Problem 20 - University Physics with Modern Physics

Question

Two long, straight wires, one above the other, are separated by a distance 2\(a\) and are parallel to the \(x\)-axis. Let the +\(y\)-axis be in the plane of the wires in the direction from the lower wire to the upper wire. Each wire carries current \(I\) in the +\(x\)-direction. What are the magnitude and direction of the net magnetic field of the two wires at a point in the plane of the wires (a) midway between them; (b) at a distance \(a\) above the upper wire; (c) at a distance \(a\) below the lower wire?

Step-by-Step Solution

Answer

(a) Zero; (b) \( \frac{2\mu_0I}{3\pi a} \) into the page; (c) \( \frac{2\mu_0I}{3\pi a} \) out of the page.

Step 1: Understand Formula for Magnetic Field of a Long Wire

The magnetic field due to a long straight wire carrying current is given by the formula: \( B = \frac{\mu_0 I}{2\pi r} \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( r \) is the distance from the wire to the point of interest.

Step 2: Analyze Magnetic Field Direction

The direction of the magnetic field is given by the right-hand rule. For a wire carrying current in the +x direction, the field direction follows a circular path in the counterclockwise direction when viewed from the wire. At the midpoint between two wires carrying the same current in the same direction, the fields due to each wire add together in magnitude but oppose each other in direction vertically.

Step 3: Calculate at Midpoint (a)

At the midpoint, the distance from each wire is \(a\). The magnetic field from each wire at this point is \( B = \frac{\mu_0 I}{2\pi a} \). The magnetic fields are in opposite directions; hence, they cancel each other out. Thus, the net magnetic field is zero.

Step 4: Calculate Field (b) Above Upper Wire

At a distance \(a\) above the upper wire, the distance from the upper wire is \(a\) and from the lower wire is \(3a\). Using the formula, the magnetic fields are: \( B_1 = \frac{\mu_0 I}{2\pi a} \) (for upper wire) and \( B_2 = \frac{\mu_0 I}{6\pi a} \) (for lower wire). The direction of \( B_1 \) using the right-hand rule is into the page, and \( B_2 \) is also into the page. They add up, so the net field is \( B_{net} = \frac{2\mu_0 I}{3\pi a} \), into the page.

Step 5: Calculate Field (c) Below Lower Wire

At a distance \(a\) below the lower wire, the distance from the lower wire is \(a\) and from the upper wire is \(3a\). The magnetic fields are: \( B_1 = \frac{\mu_0 I}{2\pi a} \) (for lower wire) and \( B_2 = \frac{\mu_0 I}{6\pi a} \) (for upper wire). The direction of \( B_1 \) is out of the page while \( B_2 \) is out of the page as well. The fields add, resulting in \( B_{net} = \frac{2\mu_0 I}{3\pi a} \), out of the page.

Key Concepts

Ampère's Law

Ampère's Law is a fundamental concept in electromagnetism that relates the circulation of a magnetic field to the electric current producing it. This law is crucial for calculating the magnetic field generated by current-carrying conductors, such as wires. It states that the line integral of the magnetic field around a closed loop is proportional to the electric current passing through the loop.

Mathematically, Ampère's Law is expressed as:

• \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \)

where:

• \( \vec{B} \) is the magnetic field vector,
• \( d\vec{l} \) is an infinitesimal element of the loop,
• \( \mu_0 \) is the permeability of free space, and
• \( I_{enc} \) is the current enclosed by the loop.

Ampère’s Law is especially useful for calculating the magnetic fields of symmetrical configurations, like our two parallel wires. In this scenario, the fields from each wire can be determined separately and understood how they will interact with each other.

Right-Hand Rule

The right-hand rule is an easy technique to determine the direction of a magnetic field created by an electric current. To use the rule for a long, straight wire:

• Point your thumb in the direction of the current flow.
• Then, curl your fingers around the wire.
• Your fingers will naturally curl in the direction of the circular magnetic field lines.

This visualization is particularly helpful when analyzing wires that carry current in the same or opposite directions.

In our specific task of two parallel wires carrying current in the same direction, the right-hand rule tells us the pattern of each wire's magnetic field:

• The magnetic field loops around each wire in a counterclockwise direction, as viewed from the end of the wire.
• At points above and below the wires, these fields add or subtract based on their orientation.

Utilizing the right-hand rule hence becomes essential not just for determining direction but also for predicting how magnetic fields combine at various points.

Magnetic Field Superposition

The concept of magnetic field superposition is based on the principle that the net magnetic field at any point in space is the vector sum of all the individual fields present.

In situations with multiple sources of magnetic fields, this principle is invaluable:

• Each wire generates its own magnetic field that can be calculated independently using the formula for a long, straight wire: \( B = \frac{\mu_0 I}{2\pi r} \)
• Once individual fields are known, they can be added together, taking into account their magnitude and direction to calculate the total field strength.

For example, halfway between the two parallel wires, the fields oppose each other vertically and thus cancel out.

Above and below each wire, however, the fields reinforce each other:

• Above the upper wire, both fields point into the page.
• Below the lower wire, both fields point out of the page.

This superposition allows us to calculate a net magnetic field by simple addition or subtraction of field vectors, giving us a clearer understanding of magnetic interactions in multi-wire systems.