91Ó°ÊÓ

A current loop is a closed circuit through which electric current flows, forming a loop-like structure, often rectangular or circular. This loop plays a crucial role in generating magnetic fields. When placed in a magnetic field, it experiences torque, which is the rotational force acting on the loop. This setup is fundamental in electromagnetism and forms the basis of many devices like motors and meters.

Key properties of a current loop include:

• Current (I): The flow of electric charge through the loop.


• Area (A): The surface area enclosed by the loop.


• Number of turns (n): Indicates how many times the wire winds around in the loop, which enhances the magnetic effect.

Each of these factors contributes to the torque experienced by the loop when placed in a magnetic field, as seen with the equation: \[ \tau = nIA \sin \theta \]where \( \tau \) is the torque and \( \theta \) is the angle between the normal to the loop and the magnetic field.

When trying to find specific angles at which a current loop's torque reaches a certain percentage of its maximum, angle calculation comes into play. The torque is dependent on the sine of the angle \( \theta \). To find angles such as when the torque is 90%, 50% or 10% of the maximum, we rearrange the torque equation:

\[ \sin \theta = \frac{\text{desired torque}}{\text{maximum torque}} \]

For a current loop, the maximum torque occurs when \( \theta = 90^\circ \), since \( \sin 90^\circ = 1 \). Therefore, to find any particular angle \( \theta \):

• Calculate the ratio of the specific torque you want to the maximum torque.


• Use the sine equation to calculate \( \sin \theta \).


• Solve for \( \theta \) using the inverse sine function.

By these steps, angles such as \( 64.16^\circ, 30^\circ, \) and \( 5.74^\circ \) are found, corresponding to 90%, 50%, and 10% of maximum torque levels, respectively.

The inverse sine function, denoted as \( \sin^{-1}(x) \) or arcsin\( (x) \), is crucial in finding angles when dealing with trigonometric equations. When you know the sine of an angle, this function helps you determine the actual angle itself.

Here's how you can use the inverse sine function efficiently:

• Given \( \sin \theta = x \), find \( \theta \) by calculating \( \sin^{-1}(x) \).


• Make sure \( x \) is within the interval \(-1 \leq x \leq 1\), since these are the bounds of the sine function.


• The result \( \theta \) will be within the range \([-90^\circ, 90^\circ]\).


• For angles outside this range, consider the periodic nature of the sine function or symmetry.

In the context of torque in a current loop, calculating \( \sin^{-1}(0.9) \), \( \sin^{-1}(0.5) \), and \( \sin^{-1}(0.1) \) helps us find angles where the torque reaches certain percentages of its maximum value, namely, around \( 64.16^\circ \), \( 30^\circ \), and \( 5.74^\circ \), respectively.