Problem 65 - University Physics with Modern Physics

Question

A magnetic field exerts a torque \(\tau\) on a round current carrying loop of wire. What will be the torque on this loop (in terms of \(\tau\) if its diameter is tripled?

Step-by-Step Solution

Answer

The torque will increase to 9 times the original torque, \( 9\tau \).

Step 1: Understanding the Relationship

Torque \( \tau \) on a current carrying loop in a magnetic field can be given by the formula \( \tau = nIAB\sin\theta \), where \( n \) is the number of turns, \( I \) is the current, \( A \) is the area of the loop, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the magnetic field and the normal to the plane of the loop. For a single circular loop perpendicular to the field, \( \theta = 90^\circ \) and \( \sin \theta = 1 \). Thus, \( \tau = IAB \).

Step 2: Calculating the Area with Tripled Diameter

The area \( A \) of the loop is \( \pi r^2 \), where \( r \) is the radius. If the diameter is tripled, the new radius \( r' \) is \( 3r \). Thus, the new area \( A' \) becomes \( \pi (3r)^2 = 9\pi r^2 \).

Step 3: Substituting New Values

Substitute the new area back into the torque equation: The new torque \( \tau' = I(9\pi r^2)B = 9(I\pi r^2 B) = 9\tau \).

Step 4: Conclusion Based on the Calculation

The torque on the loop when the diameter is tripled is nine times the original torque \( \tau \). Therefore, the new torque \( \tau' \) is \( 9\tau \).

Key Concepts

Current Carrying Loop

A current carrying loop is a fundamental concept in electromagnetism and it involves a loop of wire through which electrical current flows. This current creates a magnetic field around the loop. When this loop is placed within another external magnetic field, it experiences a force due to the interaction of these magnetic fields. This force is described as a torque that acts on the loop. The direction of this torque can be determined by the right-hand rule, which helps understand how magnetic fields and currents interact. If you curl the fingers of your right hand in the direction of the current flow, your thumb will point towards the direction of the torque. The effect of a magnetic field on a current carrying loop is not only crucial in academic studies but also has practical applications. Motors and generators use these principles to convert electrical energy into mechanical work, and vice versa. Thus, understanding how a current carrying loop operates is essential for both theoretical physics and engineering applications.

Torque Formula

The torque exerted by a magnetic field on a current carrying loop can be calculated using a specific formula. This formula is given by \( \tau = nIAB\sin\theta \). Let's break down what each component represents:

• \( n \) is the number of turns in the loop. Increasing the number of turns increases the torque.
• \( I \) is the current flowing through the loop. A higher current boosts the magnetic interaction and, consequently, the torque.
• \( A \) stands for the area of the loop. A larger area means that more magnetic field lines can interact with the loop, resulting in a larger torque.
• \( B \) is the magnetic field strength that the loop is exposed to. A stronger magnetic field imparts a greater force on the loop.
• \( \theta \) is the angle between the magnetic field and the normal (a perpendicular line) to the plane of the loop. The term \( \sin\theta \) indicates that the torque is maximized when the field is perpendicular to the loop (\( \theta = 90^\circ \)).

Analyzing the torque formula helps illustrate how each variable affects the magnitude of the torque. When the conditions achieve maximum torque, such as when the loop is aligned perfectly perpendicularly to the magnetic field, the mathematical expression simplifies to \( \tau = nIAB \). Understanding this formula is key to solving problems involving torque in magnetic fields.

Diameter and Area Relationship

The relationship between the diameter and area of a loop is a crucial aspect in understanding how changes in dimension affect the torque experienced by a current carrying loop.The area of a circular loop is calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius of the loop. If you know the diameter \( d \) of a loop, remember that it is twice the radius (\( r = \frac{d}{2} \)). Therefore, when you calculate the area in terms of the diameter, the expression becomes \( A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4} \). Consider what happens if the diameter is modified, such as being tripled. The new diameter \( d' \) is \( 3d \), meaning the new radius \( r' \) becomes \( \frac{3d}{2} \). Thus, the new area \( A' = \pi (\frac{3d}{2})^2 = 9\frac{\pi d^2}{4} \), showing the area increases as a square of the change in diameter factor. Understanding the math behind the relationship between diameter and area is key, especially in problems involving areas of loops and their resulting torque in magnetic fields. When you increase the diameter of the loop, you drastically increase its area, impacting the torque greatly because torque is directly proportional to the area in the formula \( \tau = IAB \). A tripled diameter, therefore, results in a nine-fold increase in torque, as the area increases by a factor of nine.