Problem 1 - University Physics with Modern Physics

Question

A +6.00-\(\mu\)C point charge is moving at a constant 8.00 \(\times\) 10\(^6\) m/s in the +\(y\)-direction, relative to a reference frame. At the instant when the point charge is at the origin of this reference frame, what is the magnetic- field vector \(\overrightarrow{B}\) 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, \(y =\) 0, \(z = +\)0.500 m; (d) \(x =\) 0, \(y4 = -\)0.500 m, \(z = +\)0.500 m?

Step-by-Step Solution

Answer

(a) \\( -38.4 \text{nT} \\mathbf{k} \\\), (b) \\( 0 \\\), (c) \\( 38.4 \text{nT} \\mathbf{i} \\\), (d) \\( -60.8 \text{nT} \\mathbf{i} \\\)

Step 1: Understand Biot-Savart Law

The magnetic field \( \mathbf{B} \) produced by a moving point charge can be calculated using a simplified form of the Biot-Savart Law: \[ \mathbf{B} = \frac{\mu_0}{4\pi}\frac{q (\mathbf{v} \times \mathbf{r})}{r^3} \] where \( q \) is the charge, \( \mathbf{v} \) is its velocity, \( \mathbf{r} \) is the position vector from the charge to the point, and \( r \) is the magnitude of that position vector. \( \mu_0 \) is the permeability of free space.

Step 2: Calculate Magnetic Field at Point (a)

At point \( (x = 0.5 \text{ m}, y = 0, z = 0) \), the position vector \( \mathbf{r} \) is \( (0.5, 0, 0) \). The velocity vector \( \mathbf{v} \) is \( (0, 8 \times 10^6, 0) \). Calculate \( \mathbf{v} \times \mathbf{r} = (0, 8 \times 10^6, 0) \times (0.5, 0, 0) = (0, 0, -4 \times 10^6) \). The magnitude of \( \mathbf{r} \) is \( r = 0.5 \). Substitute into the Biot-Savart Law to find \( \mathbf{B_a} = \frac{\mu_0}{4\pi}\frac{6 \times 10^{-6} \cdot (0, 0, -4 \times 10^6)}{(0.5)^3} \).

Step 3: Calculate Magnetic Field at Point (b)

For point \( (x = 0, y = -0.5, z = 0) \), \( \mathbf{r} = (0, -0.5, 0) \). Calculate \( \mathbf{v} \times \mathbf{r} = (0, 8 \times 10^6, 0) \times (0, -0.5, 0) = (0, 0, 0) \). Hence, \( \mathbf{B_b} = 0 \) because the cross product is zero.

Step 4: Calculate Magnetic Field at Point (c)

For \( (x = 0, y = 0, z = 0.5) \), \( \mathbf{r} = (0, 0, 0.5) \). Calculate \( \mathbf{v} \times \mathbf{r} = (0, 8 \times 10^6, 0) \times (0, 0, 0.5) = (4 \times 10^6, 0, 0) \). The magnitude of \( \mathbf{r} \) is \( r = 0.5 \). Therefore, \( \mathbf{B_c} = \frac{\mu_0}{4\pi}\frac{6 \times 10^{-6} \cdot (4 \times 10^6, 0, 0)}{(0.5)^3} \).

Step 5: Calculate Magnetic Field at Point (d)

For \( (x = 0, y = -0.5, z = 0.5) \), \( \mathbf{r} = (0, -0.5, 0.5) \). Calculate \( \mathbf{v} \times \mathbf{r} = (0, 8 \times 10^6, 0) \times (0, -0.5, 0.5) = (-4 \times 10^6, 0, 0) \). Here \( r = \sqrt{(0)^2 + (-0.5)^2 + (0.5)^2} = 0.707 \). Hence, \( \mathbf{B_d} = \frac{\mu_0}{4\pi}\frac{6 \times 10^{-6} \cdot (-4 \times 10^6, 0, 0)}{(0.707)^3} \).

Step 6: Calculate Expressions

Finally, use the value \( \mu_0 = 4\pi \times 10^{-7} \text{ T·m/A} \) to calculate the numerical values:- For point (a): \( \mathbf{B_a} = -38.4 \text{nT} \mathbf{k} \)- For point (b): \( \mathbf{B_b} = 0 \)- For point (c): \( \mathbf{B_c} = 38.4 \text{nT} \mathbf{i} \)- For point (d): \( \mathbf{B_d} = -60.8 \text{nT} \mathbf{i} \)

Key Concepts

Magnetic Field Calculation

To find the magnetic field created by a moving point charge, we use the Biot-Savart Law. This law provides a formula that helps determine how a charge's movement produces a magnetic field in its vicinity.
The simplified version of this law, relevant for a point charge, is given by: \[\mathbf{B} = \frac{\mu_0}{4\pi}\frac{q (\mathbf{v} \times \mathbf{r})}{r^3}\] Here, \(\mathbf{B}\) is the magnetic field vector. The terms \(q\), \(\mathbf{v}\), and \(\mathbf{r}\) represent the charge, velocity, and position vector from the source charge to the observation point, respectively.
The term \(r\) stands for the magnitude of the position vector. This equation highlights how magnetic fields depend on the charge, speed, and relative position to the point of observation.

Vector Cross Product

The cross product is a mathematical operation performed on two vectors. It is crucial when calculating a magnetic field using the Biot-Savart Law, as it influences the direction and magnitude of the result.
For vectors \(\mathbf{v}\) and \(\mathbf{r}\), the cross product is represented as \(\mathbf{v} \times \mathbf{r}\). This operation returns a new vector that is orthogonal to both original vectors.
The magnitude of this new vector is determined by the product of the magnitudes of the original vectors and the sine of the angle between them.

• The direction of the resulting vector follows the right-hand rule.
• Using the right hand, point fingers in the direction of the first vector, \(\mathbf{v}\), and curl towards the second vector, \(\mathbf{r}\).
• Your thumb then points in the direction of \(\mathbf{v} \times \mathbf{r}\).

The cross product is what ensures magnetic fields form circular or spiral paths around moving charges.

Permeability of Free Space

The permeability of free space, denoted as \(\mu_0\), is a fundamental constant in electromagnetism. It reflects how much magnetic field a material can support in the vacuum.
The standard value of \(\mu_0\) is \(4\pi \times 10^{-7}\, \text{T·m/A}\) (Tesla meter per Ampere). This value establishes the relationship between electric current and magnetic field in a vacuum.
In the Biot-Savart Law equation, \(\mu_0\) scales the magnetic field's strength, ensuring that units and measurements are consistent and comparable across calculations.
Understanding \(\mu_0\) is essential, as it lays the groundwork for how we model electromagnetic forces in theoretical physics as well as practical applications. It is prominently used in calculations involving circuits, coils, and magnetic fields around charged particles.

Point Charge Dynamics

Point charge dynamics involve understanding how a single, highly localized charge moves and interacts with its environment. This concept is vital for predicting how charges create electric and magnetic fields.
A point charge is treated as an infinitely small particle with a specific charge value. In our scenario, the charge moves at a constant velocity, translating into linear motion in a given direction.
Here are some key insights:

• Motion dynamics of the point charge affect the resultant magnetic field's shape and direction.
• When defining a reference point, use coordinates that clearly show the position of the moving charge.
• The origin of the reference is where the point charge is located at a given instant.

The point charge's velocity and position, coupled with its charge, allow us to compute magnetic fields at various points in space using the Biot-Savart Law. This ability is crucial for applications ranging from measuring electromagnetic influences in scientific explorations to practical engineering tasks.