Problem 35 - College Physics

Question

\(\cdot\) The plane of a 5.0 \(\mathrm{cm}\) by 8.0 \(\mathrm{cm}\) rectangular loop of wire is parallel to a 0.19 T magnetic field, and the loop carries a cur- rent of 6.2 \(\mathrm{A}\) . (a) What torque acts on the loop? (b) What is the magnetic moment of the loop?

Step-by-Step Solution

Answer

Torque is 0.004712 Nm. Magnetic moment is 0.0248 Am sup{2}.

Step 1: Understand the Problem

We need to find the torque acting on a rectangular loop of wire in a magnetic field and the magnetic moment of the loop. The given values are: Width of the loop = 5.0 cm, Length of the loop = 8.0 cm, Magnetic field strength ( B ) = 0.19 T, Current ( I ) = 6.2 A.

Step 2: Calculate the Area of the Loop

The area (A) of the rectangular loop is calculated as:\[ A = ext{Length} \times ext{Width} \]Convert dimensions to meters, 5.0 cm = 0.05 m and 8.0 cm = 0.08 m. Thus,\[ A = 0.08 \, m \times 0.05 \, m = 0.004 \, m^2 \]

Step 3: Find the Magnetic Moment

The magnetic moment (\mu) of the loop is given by:\[ \mu = I \times A \]Substitute the known values:\[ \mu = 6.2 \, A \times 0.004 \, m^2 = 0.0248 \, Am^2 \]

Step 4: Calculate the Torque on the Loop

The torque (\tau) acting on the loop is given by:\[ \tau = \mu \times B \times \sin(\theta) \]Since the plane of the loop is parallel to the magnetic field, \(\theta = 90^\circ\) and \( \sin(90^\circ) = 1 \). Thus,\[ \tau = 0.0248 \, Am^2 \times 0.19 \, T \times 1 = 0.004712 \, Nm \]

Key Concepts

Torque on a Loop

The concept of torque on a current-carrying loop in a magnetic field is essential to grasp when dealing with electromagnetic systems. Torque, in this context, is the tendency of a force to rotate an object about an axis. For a rectangular loop with sides of lengths 5.0 cm and 8.0 cm, placed parallel to a magnetic field of 0.19 Tesla, we must consider how the magnetic field interacts with the current flowing through the wire loop.

Torque (\( \tau \)) on a loop is derived from the equation:\[ \tau = \mu \times B \times \sin(\theta) \]where \( \mu \) is the magnetic moment, \( B \) is the magnetic field, and \( \theta \) is the angle between \( \mu \) and \( B \).

• If the loop is parallel to the magnetic field, as given, \( \theta \) is \( 90^\circ \)
• This makes \( \sin(90^\circ) = 1 \), simplifying the torque calculation

Ultimately, this gives us an intuitive explanation. The torque attempts to align the loop's magnetic moment with the applied magnetic field. This is a critical factor in devices like electric motors, where torque translates into mechanical rotation.

Rectangular Loop in Magnetic Field

A rectangular loop in a magnetic field presents an intriguing scenario for examining concepts such as magnetic flux, torque, and loop behavior under electromagnetic forces. A loop, with a width of 5.0 cm and a length of 8.0 cm, is placed in a uniform magnetic field of 0.19 Tesla. The orientation becomes crucial here.

Since the loop is perfectly parallel to the field, it experiences the maximum interaction as governed by its physical dimensions and current.

• Area Calculation: The area \( A \) of the loop can be calculated using \( A = \text{Length} \times \text{Width} \).
• Convert this to meters for consistency, leading to an area of \( 0.004 \, m^2 \).
• The loop's effective exposure to the magnetic field influences its electromagnetic properties like torque and magnetic moment.

This setup is foundational in electromagnetic applications, as it highlights how altering angles or dimensions affects the resulting forces on the loop.

Current-Carrying Loop

An understanding of a current-carrying loop in a magnetic field is pivotal to analyzing systems such as generators and electric motors. Consider a loop that carries a 6.2 A current as given in the problem. This current influences the magnetic moment and, consequently, the electromagnetic interactions with the magnetic field.

To calculate the magnetic moment \( \mu \), we utilize the formula:\[ \mu = I \times A \]

• Here, \( I \) is the current (6.2 A), and \( A \) is the area (0.004 \, m^2).
• This yields a magnetic moment of \( 0.0248 \, Am^2 \).

The magnetic moment serves as a measure of the strength and orientation of the loop's magnetic field. When placed in an external magnetic field, this moment interacts dynamically, seeking alignment with the external field.
This principle is what enables motors to convert electrical energy into mechanical movement, manifesting the practical importance of understanding current-carrying loops in magnetic fields.