Problem 47 - University Physics with Modern Physics

Question

A coil with magnetic moment 1.45 \(\mathrm{A} \cdot \mathrm{m}^{2}\) is oriented initially with its magnetic moment antiparallel to a uniform \(0.835-\mathrm{T}\) magnetic field. What is the change in potential energy of the coil when it is rotated \(180^{\circ}\) so that its magnetic moment is parallel to the field?

Step-by-Step Solution

Answer

The change in potential energy is \(-2.4205\,\text{J}\).

Step 1: Understand the Concept of Potential Energy in a Magnetic Field

The potential energy (U) of a magnetic dipole moment (\(\mu\)) in a magnetic field (\(B\)) is given by \[ U = -\mu \cdot B \cdot \cos(\theta) \] where \(\theta\) is the angle between the magnetic moment and the magnetic field. A rotation from antiparallel to parallel changes \(\theta\) from \(180^\circ\) to \(0^\circ\).

Step 2: Calculate Initial Potential Energy

Initially, the coil's magnetic moment is antiparallel to the field, which means \(\theta = 180^\circ\). The initial potential energy is \[ U_\text{initial} = -\mu \cdot B \cdot \cos(180^\circ) \] Since \(\cos(180^\circ) = -1\), it simplifies to: \[ U_\text{initial} = \mu \cdot B \]

Step 3: Calculate Final Potential Energy

After rotating the coil \(180^\circ\), the magnetic moment is parallel to the field, so \(\theta = 0^\circ\). The final potential energy is \[ U_\text{final} = -\mu \cdot B \cdot \cos(0^\circ) \] Since \(\cos(0^\circ) = 1\), it simplifies to: \[ U_\text{final} = -\mu \cdot B \]

Step 4: Determine the Change in Potential Energy

The change in potential energy \(\Delta U\) as the coil rotates is calculated by the formula: \[ \Delta U = U_\text{final} - U_\text{initial} \] Substituting the expressions for \(U_\text{final}\) and \(U_\text{initial}\): \[ \Delta U = (-\mu \cdot B) - (\mu \cdot B) = -2 \cdot \mu \cdot B \]

Step 5: Plug in Known Values

Given \(\mu = 1.45\,\mathrm{A} \cdot \mathrm{m}^2\) and \(B = 0.835\,\mathrm{T}\), substitute into the expression for \(\Delta U\):\[ \Delta U = -2 \cdot 1.45 \cdot 0.835 = -2.4205 \,\text{J} \] Thus, the change in potential energy is \(-2.4205\,\text{J}\).

Key Concepts

Magnetic Dipole Moment

A magnetic dipole moment is like a tiny bar magnet inside an object or particle, which has a north and south pole. This characteristic of an object, often a current loop such as a coil, describes how strongly it can get magnetized in the presence of a magnetic field. The magnetic dipole moment (\(\mu\)) is a vector quantity, which means it has both a magnitude and direction.
Its unit is \(\mathrm{A} \cdot \mathrm{m}^2\), which represents an ampere-loop of area. Think of it as a measure of how effective an object is at creating a magnetic field in its surrounding space.
For example, in this exercise, a coil with a magnetic dipole moment of \(1.45 \mathrm{A} \cdot \mathrm{m}^2\) is being analyzed to understand the change in energy when aligned with a magnetic field.
Some important points about magnetic dipole moments:

• Magnetic dipole moments arise from either the spin of charged particles or the movement of electrons in atoms.
• Larger dipole moments imply stronger magnetic behavior in an external field.
• The direction of the magnetic dipole moment vector is perpendicular to the plane of the loop in the coil.

Understanding this concept is essential for analyzing the effects of magnetic fields on electrical components.

Magnetic Field Strength

Magnetic field strength, often denoted as \(B\), represents the force exerted by a magnetic field on a moving charged particle or magnetic dipole. It is a vector field, which means it has both a direction and a magnitude. The unit of magnetic field strength is the tesla (\(\mathrm{T}\)), which is equivalent to one newton per ampere-meter.
When a magnetic field interacts with a magnetic dipole moment, it can exert forces and torques on the dipole. This is highly relevant in scenarios like our coil rotating in the magnetic field because the interaction changes its potential energy.
Key aspects of magnetic field strength include:

• The direction of \(B\) is defined as the direction a north magnetic pole would be pushed.
• It varies in strength, determined by the source of the magnetic field, such as permanent magnets or electric currents.
• Uniform magnetic fields, such as the one described in our exercise (\(0.835 \mathrm{T}\)), apply a consistent force throughout a defined region.

Understanding magnetic field strength is crucial for grasping how magnetic fields interact with objects possessing magnetic dipole moments.

Potential Energy Calculation

The potential energy of a magnetic dipole in a magnetic field is influenced by its orientation relative to the field. This energy is given by the formula: \[ U = -\mu \cdot B \cdot \cos(\theta) \] where \(\mu\) is the magnetic dipole moment, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the magnetic dipole and the magnetic field direction.
The formula indicates how energy changes based on orientation. When a coil is rotated from anti-parallel (\(180^\circ\)) to parallel (\(0^\circ\)) with the field, the change in angle results in a shift in potential energy.
Let's break down the process:

• Initially, when the dipole is antiparallel (\(\cos(180^\circ) = -1\)), the potential energy is maximized as \(\mu \times B\).
• When aligned parallel to the field (\(\cos(0^\circ) = 1\)), the potential energy is minimized at \(-\mu \times B\).
• The change, given by \(-2 \mu B\), signifies that energy decreases as the coil aligns with the field.

This understanding explains the calculated energy shift of \(-2.4205\) Joules for the coil, highlighting the impact of position within a magnetic field.