Problem 36 - College Physics

Question

A circular coil of wire 8.6 \(\mathrm{cm}\) in diameter has 15 turns and carries a current of 2.7 \(\mathrm{A}\) . The coil is in a region where the magnetic field is 0.56 \(\mathrm{T}\) (a) What orientation of the coil gives the maximum torque on the coil, and what is this maximum torque? (b) For what orientation of the coil is the magnitude of the torque 71\(\%\) of the maximum found in part (a)?

Step-by-Step Solution

Answer

(a) Perpendicular orientation; max torque is 0.131 Nm. (b) 45.34° angle with field for 71% torque.

Step 1: Calculate the Area of the Coil

The first step is to convert the diameter of the coil into the radius. Given the diameter is 8.6 cm, the radius is half of that: \( \frac{8.6}{2} = 4.3 \) cm, or 0.043 m. The area \( A \) of the coil is found using the formula \( A = \pi r^2 \). Therefore, \( A = \pi (0.043)^2 \approx 5.8 \times 10^{-3} \) m².

Step 2: Calculate Maximum Torque

The maximum torque \( \tau_{max} \) on a coil in a magnetic field occurs when the coil is perpendicular to the field. The formula for the torque \( \tau \) is \( \tau = nIBA \), where \( n \) is the number of turns, \( I \) is the current, \( B \) is the magnetic field strength, and \( A \) is the area. Substitute the values: \( \tau_{max} = 15 \times 2.7 \times 0.56 \times 5.8 \times 10^{-3} \). After calculation, \( \tau_{max} \approx 0.131 \) Nm.

Step 3: Determine Orientation for Specific Torque

The torque for a specific orientation is given by \( \tau = \tau_{max} \sin{\theta} \). To find when \( \tau = 0.71 \tau_{max} \), equate \( \tau = 0.71 \times 0.131 \) Nm and solve for \( \theta \) using \( \sin{\theta} = 0.71 \). Since \( 0.71 \) corresponds to \( \sin{45.34^\circ} \), the coil needs to be at this angle with the magnetic field to achieve 71% of maximum torque.

Key Concepts

Magnetic Field

A magnetic field is a region in space where magnetic forces can be observed. These fields are generated by moving electric charges, such as those in a current-carrying wire.
The strength of the magnetic force experienced by an object depends on the magnitude and orientation of the field. This is typically represented by the symbol \(B\), and its unit is the Tesla (T). In the context of a coil in a magnetic field, the field exerts a force on the coil that can produce torque, causing the coil to rotate. This effect is especially pronounced when dealing with loops or coils because they consist of turns of wire that amplify the interaction with the magnetic field.
Understanding the interplay between a magnetic field and a current-carrying coil helps us in multiple applications, from electric motors to generators.

Current in a Coil

When electric current flows through a coil of wire, a magnetic field is created.
This phenomenon occurs due to the movement of charges within the wire, as the flow of electrons generates a circular magnetic field around the wire. This interaction is the foundation of electromagnetism. Within the context of the problem, our coil carries a current of 2.7 A.
The current has an impact on the resulting magnetic field strength, as well as the produced torque. The magnitude of torque on the coil is dependent on several factors, including the number of turns (or loops) in the coil, the strength of the magnetic field it is placed in, the size of the coil, and the amount of current flowing through it, based on the formula \(\tau = nIBA\). This direct relationship between current and torque means that the greater the current, the stronger the interaction with the magnetic field, and consequently, the greater the torque.

Orientation of Coil

The orientation of a coil in relation to a magnetic field is crucial in determining the torque it experiences. When the plane of the coil is perpendicular to the magnetic field lines, the torque is at its maximum.
This condition maximizes the magnetic forces acting on the current-carrying wires of the coil. Mathematically, this maximum torque \(\tau_{max}\) can be calculated by \(\tau_{max} = nIAB\), where \(n\) is the number of turns, \(I\) is the current, \(A\) is the area of the coil, and \(B\) is the magnetic field strength.
In situations where a specific percentage of this maximum torque is desired, such as 71% in our example, the orientation might need to be adjusted accordingly. This can be calculated using the sine function as \(\tau = \tau_{max} \sin{\theta}\), where \(\theta\) is the angle between the normal to the plane of the coil and the magnetic field lines.

Electric Circuits

Electric circuits are paths through which electricity flows. They consist of a network of components, including wires, resistors, and often coils, to manage the flow of current.
The covered content about coils directly relates to electric circuits as it exemplifies how components in a circuit can affect the magnetic properties and forces experienced. In electric circuits involving coils, the coil can either enhance or hinder the circuit functionality by influencing magnetic interactions.
Practical applications of circuits with coils include transformers and inductors, which utilize the properties of magnetic fields to transfer or store energy. Understanding the behavior of current in a coil and its effects is essential for designing efficient and functional electric circuits.