Problem 88 - University Physics with Modern Physics

Question

A circular loop of wire with area \(A\) lies in the \(x y\) -plane. As viewed along the \(z\) -axis looking in the \(- z\) -direction toward the origin, a current \(I\) is circulating clockwise around the loop. The torque produced by an external magnetic field \(\vec { B }\) is given by \(\vec { \tau } = D ( 4 \hat { \imath } - 3 \hat { \jmath } ) ,\) where \(D\) is a positive constant, and for this orientation of the loop the magnetic potential energy \(U = - \vec { \mu } \cdot B\) is negative. The magnitude of the magnetic field is \(B _ { 0 } = 13 D / I A\) . (a) Determine the vector magnetic moment of the current loop. (b) Determine the components \(B _ { x } , B _ { y }\) and \(B _ { z }\) of \(\vec { B } .\)

Step-by-Step Solution

Answer

(a) \(\vec{\mu} = -IA\hat{k}\); (b) \( B_x = \frac{3D}{IA}, B_y = \frac{4D}{IA}, B_z = \frac{12D}{IA} \).

Step 1: Express the Magnetic Moment

The magnetic moment \( \vec{\mu} \) of a current-carrying loop is given by \( \vec{\mu} = I \cdot \vec{A} \). Since the loop lies in the \(xy\)-plane and has area \(A\), the magnetic moment will be aligned along the \(z\)-axis in the direction of the surface normal. However, since the current is circulating clockwise when viewed from the \(z\)-axis looking in the \(-z\) direction, \(\vec{\mu} = - I \cdot A \hat{k} \).

Step 2: Relate Torque to Magnetic Moment and Magnetic Field

The torque \( \vec{\tau} \) on a current loop in a magnetic field is given by \( \vec{\tau} = \vec{\mu} \times \vec{B} \). We have \( \vec{\tau} = D(4\hat{\imath} - 3\hat{\jmath}) \). Substitute \( \vec{\mu} = -IA\hat{k} \) into this equation: \( -IA\hat{k} \times \vec{B} = D(4\hat{\imath} - 3\hat{\jmath}) \).

Step 3: Determine Magnetic Field Components using Cross Product

Perform the cross product \( -IA\hat{k} \times (B_x\hat{i} + B_y\hat{j} + B_z\hat{k}) = D(4\hat{i} - 3\hat{j}) \). Simplifying gives: \( IA(B_y)\hat{i} - IA(B_x)\hat{j} = D(4\hat{i} - 3\hat{j}) \). Thus, \( IA B_y = 4D \) and \( -IA B_x = -3D \). Solve these to get \( B_y = \frac{4D}{IA} \) and \( B_x = \frac{3D}{IA} \).

Step 4: Find Magnetic Field Along z-axis

Given the magnitude of the magnetic field \( B_0 = 13D/IA \), relate it to the components: \( B_0 = \sqrt{B_x^2 + B_y^2 + B_z^2} \). Substitute \( B_x = \frac{3D}{IA}, \ \; B_y = \frac{4D}{IA} \) into the equation to solve for \( B_z \).

Step 5: Solve for Component B_z

Substitute the expressions for \( B_x \) and \( B_y \) into the magnitude equation: \( \left( \frac{3D}{IA} \right)^2 + \left( \frac{4D}{IA} \right)^2 + B_z^2 = \left( \frac{13D}{IA} \right)^2 \). Simplify to find \( B_z^2 = \left( \frac{13D}{IA} \right)^2 - \left( \frac{3D}{IA} \right)^2 - \left( \frac{4D}{IA} \right)^2 \).\Solve for \( B_z \).

Step 6: Calculate Magnetic Field Components

Perform the calculations: \( \left( \frac{3}{IA} \right)^2 = \frac{9}{I^2A^2}, \; \left( \frac{4}{IA} \right)^2 = \frac{16}{I^2A^2} \), so sum these and subtract from \( \left( \frac{13}{IA} \right)^2 = \frac{169}{I^2A^2} \) to find \( B_z^2 \). Simplification shows \( B_z^2 = \frac{144}{I^2A^2} \), thus \( B_z = \frac{12D}{IA} \).

Step 7: Summarize Results for Magnetic Moment

Using previous findings of \( \vec{\mu} = -IA\hat{k} \), we can deduce its direction is negative \(z\)-axis and magnitude is \(IA\), given the relationship with the torque and external magnetic field.

Step 8: Confirm Answers for Magnetic Field Components

The magnetic field components that satisfy the problem conditions are \( B_x = \frac{3D}{IA} \), \( B_y = \frac{4D}{IA} \), and \( B_z = \frac{12D}{IA} \). These values correctly solve the torque equation and match the field magnitude specified.

Key Concepts

Torque in Magnetic Fields

When a loop carrying a current is placed in a magnetic field, it experiences a torque. This torque, denoted as \(\vec{\tau}\), is directly related to the magnetic moment of the loop. The concept of torque in magnetic fields can be understood through its formula:

• \(\vec{\tau} = \vec{\mu} \times \vec{B} \)

Here, \(\vec{\mu}\) is the magnetic moment, and \(\vec{B}\) is the magnetic field. The cross product implies that the torque direction is perpendicular to both the magnetic moment and the magnetic field.
The torque will try to align the magnetic moment with the magnetic field. This is analogous to the way a compass needle aligns with Earth's magnetic field, seeking a minimum energy state. In our scenario, the given torque \(D(4\hat{\imath} - 3\hat{\jmath})\) showcases how a current loop undergoes a rotational effect due to the magnetic field's components.

Magnetic Field Components

The magnetic field \(\vec{B}\) is characterized by its components in space: \(B_x\), \(B_y\), and \(B_z\). These components define the field's intensity in the \(x\), \(y\), and \(z\) directions respectively. Each component has a unique influence on the magnetic moment in terms of torque and potential energy.
The steps to find these components involve using the cross product from the torque equation:

• Given \(\vec{\tau} = D(4\hat{\imath} - 3\hat{\jmath})\)
• By relating it to \(-IA\hat{k} \times (B_x \hat{i} + B_y \hat{j} + B_z \hat{k})\), each component is determined.

Through solving the equations, we find:

• \(B_x = \frac{3D}{IA}\)
• \(B_y = \frac{4D}{IA}\)
• \(B_z = \frac{12D}{IA}\)

These components are essential to describe how the magnetic field influences both the loop's torque and alignment.

Magnetic Potential Energy

Magnetic potential energy relates to the orientation of a magnetic moment within a magnetic field. The potential energy \(U\) of a magnetic moment \(\vec{\mu}\) in a magnetic field \(\vec{B}\) is expressed as:

• \(U = - \vec{\mu} \cdot \vec{B} \)

Here, the negative sign indicates a desire for the system to reach lower energy states by aligning the moment with the field.
In our specific scenario, the negative potential energy implies that the loop is already near this stable orientation, despite resisting forces like torque that might seek to misalign it. The components \(B_x\), \(B_y\), and \(B_z\) contribute to this energy, dictating how strong the alignment tendency is in each spatial direction.
Understanding magnetic potential energy is vital for comprehending why objects like magnets and loops rotate within fields—they are seeking the lowest possible energy configuration.