Problem 57 - Fundamentals of Physics

Question

A circular coil of 160 turns has a radius of \(1.90 \mathrm{~cm} .\) (a) Calculate the current that results in a magnetic dipole moment of magnitude \(2.30 \mathrm{~A} \cdot \mathrm{m}^{2}\). (b) Find the maximum magnitude of the torque that the coil, carrying this current, can experience in a uniform \(35.0 \mathrm{mT}\) magnetic field.

Step-by-Step Solution

Answer

(a) Current \( I = 1.54 \; \mathrm{A} \). (b) Maximum torque \( \tau_{\text{max}} = 0.0805 \; \mathrm{Nm} \).

Step 1: Understand the Magnetic Dipole Moment Formula

The magnetic dipole moment \( \mu \) of a coil is given by the formula \( \mu = nIA \), where \( n \) is the number of turns, \( I \) is the current, and \( A \) is the area of the coil. Our task is to find the current \( I \).

Step 2: Calculate the Area of the Coil

The area \( A \) of a circular coil is given by \( A = \pi r^2 \). Here, the radius is \( r = 1.90 \) cm which we convert to meters: \( r = 0.019 \; \mathrm{m} \). Thus, \( A = \pi (0.019)^2 \; \mathrm{m}^2 \).

Step 3: Calculate the Current

Rearrange the magnetic dipole moment formula \( \mu = nIA \) to find the current: \( I = \frac{\mu}{nA} \). Substitute \( \mu = 2.30 \; \mathrm{A \cdot m^2} \), \( n = 160 \), and the area (computed previously). Calculate \( I \).

Step 4: Compute the Torque

The torque \( \tau \) experienced by the coil in a magnetic field is given by \( \tau = \mu B \sin\theta \). The maximum torque occurs when \( \sin\theta = 1 \), thus \( \tau_{\text{max}} = \mu B \). Substitute \( \mu = 2.30 \; \mathrm{A \cdot m^2} \) and \( B = 35.0 \times 10^{-3} \; \mathrm{T} \) to find the maximum torque.

Key Concepts

Current Calculation

To determine the current required for a given magnetic dipole moment in a coil, we start by revisiting an important relationship in electromagnetism. This relationship connects the magnetic dipole moment \( \mu \) with the number of turns in the coil \( n \), the current \( I \), and the area \( A \) through the formula: \[\mu = nIA\]Here, each component plays a crucial role.

• The **magnetic dipole moment** \( \mu \) represents the strength and direction of a magnetic source—here, our coil's magnetic strength.
• The **number of turns** \( n \) in the coil amplifies its magnetic properties proportionally.
• The **area** \( A \) of the coil affects how much magnetic field it can generate.

Given values: - Magnetic dipole moment \( \mu = 2.30 \, \mathrm{A \cdot m^2} \)- Number of turns \( n = 160 \)- Radius of the coil \( r = 1.90 \, \mathrm{cm} = 0.019 \, \mathrm{m} \) First, the area of the coil is calculated as:\[A = \pi r^2 = \pi (0.019)^2 \, \text{m}^2\]Next, rearrange the magnetic dipole formula to solve for the current \( I \): \[I = \frac{\mu}{nA}\]Substitute the known values to determine the current, ensuring all units align correctly. This current sets up the necessary magnetic dipole moment for the coil.

Torque in Magnetic Field

The torque experienced by a magnetic dipole in an external magnetic field is an important concept. Torque is the rotational force that the magnetic field exerts on the coil, and it depends on several factors. The formula to calculate the torque \( \tau \) is: \[\tau = \mu B \sin\theta\]Where:

• \( \mu \) is the magnetic dipole moment.
• \( B \) is the magnetic field strength.
• \( \theta \) is the angle between the magnetic field and the normal to the plane of the coil.

For maximum torque, the sine term reaches its maximum value of 1, which occurs when \( \theta = 90^\circ \). The formula for maximum torque then simplifies to: \[\tau_{\text{max}} = \mu B\]Substitute \( \mu = 2.30 \, \mathrm{A \cdot m^2} \) and \( B = 35.0 \times 10^{-3} \, \mathrm{T} \) into this equation. This will give you the maximum torque that the coil could experience under optimal conditions.

Circular Coil Properties

A circular coil is a fundamental component in various electro-mechanical systems. Its properties significantly influence how it interacts with magnetic fields. Here's a breakdown of the key properties:- **Number of Turns (n):** This indicates how many loops of wire the coil contains. More turns mean a stronger magnetic effect, as each loop contributes to the overall magnetic dipole moment.- **Radius (r):** The radius of the coil affects its area \( A \), which is critical for calculating both the current to achieve a certain dipole moment and how the coil immerses in the magnetic field. \[A = \pi r^2\] A larger area can contribute to a stronger magnetic field interaction.- **Material:** Though not explicitly stated in the formulae, the material of the coil affects its electrical resistance and the resultant current flow. These properties work together to define the coil's ability to generate and respond to magnetic forces. In applications, understanding these properties enables better design and use for achieving desired electromagnetic behavior.