Problem 40 - Physics

Question

A coil carries a current and experiences a torque due to a magnetic field. The value of the torque is \(80.0 \%\) of the maximum possible torque. (a) What is the smallest angle between the magnetic field and the normal to the plane of the coil? (b) Make a drawing, showing how this coil would be oriented relative to the magnetic field. Be sure to include the angle in the drawing.

Step-by-Step Solution

Answer

The angle is approximately \(53.13^\circ\).

Step 1: Understand the relationship between torque and angle

The torque \( \tau \) experienced by a coil in a magnetic field is given by \( \tau = \tau_{max} \sin \theta \), where \( \tau_{max} \) is the maximum possible torque and \( \theta \) is the angle between the normal to the plane of the coil and the magnetic field. In this problem, \( \tau = 0.8 \times \tau_{max} \).

Step 2: Set up the equation

Since \( \tau = 0.8 \tau_{max} \), we can set the equation \( 0.8 = \sin \theta \). We need to find the angle \( \theta \) such that this equation holds true.

Step 3: Solve for the angle \( \theta \)

To find \( \theta \), take the inverse sine (arcsine) of both sides: \[ \theta = \arcsin(0.8) \]Calculate \( \theta \) using a calculator to find:\[ \theta \approx 53.13^\circ \]

Step 4: Create a diagram

Draw the coil as a flat loop or circle. Draw a line perpendicular to the plane of the coil to represent the normal. Then, show the magnetic field as a vector. The angle formed between the normal and the magnetic field vector should be \( 53.13^\circ \).

Key Concepts

Coil in Magnetic Field

When a coil carrying an electric current is placed in a magnetic field, it experiences a force known as magnetic torque. This torque can cause the coil to rotate. The fundamental idea is that as current flows through the coil, a magnetic moment is generated, interacting with the external magnetic field. This creates a push or pull effect on the coil.
The torque (\( \tau \)) applied to the coil is determined by three main factors:

• The strength of the magnetic field (\( B \)).
• The current (\( I \)) flowing through the coil.
• The area of the coil (\( A \)).

The formula for calculating the torque is given by: \( \tau = nIAB \sin \theta \), where \( n \) is the number of turns in the coil and \( \theta \) is the angle between the normal to the coil and the magnetic field. This highlights the dependence of torque on both physical dimensions and orientation of the coil relative to the field.
The coil will experience maximum torque when the plane of the coil is perpendicular to the magnetic field, which occurs when the angle \( \theta \) is \( 90^\circ \). Understanding these principles is crucial for applications like electric motors and generators, where precise control of coil orientation in magnetic fields is required.

Angle in Magnetic Fields

The angle between the magnetic field and the normal to the plane of the coil plays a critical role in determining the magnetic torque experienced by the coil. This angle, \( \theta \), dictates how effectively the magnetic field can interact with the coil's magnetic moment.
The equation for torque, \( \tau = \tau_{max} \sin \theta \), demonstrates this relationship. Here, \( \tau_{max} \) represents the maximum possible torque, achieved when \( \theta = 90^\circ \).
In the given exercise, the torque experienced is 80% of the maximum torque, implying that \( \sin \theta = 0.8 \). To find \( \theta \), use the inverse sine function (arcsine):

• \( \theta = \arcsin(0.8) \approx 53.13^\circ \)

This calculated angle indicates the inclination required for the normal of the coil relative to the magnetic field to achieve the specified torque. Within an educational setting, it's helpful to sketch this setup to visualize the angle clearly.

Maximum Torque

Maximum torque occurs when the coil is perfectly aligned such that its plane is perpendicular to the magnetic field. At this point, the angle \( \theta \) between the normal to the coil and the magnetic field is \( 90^\circ \). Hence, the sine of this angle is 1, maximizing the torque in the equation:\( \tau = \tau_{max} \sin \theta \).
For practical purposes, achieving maximum torque is critical in devices such as electric motors. When the orientation of the coil is perpendicular to the magnetic field, the device performs efficiently since all available magnetic force is utilized to produce motion.
To adjust a coil for maximum torque, consider:

• Rotating the coil until its plane aligns perpendicularly to the magnetic field.
• Ensuring no slippage or misalignment occurs, which could reduce torque.

By comprehending the conditions for maximum torque, one gains valuable insights into the optimal operation of various electromagnetic devices used in technology and engineering.