^3}\]where \(\mathbf{B}\) is the magnetic field, \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7} \; \text{T}\cdot \text{m/A}\)), \(q\) is the charge, \(\mathbf{v}\) is the velocity vector of the charge, and \(\mathbf{r}\) is the position vector from the charge to the point of interest, with magnitude \(r\).

Step 2: Set up the Problem

For each point, determine the position vector \(\mathbf{r}\) and its magnitude \(r\) from the charge to the point. The charge is at the origin moving in the positive \(x\)-direction, so \(\mathbf{v} = 6.80 \times 10^5 \; \hat{i} \; \text{m/s}\). Calculate the cross product \(\mathbf{v} \times \mathbf{r}\) for each point.

Step 3: Calculate for Point (a)

For point (a) \((x=0.500 \; \text{m}, y=0, z=0)\), the position vector is \(\mathbf{r} = 0.500\; \hat{i}\). Thus, the cross product \(\mathbf{v} \times \mathbf{r} = (6.80 \times 10^5 \; \hat{i}) \times (0.500 \; \hat{i}) = \mathbf{0}\). The magnetic field \(\mathbf{B}\) is zero because the velocity and position vectors are parallel.

Step 4: Calculate for Point (b)

For point (b) \((x=0, y=0.500 \; \text{m}, z=0)\), \(\mathbf{r} = 0.500 \; \hat{j}\). Here, \(\mathbf{v} \times \mathbf{r} = (6.80 \times 10^5 \; \hat{i}) \times (0.500 \; \hat{j}) = 3.40 \times 10^5 \; \hat{k}\).The magnitude of \(r\) is 0.500 m: \[\mathbf{B} = \frac{(4\pi \times 10^{-7}) \times (-4.80 \times 10^{-6}) \times (3.40 \times 10^5) \; \hat{k}}{(0.500)^3} \approx -1.31 \times 10^{-6} \; \hat{k} \; \text{T}.\]

Step 5: Calculate for Point (c)

For point (c) \((x=0.500 \; \text{m}, y=0.500 \; \text{m}, z=0)\), \(\mathbf{r} = 0.500 \; \hat{i} + 0.500 \; \hat{j}\). The cross product \(\mathbf{v} \times \mathbf{r} = (6.80 \times 10^5 \; \hat{i}) \times (0.500 \; \hat{i} + 0.500 \; \hat{j}) = 3.40 \times 10^5 \; \hat{k}\).The magnitude of \(r\) is \(\sqrt{(0.500)^2 + (0.500)^2} = 0.707 \; \text{m}\):\[\mathbf{B} = \frac{(4\pi \times 10^{-7}) \times (-4.80 \times 10^{-6}) \times (3.40 \times 10^5) \; \hat{k}}{(0.707)^3} \approx -6.57 \times 10^{-7} \; \hat{k} \; \text{T}.\]

Step 6: Calculate for Point (d)

For point (d) \((x=0, y=0, z=0.500 \; \text{m})\), \(\mathbf{r} = 0.500 \; \hat{k}\). Here, \(\mathbf{v} \times \mathbf{r} = (6.80 \times 10^5 \; \hat{i}) \times (0.500 \; \hat{k}) = -3.40 \times 10^5 \; \hat{j}\).The magnitude of \(r\) is 0.500 m:\[\mathbf{B} = \frac{(4\pi \times 10^{-7}) \times (-4.80 \times 10^{-6}) \times (-3.40 \times 10^5) \; \hat{j}}{(0.500)^3} \approx 1.31 \times 10^{-6} \; \hat{j} \; \text{T}.\]

Step 7: Summarize the Results

At (a) the magnetic field is zero. At (b), the magnetic field is approximately \(-1.31 \times 10^{-6} \; \hat{k} \; \text{T}\). At (c), it is approximately \(-6.57 \times 10^{-7} \; \hat{k} \; \text{T}\). At (d), the field is approximately \(1.31 \times 10^{-6} \; \hat{j} \; \text{T}\).

Key Concepts

Magnetic Field

In physics, a magnetic field is an invisible force field that surrounds magnetic materials and is caused by electric currents. It is represented by the symbol \( \mathbf{B} \) and is measured in teslas (T). Magnetic fields influence moving charges and other magnets in their vicinity, creating forces that can cause movement.

• The direction of a magnetic field is indicated by magnetic field lines, flowing from the north to the south pole of a magnet.
• Magnetic field strength is determined by the density of these lines. A stronger magnetic field has more lines close together.
• This field is fundamental in creating electromagnets, as when a current flows through a wire, it generates a magnetic field around it.

Understanding magnetic fields is crucial for exploring forces created by moving charges, like those in electrons or electric currents.

Moving Charge

A moving charge, such as an electron in motion, generates a magnetic field. This is the basic principle behind many electrical devices, like electromagnets and motors. When a charge moves, its motion creates a circulating magnetic field around its path.

• A charged particle's velocity and direction determine the configuration of the magnetic field it creates.
• The strength of the magnetic field depends on the speed and the amount of charge; higher values result in stronger fields.
• In our scenario, the charge is moving in the positive x-direction, producing varied magnetic fields at different points in space.

Exploring how moving charges influence magnetic fields helps in designing electric circuits and understanding electromagnetic principles.

Permeability of Free Space

Permeability of free space, denoted by \( \mu_0 \), is a physical constant that measures the ability of the vacuum to support a magnetic field. It is crucial in magnetic field calculations, such as those using the Biot-Savart Law.

• The value of \( \mu_0 \) is approximately \( 4\pi \times 10^{-7} \, \text{T}\cdot \text{m/A} \).
• Facilitating the interaction between magnetic fields and currents in a vacuum, it appears in many electromagnetic equations.
• It serves as a proportional constant in the Biot-Savart Law, connecting the physical movement of a charge to the resultant magnetic field.

The concept of permeability is essential for understanding how various environments can affect magnetic field behaviors.

Cross Product

The cross product is a mathematical operation on two vectors in three-dimensional space that yields a third vector, orthogonal to both original ones. In the context of magnetic fields, it helps in determining the direction and magnitude of the field due to moving charges.

• If two vectors are \( \mathbf{a} \) and \( \mathbf{b} \), their cross product \( \mathbf{a} \times \mathbf{b} \) is perpendicular to the plane containing both.
• The magnitude of the result depends on the sine of the angle between \( \mathbf{a} \) and \( \mathbf{b} \), and the magnitudes of the vectors themselves.
• A zero cross product indicates parallel vectors, while maximum magnitude occurs when the vectors are perpendicular.

This operation is instrumental in calculating the magnetic field produced by moving charges as per the Biot-Savart Law.

Vector Analysis

Vector analysis involves the study of vectors, which are mathematical objects possessing both magnitude and direction. It plays a crucial role in fields like physics and engineering, particularly in understanding forces and motion.

• Vectors can represent a variety of physical quantities such as displacement, velocity, and force.
• Operations such as addition, subtraction, and cross products are used to manipulate vectors for analysis.
• In magnetic field problems, vectors express quantities like the velocity of moving charges and the position relating to observation points.

Mastery in vector analysis provides insights into graphical and mathematical representations of physical phenomena, enabling better comprehension of dynamic systems like moving charges creating magnetic fields.