Problem 74 - Fundamentals of Physics

Question

The magnitude of the magnetic field at a point \(88.0 \mathrm{~cm}\) from the central axis of a long straight wire is \(7.30 \mu \mathrm{T}\). What is the current in the wire?

Step-by-Step Solution

Answer

The current in the wire is approximately 32.06 A.

Step 1: Understand the Problem

We need to find the current in a long straight wire given the magnitude of the magnetic field at a certain distance from the wire. The distance is given as \(88.0 \; \text{cm}\) and the magnetic field is \(7.30 \; \mu\text{T}\). We will use Ampère's law to solve this.

Step 2: Recall the Formula

According to Ampère's Law, the magnetic field around a long straight current-carrying wire can be calculated using the formula: \[ B = \frac{\mu_0 I}{2 \pi r} \]where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space \(= 4\pi \times 10^{-7} \; \text{N} \text{A}^{-2}\), \(I\) is the current, and \(r\) is the distance from the wire.

Step 3: Convert Units

Convert the given quantities to standard units. Distance \(r = 88.0 \; \text{cm} = 0.88 \; \text{m}\), and the magnetic field \(B = 7.30 \; \mu\text{T} = 7.30 \times 10^{-6} \; \text{T}\).

Step 4: Rearrange the Formula to Solve for Current

Rearrange the formula to solve for the current \(I\):\[ I = \frac{2 \pi r B}{\mu_0} \]

Step 5: Substitute Values into the Formula

Substitute the known values into the rearranged formula:\[ I = \frac{2 \pi (0.88) (7.30 \times 10^{-6})}{4\pi \times 10^{-7}} \]

Step 6: Perform the Calculation

Calculate the numerical value:\[ I = \frac{2 \times 0.88 \times 7.30 \times 10^{-6}}{4 \times 10^{-7}} \]Simplify the expression:\[ I = \frac{12.824 \times 10^{-6}}{4 \times 10^{-7}} = 32.06 \; \text{A} \]

Step 7: Conclusion

The current flowing through the wire that generates the given magnetic field at the specified distance is approximately \(32.06\; \text{A}\).

Key Concepts

Magnetic Field

The magnetic field is a crucial concept in physics, especially when discussing electromagnetic phenomena. It describes the area around a magnet or current-carrying wire where magnetic forces can be detected. The strength and direction of the magnetic field can vary based on the source's distance and current.

• Visualization: Imagine the space around a wire as a series of concentric circles of influence. Each circle represents the magnetic field lines, which are invisible but can be visualized as lines following the direction the field would exert a force on a positive north magnetic pole.
• Magnetic Field Direction: For a straight wire, the right-hand grip rule helps determine the field's direction: if you grip the wire with your right hand, with your thumb pointing in the direction of the current, your fingers will curl in the direction of the magnetic field lines.

Understanding the magnetic field is essential for comprehending how electricity and magnetism interact. It forms the backbone of technologies relying on electromagnetism, from electric motors to MRI machines.

Current in a Straight Wire

Current flowing through a straight wire generates a magnetic field. This connection is fundamental in electromagnetism and can be described using Ampère's Law. When a current flows through the wire, it creates circular magnetic fields around it. These fields can be measured at a specific distance from the wire, providing insight into the current's strength.

• Current and Field Strength Relationship: The magnetic field's strength at a given distance is directly proportional to the current. This relationship is a core part of understanding how the magnetic field behaves around conductors.
• Ampère’s Law Application: This law states that the line integral of the magnetic field B around a closed path is proportional to the current I enclosed by the path. For a straight wire, it's expressed as: \( B = \frac{\mu_0 I}{2 \pi r} \), showing the relationship between magnetic field B, current I, distance r from the wire, and the permeability of free space \(\mu_0\).

This relationship helps predict how electricity behaves in circuits and is key in designing electrical and electronic systems.

Permeability of Free Space

The permeability of free space, represented as \(\mu_0\), is a fundamental constant in physics relating magnetic fields to the current producing them. It defines the extent to which a material can become magnetized in response to an applied magnetic field. In a vacuum, it's typically a measure of how much resistance the vacuum offers against the formation of a magnetic field.

• Physics Constants: The standard value of \(\mu_0\) is \(4\pi \times 10^{-7} \, \text{N/A}^2\). This value is crucial when calculating magnetic fields generated by currents in a vacuum or air.
• Role in Calculations: When using equations like Ampère's Law, \(\mu_0\) allows physicists and engineers to calculate the magnetic field resulting from a known current. This concept is essential not just for theoretical physics but for practical applications where electromagnetism is involved.

This constant also serves as a basis for understanding the intrinsic properties of other materials besides a vacuum, which can alter the magnetic field they are exposed to based on their permeability.