Problem 19 - University Physics with Modern Physics

Question

A long, straight wire lies along the \(y\) -axis and carries a current \(I=8.00\) A in the \(-y\) -direction (Fig. 28.39\()\) . In addition to the magnetic field due to the current in the wire, a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) with magnitude \(1.50 \times 10^{-6} \mathrm{T}\) is in the \(+x\) -direction \(\mathrm{What}\) is the total field (magnitude and direction) at the following points in the \(x z\) -plane: \((a) x=0, z=1.00 \mathrm{m}\) (b) \(x=1.00 \mathrm{m}, z=0 ;(\mathrm{c}) x=0\) \(z=-0.25 \mathrm{m} ?\)

Step-by-Step Solution

Answer

(a) 2.20 μT at 46.13° from x-axis; (b) 2.20 μT at 46.13°; (c) 6.56 μT at 76.09°.

Step 1: Understanding the Problem

We have a long straight wire carrying a current of 8.00 A in the -y direction. We also have a uniform magnetic field, \( \overrightarrow{\boldsymbol{B}}_0 \), of magnitude \( 1.50 \times 10^{-6} \, \text{T} \) in the +x direction. We need to find the magnitude and direction of the total magnetic field at given points in the \(x-z\) plane.

Step 2: Magnetic Field Due to a Wire

The magnetic field due to a long straight wire at a distance \( r \) is given by the formula: \( B = \frac{{\mu_0 I}}{{2 \pi r}} \), where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \). This field is circular around the wire, and at any point in the \(x-z\) plane, it will be directed along the \(y\)-axis.

Step 3: Finding Total Magnetic Field at Point (a)

For point (a) at \((x=0, z=1.00 \text{m})\):- Distance from wire, \( r = 1.00 \text{m} \).- Magnetic field due to wire, \( B_{\text{wire}} = \frac{{4\pi \times 10^{-7} \times 8}}{{2\pi \times 1}} = 1.60 \times 10^{-6} \, \text{T} \) in the \(+y\) direction.- Total magnetic field, combining \( B_0 \) in the \(+x\) direction, requires vector addition: \( B_{\text{total}} = \sqrt{(1.50 \times 10^{-6})^2 + (1.60 \times 10^{-6})^2} = 2.20 \times 10^{-6} \, \text{T} \).- Direction angle \( \theta = \tan^{-1}\left(\frac{B_{ ext{wire}}}{B_{0}}\right) \approx 46.13^\circ \) relative to the x-axis.

Step 4: Finding Total Magnetic Field at Point (b)

For point (b) at \((x=1.00 \text{m}, z=0)\):- Distance from wire, \( r = 1.00 \text{m} \).- Magnetic field due to wire, \( B_{\text{wire}} = 1.60 \times 10^{-6} \, \text{T} \) in the \( -z \) direction.- Already existing magnetic field \( B_{0} = 1.50 \times 10^{-6} \, \text{T} \) in the \(+x\) direction.- Resultant magnetic field, \( B_{\text{total}} = \sqrt{(1.50 \times 10^{-6})^2 + (1.60 \times 10^{-6})^2} = 2.20 \times 10^{-6} \, \text{T} \).- Direction angle \( \theta = \tan^{-1}\left(\frac{B_{ ext{wire}}}{B_{0}}\right) \approx 46.13^\circ \) relative to the x-axis.

Step 5: Finding Total Magnetic Field at Point (c)

For point (c) at \((x=0, z=-0.25 \text{m})\):- Distance from wire, \( r = 0.25 \text{m} \).- Magnetic field due to wire, \( B_{\text{wire}} = \frac{{4\pi \times 10^{-7} \times 8}}{{2\pi \times 0.25}} = 6.40 \times 10^{-6} \, \text{T} \) in the \(-y\) direction.- The magnetic field \( B_0 = 1.50 \times 10^{-6} \, \text{T} \) in the \(+x\) direction.- Resultant magnetic field, \( B_{\text{total}} = \sqrt{(1.50 \times 10^{-6})^2 + (6.40 \times 10^{-6})^2} = 6.56 \times 10^{-6} \, \text{T} \).- Direction angle \( \theta = \tan^{-1}\left(\frac{B_{ ext{wire}}}{B_{0}}\right) \approx 76.09^\circ \) relative to the x-axis.

Key Concepts

Ampere's Law

Ampere's Law is a fundamental law used in the calculation of magnetic fields generated by currents. It provides a simple equation that relates the magnetic field around a closed loop to the electric current passing through the loop. This law states that the integral of the magnetic field along a closed path is equal to the permeability of free space, \( \mu_0 \), times the current enclosed by the path. The expression for Ampere's Law is:\[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}\]where:

• \( \mathbf{B} \) is the magnetic field vector.
• \( d\mathbf{l} \) is the differential length vector along the closed path.
• \( \mu_0 \) is the permeability of free space (\(4\pi \times 10^{-7}\, \text{T m/A}\)).
• \( I_{\text{enc}} \) is the current enclosed by the path.

This law is especially useful for calculating the magnetic field due to long straight wires, solenoids, and toroids. In the situation presented in the exercise, we use Ampere's Law to understand the magnetic field created by a current in a long straight wire. However, the calculations are often simplified by using symmetry and geometric considerations, which is what makes this law particularly powerful in such configurations. This allows us to easily determine the magnetic field at specific points around the wire.

Vector Addition

Vector addition is a critical mathematical operation used when combining multiple vectors, including magnetic fields. Magnetic fields are vector quantities, which means they have both magnitude and direction. When determining the total magnetic field in a scenario where multiple magnetic fields intersect, vector addition is performed to find both the resultant magnitude and direction. In our exercise, we have a magnetic field produced by a wire and a uniform magnetic field. To find the total field at any point, we add these vectors using:

• The Pythagorean theorem to calculate the magnitude of the resultant vector:\[B_{\text{total}} = \sqrt{B_x^2 + B_y^2}\]
• The tangent function to determine the angle \( \theta \), which provides the direction of the resultant vector relative to an axis:\[\theta = \tan^{-1}\left(\frac{B_y}{B_x}\right)\]

This results in a complete picture of the combined effect of different magnetic fields, offering insights into how they influence the observed magnetic field at various points. By performing this operation for different coordinates, you identify how the total field vector orientation and magnitude change based on distance and field direction.

Magnetic Field Due to a Wire

The magnetic field generated by a long straight current-carrying wire is a classic application of physics that is commonly encountered. According to the Biot-Savart Law, a wire carrying a current produces a circular magnetic field around it. The strength of this magnetic field decreases as you move further from the wire.To calculate this magnetic field at a distance \( r \) from the wire, we use the formula:\[B = \frac{\mu_0 I}{2\pi r}\]where:

• \( B \) is the magnetic field strength.
• \( \mu_0 \) is the permeability of free space.
• \( I \) is the current through the wire.
• \( r \) is the radial distance from the wire.

This field is oriented perpendicular to the wire and follows a right-hand rule for direction: if you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field lines.In the exercise, the current is given as moving in the \(-y\) direction, dictating that the magnetic field circles around the \( y \)-axis. Calculating it using the above formula gives the field's magnitude at any specific point in space. Understanding how the magnetic field due to a wire behaves is essential for predicting how it interacts with other magnetic fields, such as a uniform magnetic field present in other directions.