Problem 65 - University Physics with Modern Physics

Question

A circular wire loop of radius \(a\) has \(N\) turns and carries a current \(I .\) A second loop with \(N^{\prime}\) turns of radius \(a^{\prime}\) carries current \(I^{\prime}\) and is located on the axis of the first loop, a distance \(x\) from the center of the first loop. The second loop is tipped so that its axis is at an angle \(\theta\) from the axis of the first loop. The distance \(x\) is large compared to both \(a\) and \(a^{\prime}\) (a) Find the magnitude of the torque exerted on the second loop by the first loop. (b) Find the potential energy for the second loop due to this interaction. (c) What simplifications result from having \(x\) much larger than \(a ?\) From having \(x\) much larger than \(a^{\prime} ?\)

Step-by-Step Solution

Answer

Find \( \tau = \frac{\mu_0 N N' I I' a^2 (a')^2 \sin\theta}{2x^3} \) and \( U = -\frac{\mu_0 N N' I I' a^2 (a')^2 \cos\theta}{2x^3} \). Approximations apply when \(x \gg a\) and \(x \gg a'\).

Step 1: Setup the Problem

We have two circular loops, each with their own set of parameters like number of turns, current, and radius. We need to find the torque and the potential energy affecting the second loop due to the first, with consideration of large distances.

Step 2: Determine Magnetic Field of First Loop

For a circular loop of radius \(a\), carrying current \(I\), the magnetic field at a point on the axis of the loop, at a distance \(x\) from its center, is given by the formula \( B = \frac{\mu_0 I a^2}{2(x^2 + a^2)^{3/2}} \). Since \(x \gg a\), we approximate it as \( B \approx \frac{\mu_0 I a^2}{2x^3} \).

Step 3: Calculate the Torque on the Second Loop

The torque \(\tau\) on a loop due to an external magnetic field \(B\) is \(\tau = NI\pi r^2 B \sin\theta\). Substituting the magnetic field from Step 2, we find \(\tau = N'I'\pi (a')^2 \frac{\mu_0 N I a^2}{2x^3} \sin\theta\).

Step 4: Compute Potential Energy of the Second Loop

The potential energy \(U\) for a magnetic moment \(\mathbf{m}\) in a magnetic field \(\mathbf{B}\) is given by \(U = -\mathbf{m} \cdot \mathbf{B}\). Here, \( U = -N'I'\pi (a')^2 \frac{\mu_0 N I a^2}{2x^3} \cos\theta \), since the loop's magnetic moment aligns with the field of the first loop.

Step 5: Simplifications from Large Distances

If \(x \gg a\), the factor \(a^2\) becomes negligible in comparison, simplifying the magnetic field expression. With \(x \gg a'\), higher order terms involving \(a'/x\) diminish, justifying approximations in the calculations for torque and potential energy.

Key Concepts

Torque in Magnetic Fields

Torque in magnetic fields describes the twisting force acting on a magnetic dipole within an external magnetic field. Here, the second loop is affected by the magnetic field generated by the first loop. The torque \( \tau \) can be calculated using the formula \( \tau = \mathbf{m} \times \mathbf{B} \), where \( \mathbf{m} \) is the magnetic moment and \( \mathbf{B} \) is the magnetic field created by the first loop.
The direction of this torque is perpendicular to the plane formed by \( \mathbf{m} \) and \( \mathbf{B} \), causing the second loop to rotate towards alignment with the field. This interaction depends on several factors:

• The number of turns \( N' \) in the second loop amplifies the torque.
• The radius \( a' \) affects the strength of the dipole moment.
• The angle \( \theta \) determines the component of \( \mathbf{m} \) perpendicular to \( \mathbf{B} \).

These factors contribute to understanding why positioning, current strength, and geometric dimensions influence the torque experienced by magnetic dipoles.

Potential Energy in Magnetic Fields

Potential energy in magnetic fields involves the energy stored due to the alignment of a magnetic dipole in an external magnetic field. For the second loop, its potential energy \( U \) is calculated by \( U = -\mathbf{m} \cdot \mathbf{B} \). This formula uses the dot product, indicating how closely aligned the magnetic moment of the second loop is with the magnetic field of the first loop.
The potential energy is minimized when the loop's magnetic moment aligns completely with the magnetic field. This alignment represents a state of lower energy, which is stable and favorable. On the flip side, as the dipole opposes the field, the potential energy increases, representing an unstable state.

• The cosine function in the equation shows how orientation affects potential energy.
• A smaller potential energy means a more stable configuration.
• Understanding potential energy helps explain why objects in magnetic fields prefer specific orientations.

Magnetic Dipole Moment

The magnetic dipole moment is a vector describing the magnetic strength and orientation of a magnetic dipole, such as a loop of wire carrying a current. This concept forms the foundation of interactions in magnetic fields for both scientific and practical applications.
The dipole moment \( m \) in this scenario is given by \( m = N'I'\pi (a')^2 \), where \( N' \) is the number of turns in the loop, \( I' \) is the current, and \( a' \) is the radius. These factors combine to indicate how strong and in which direction the loop acts like a bar magnet.

• Higher currents result in stronger magnetic fields, increasing the dipole moment.
• A larger number of turns reinforces the magnetic effect, enhancing the dipole moment.
• The circular nature of the coil provides uniform magnetic properties across its surface.

Understanding magnetic dipoles is essential, as it is a fundamental concept in electromagnetism affecting everything from magnetic storage devices to electromechanical systems.

Circular Current Loops

Circular current loops are loops of wire that generate magnetic fields when electricity flows through them. They are essential components in many devices, such as electromagnets, transformers, and motors.
The loop's geometry influences its magnetic field. Specifically, a circular loop produces a symmetrical magnetic field around its axis. In the given problem, both loops create magnetic fields, with their characteristics influenced by the radius \( a \) and \( a' \) as well as by their relative positions.

• The magnetic field inside a loop is strongest at the center and diminishes with distance.
• Increasing the loop's radius decreases the field strength at any given point outside the loop.
• The symmetry of the loop simplifies calculations of the magnetic field along the axis.

Circular current loops are a staple in electromagnetism, providing a straightforward yet powerful means to generate and manipulate magnetic fields.

Physics Problem Solving

Physics problem solving involves systematically analyzing and solving physics-related tasks, often starting with defining the problem, setting up equations, and simplifying complex situations. In scenarios with magnetic fields, it’s crucial to break the problem into understandable parts.
Effective problem solving in the context of these loops includes:

• Identifying known parameters such as current, direction, and distance between loops.
• Applying relevant formulas like those for magnetic fields and dipole moments.
• Acknowledging assumptions, such as \( x \gg a \), which simplifies calculations by removing terms that have minimal impact.

Gaining proficiency in these steps sharpens skills not only for textbook problems but also for real-world applications, where magnetic field considerations are crucial in technology and engineering fields.