Problem 39 - College Physics

Question

\bullet A circular coil of 50 loops and diameter 20.0 \(\mathrm{cm}\) is lying flat on a tabletop, and carries a clockwise current of 2.50 A. A magnetic field of 0.450 \(\mathrm{T}\) , directed to the north and at an angle of \(45.0^{\circ}\) from the vertical down through the coil and into the tabletop is turned on. (a) What is the torque on the coil, and (b) which side of the coil (north or south) will tend to rise from the tabletop?

Step-by-Step Solution

Answer

(a) Torque is 1.25 Nm. (b) North side of the coil will rise.

Step 1: Calculate the Area of the Coil

The coil is circular, so its area can be calculated using the formula for the area of a circle: \[ A = \pi r^2 \]where \( r \) is the radius of the coil. The diameter is given as 20.0 cm, thus the radius \( r \) is 10.0 cm or 0.10 m.Substituting this into the formula gives:\[ A = \pi (0.10)^2 = \pi \times 0.01 = 0.0314 \, \text{m}^2 \]

Step 2: Determine the Magnetic Moment (μ)

The magnetic moment \( \mu \) of the coil is given by:\[ \mu = NIA \]where:- \( N = 50 \) is the number of loops,- \( I = 2.50 \) A is the current,- \( A = 0.0314 \, \text{m}^2 \) is the area we calculated.Substitute these values into the formula:\[ \mu = 50 \times 2.50 \times 0.0314 = 3.925 \, \text{Am}^2 \]

Step 3: Calculate Torque Experienced by Coil

The torque \( \tau \) on a current-carrying coil in a magnetic field is given by:\[ \tau = \mu B \sin \theta \]where:- \( \mu = 3.925 \, \text{Am}^2 \) is the magnetic moment,- \( B = 0.450 \, \text{T} \) is the magnetic field,- \( \theta = 45.0^{\circ} \) is the angle between the magnetic moment and the magnetic field.Calculate the torque:\[ \tau = 3.925 \times 0.450 \times \sin 45.0^{\circ} = 3.925 \times 0.450 \times 0.7071 = 1.2498 \, \text{Nm} \]

Step 4: Determine Direction of Coil's Movement

Using the right-hand rule, determine the direction of the magnetic force on the coil. The current direction is clockwise, and the magnetic field points north at a 45-degree angle from vertical, downward into the table. Since the magnetic moment aligns with the current direction, applying the right-hand rule reveals that the north side of the coil will feel an upward torque. Therefore, the north side will rise.

Key Concepts

Magnetic Moment

The magnetic moment (\(\mu\)) serves as a measure of the coil's ability to produce magnetic effects. You can think of it as the strength of the coil's magnetic field. For a circular coil, the magnetic moment is given by the product of the number of turns (\(N\)), the current in the coil (\(I\)), and the area (\(A\)) of the coil: \[\mu = NIA\]. In the exercise, we found that the magnetic moment of the coil with 50 turns carrying a current of 2.50 A across an area of 0.0314 \(\text{m}^2\) was 3.925 \(\text{Am}^2\). That means the coil can generate a reasonable magnetic effect due to its current and geometry. Understanding the magnetic moment helps in predicting how the coil will interact with external magnetic fields.

Magnetic Field and Torque

When a coil carrying current is placed in a magnetic field, it experiences a twisting force, known as torque. This occurs due to the interaction between the coil's magnetic moment and the external magnetic field. The amount of torque experienced by the coil can be calculated using the formula: \[\tau = \mu B \sin \theta\] where \(\mu\) is the magnetic moment, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the magnetic moment and the magnetic field. From the exercise, the torque on the coil was calculated to be 1.2498 Nm. This result shows how the coil rotates or tends to rotate under the influence of the magnetic field, which plays an important role in many devices, such as electric motors.

Circular Coil Physics

The circular coil, due to its symmetrical shape, serves as a fundamental component in numerous electromagnetic applications. Its operation relies heavily on magnetic principles. In this exercise, the circular coil, with 50 loops and a diameter of 20 cm, lies flat. This setup allows it to be influenced by external magnetic fields effectively. Key features of circular coils in physics include:

• Generation of magnetic fields when carrying current
• Interacting with external magnetic fields to produce torque
• Relying on the geometry, or number of turns and area, to determine magnetic moment

By examining these features in the exercise, you gain insight into how circular coils function in magnetic environments and their resultant movements due to torque.

Right-Hand Rule in Physics

The right-hand rule is a simple yet powerful tool to determine the direction of magnetic effects in physics. It helps us visualize how forces act on current-carrying wires. Here's how it applies:

• Point your thumb in the direction of the conventional current (positive to negative)
• Extend your fingers in the direction of the magnetic field (from the source pole to the opposite pole)
• Your palm points in the direction of the resultant force experienced by the wire or coil

In our exercise, we used the right-hand rule to determine which side of the coil would lift. Applying this rule reveals that the north side of the coil rises, indicating that the torque pushes it upward. This rule is indispensable for quickly determining the orientation of forces and movements in various physics problems.