Problem 14 - University Physics with Modern Physics

Question

A square wire loop 10.0 cm on each side carries a clockwise current of 8.00 A. Find the magnitude and direction of the magnetic field at its center due to the four 1.20-mm wire segments at the midpoint of each side.

Step-by-Step Solution

Answer

The magnetic field at the center is \\(7.70 \times 10^{-6}\, \text{T}\) out of the loop's plane.

Step 1: Understand the Setup

The problem involves a square loop of wire with sides of 10.0 cm. The wire carries a current of 8.00 A in a clockwise direction. We are told to consider the magnetic field at the center of the square due to 1.20-mm wire segments located at the midpoint of each side.

Step 2: Use Biot-Savart Law

The Biot-Savart Law gives the magnetic field \(d\mathbf{B}\) due to a small segment of current \(d\mathbf{I}\) as \[d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{s} \times \mathbf{\hat{r}}}{r^2}\]where \(\mu_0\) is the permeability of free space, \(d\mathbf{s}\) is the length element, \(\mathbf{\hat{r}}\) is the unit vector in the direction of the point from the current element, and \(r\) is the distance from the element to the point of interest.

Step 3: Calculate the Contribution from One Element

Each current segment at the midpoint of a side is perpendicular to \(\mathbf{r}\), simplifying the cross-product in Biot-Savart. The segment length \(d\mathbf{s} = 1.20\, \text{mm} = 0.0012\, \text{m}\). The distance \(r\) from any midpoint to the center of the square is \(5.0 \div \sqrt{2}\ \text{cm} = 0.05\div\sqrt{2}\, \text{m}\).

Step 4: Compute Single Element Field Value

Using the Biot-Savart Law: \[dB = \frac{(4\pi \times 10^{-7}\, \text{T}\cdot\text{m/A})(8.00) (0.0012)}{4\pi (0.05/\sqrt{2})^2}\]Solving gives \(dB = 1.925 \times 10^{-6}\, \text{T}\) for each segment.

Step 5: Total Magnetic Field at the Center

There are four wire segments contributing equally to the total magnetic field at the center. Because the current is clockwise, use the right-hand rule to determine that each segment contributes to the field in the same direction at the center. Thus, we multiply the field from a single segment by four: \[B_{\text{total}} = 4 \times 1.925 \times 10^{-6}\, \text{T} = 7.70 \times 10^{-6}\, \text{T}\].

Step 6: Determine Magnetic Field Direction

With the right-hand rule, a clockwise current on each wire segment generates a magnetic field out of the plane at the center. Therefore, the magnetic field direction is outwards perpendicular to the plane of the loop at its center.

Key Concepts

Magnetic Field

A magnetic field is an invisible field that exerts a force on particles that are charged and in motion. Its presence can be detected using a compass or various sensors, and it plays a crucial role in the realm of electromagnetism. When an electric current flows through a wire, it creates a circular magnetic field around it. The strength and direction of this field depend on several factors, such as the current's strength, the shape of the wire, and the permeability of the surrounding medium.

• The direction of the magnetic field can be determined using the right-hand rule: point your thumb in the direction of the current and your fingers will curl in the direction of the magnetic field.
• Magnetic fields are measured in teslas (T), a measure of field strength.

For wires in specific configurations, such as loops or coils, the interaction of different segment fields can be calculated using the Biot-Savart Law, which is critical in understanding and predicting the behavior of these fields at any given point.

Square Loop

A square loop in electromagnetism refers to a closed loop of wire arranged in a square shape. The configuration of the current-carrying wire within the square significantly affects the resulting magnetic field. In a square loop, the magnetic field produced by different sections of the wire interacts at the loop's center.

• The shape of the loop causes contributions from different segments of wire to potentially combine constructively or destructively, altering the total field experienced at any point.
• For a square loop, symmetry helps us predict that each segment contributes equally to the magnetic field at the center.
• In our example, each side is exactly 10 cm, forming a perfect square which simplifies calculations as symmetry plays a strong role in the field distribution.

Using techniques like the Biot-Savart Law and symmetry can simplify the process of determining the resultant magnetic field at specific points like the center of a square loop.

Current in a Wire

Current flowing through a wire creates a magnetic field. This is fundamental to understanding electromagnetism. In the case of a square loop, the current flows consistently through each section of the wire.

• The strength of the current, measured in amperes (A), directly affects the magnetic field's magnitude.
• In our example, the wire carries a consistent current of 8.00 A in a clockwise direction.
• Each segment of the wire contributes to the field at the center of the square.

The route of the current helps determine the magnetic field's direction using the right-hand rule. The consistency of the current all around the loop ensures that each wire segment contributes a magnetic field in the same overall direction if symmetry is maintained.

Permeability of Free Space

The permeability of free space (\( \mu_0 \)) is a constant that describes how much the magnetic field can penetrate a vacuum. It is essential in quantifying the strength of a magnetic field created by an electric current.

• The constant \( \mu_0 \) has a defined value of \( 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \).
• This value is crucial when applying the Biot-Savart Law to calculate magnetic fields as it links the magnetic field strength to the source current.
• The permeability tells us how easy it is for the magnetic field lines to pass through free space or vacuum.

In the context of our square loop problem, using \( \mu_0 \) allows us to accurately compute the magnetic field contribution from each segment of current-carrying wire to the total field experienced at the loop's center. It highlights how fundamental constants of nature are utilized in electrical engineering and physics to predict electromagnetic phenomena.