Problem 11 - University Physics with Modern Physics

Question

Along, straight wire lies along the \(z\) -axis and carries a \(4.00-\mathrm{A}\) current in the \(+z\) -direction. Find the magnetic field (magnitude and direction) produced at the following points by a \(0.500-\mathrm{mm}\) segment of the wire centered at the origin: (a) \(x=2.00 \mathrm{m}, y=0\) , \(z=0 ;(b) x=0, y=2.00 \mathrm{m}, z=0 ;(\mathrm{c}) x=2.00 \mathrm{m}, y=2.00 \mathrm{m}\) \(z=0 ;(\mathrm{d}) x=0, y=0, z=2.00 \mathrm{m}\)

Step-by-Step Solution

Answer

At (a) \(1 \times 10^{-7}\, \text{T}\), +y; (b) \(1 \times 10^{-7}\, \text{T}\), -x; (c) \(\frac{1}{2\sqrt{2}} \times 10^{-7}\, \text{T}\); (d) \(0\, \text{T}\).

Step 1: Understand the Problem

We need to determine the magnetic field created by a current segment of a wire using the Biot-Savart Law, which relates the magnetic field \(\mathbf{B}\) to the current \(I\), the distance \(r\), and the segment length \(d\mathbf{l}\).

Step 2: Biot-Savart Law Formula

The Biot-Savart Law is given by: \[\mathbf{B} = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot d\mathbf{l} \times \mathbf{r}}{r^3}\] where \(\mu_0 = 4\pi \times 10^{-7} \, \text{T} \, \text{m/A}\) is the permeability of free space, \(I\) is the current, \(d\mathbf{l}\) is the infinitesimal length of the wire, and \(\mathbf{r}\) is the position vector from the segment to the point.

Step 3: Determine the Common Parameters

For all parts of the problem, the current \(I = 4.00\, \text{A}\), \(d\mathbf{l} = 0.500\, \text{mm} = 0.0005\, \text{m}\), and the origin is at (0,0,0). The unit vector of \(d\mathbf{l}\) is in the \(z\)-direction, i.e., \([0, 0, d\mathbf{l}]\).

Step 4: Calculate for Point (a)

For point \(x = 2.00\, \text{m}, y = 0, z = 0\): - Position vector \(\mathbf{r} = [2, 0, 0]\)- Magnitude of \(\mathbf{r} = 2.00\, \text{m}\)- Using the right-hand rule, \(d\mathbf{l} \times \mathbf{r} = [0, 0, 0.0005] \times [2, 0, 0] = [0, 0.001, 0]\)- \(\mathbf{B} = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{4 \cdot 0.001}{8} = 1 \times 10^{-7} \, \text{T}\) in the \(+y\) direction.

Step 5: Calculate for Point (b)

For point \(x = 0, y = 2.00\, \text{m}, z = 0\): - Position vector \(\mathbf{r} = [0, 2, 0]\)- Magnitude of \(\mathbf{r} = 2.00\, \text{m}\)- \(d\mathbf{l} \times \mathbf{r} = [0, 0, 0.0005] \times [0, 2, 0] = [-0.001, 0, 0]\)- \(\mathbf{B} = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{4 \cdot 0.001}{8} = 1 \times 10^{-7} \, \text{T}\) in the \(-x\) direction.

Step 6: Calculate for Point (c)

For point \(x = 2.00\, \text{m}, y = 2.00\, \text{m}, z = 0\): - Position vector \(\mathbf{r} = [2, 2, 0]\)- Magnitude of \(\mathbf{r} = \sqrt{8}\)- \(d\mathbf{l} \times \mathbf{r} = [0, 0, 0.0005] \times [2, 2, 0] = [-0.001, 0.001, 0]\)- \(\mathbf{B} = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{4 \cdot \sqrt{2} \cdot 0.001}{8\sqrt{8}} = \frac{1}{2\sqrt{2}} \times 10^{-7} \, \text{T}\) in the direction determined by vector \([-1, 1, 0]\).

Step 7: Calculate for Point (d)

For point \(x = 0, y = 0, z = 2.00\, \text{m}\): - Position vector \(\mathbf{r} = [0, 0, 2]\)- \(d\mathbf{l}\) and \(\mathbf{r}\) are parallel, \(d\mathbf{l} \times \mathbf{r} = [0, 0, 0.0005] \times [0, 0, 2] = [0, 0, 0]\)- Therefore, \(\mathbf{B} = 0 \, \text{T}\).

Step 8: Consider the Direction

The direction can be determined using the right-hand rule: point your thumb in the direction of \(d\mathbf{l}\) and your fingers towards \(\mathbf{r}\), the field direction is given by the direction your palm's push indicates orthogonal to both \(d\mathbf{l}\) and \(\mathbf{r}\).

Key Concepts

Magnetic Field

The magnetic field (\(\mathbf{B}\)) is a vector field surrounding a magnetic material or a moving electric charge. This field has both a direction and a magnitude. The role of a magnetic field is to exert a force on other moving charges or magnetic moments in its vicinity. According to classical physics, whenever a current flows through a conductor, it generates a magnetic field around it.

The Biot-Savart Law is one way to quantitatively describe the magnetic field produced by an electric current. For a small segment of wire with current (\(I\)), the resulting magnetic field at a particular point is not uniform and is typically estimated by vector calculus. The principles behind the Biot-Savart Law allow us to determine how the magnetic field behaves at various points in space, influenced by the position of the current carrying wire.

Current in a Wire

Current is defined as the flow of electric charge and is typically measured in amperes (\(\text{A}\)). In a wire, this means the movement of electrons along the conductor. Current always has a direction, and in problems like the one given, we imagine it flowing through a straight wire. The direction is crucial, as it affects the orientation of the magnetic field that is generated.

When analyzing the current in a wire, it's essential to comprehend how this movement dictates the presence and direction of associated magnetic fields. A significant practical concept is that of a linear current element, an infinitesimal segment showing the local characteristics of the wire, contributing to the overall magnetic field. In our exercise, the current in the wire is constant, extending along the z-axis, affecting fields at nearby points.

Permeability of Free Space

One fundamental constant in electromagnetism is the permeability of free space (\(\mu_0\)), symbolizing the ability of a vacuum to support the formation of a magnetic field. The value of \(\mu_0\) is \(4\pi \times 10^{-7} \, \text{T}\, \text{m/A}\). It's a pivotal measure that relates magnetic field strength to physical dimensions and system characteristics in a vacuum.

The permeability of free space appears in many equations dealing with magnetic fields, including the Biot-Savart Law. Understanding its role helps us bridge the mathematical relationship between electrical currents and the magnetic fields they produce. It's the baseline for material behavior regarding magnetic field generation, essentially setting a standard measure for how magnetic fields can "spread" throughout space.

Right-Hand Rule

The right-hand rule is a straightforward mnemonic for determining the direction of vectors in physics. It's used extensively in electromagnetism, particularly when dealing with magnetic fields generated by currents. When applying the right-hand rule, you orient your right hand such that your thumb points in the direction of current flow (\(d\mathbf{l}\)) and your fingers along the vector pointing from the current to the point of interest (\(\mathbf{r}\)).

Once you've positioned your hand correctly, the direction your palm pushes towards is the direction of the magnetic field (\(\mathbf{B}\)). This rule is crucial for visualizing the orientation and relationship between current and the resultant magnetic field. In the given exercise, it helps determine the vector direction of the magnetic field at various points, ensuring that the orientation aligns with physical observations and laws of physics.