Problem 41 - Master Resource Book in JEE Main Physics

Question

A square coil of side \(10 \mathrm{~cm}\) consists of 20 turns and carries a current of \(12 \mathrm{~A}\). The coil is suspended vertically and the normal to the plane of the coil makes an angle of \(30^{\circ}\) with the direction of a uniform horizontal magnetic field of magnitude \(0.80 \mathrm{~T}\). What is the magnitude of torque experienced by the coil? (a) \(0.96 \mathrm{~N}-\mathrm{m}\) (b) \(2.06 \mathrm{~N}-\mathrm{m}\) (c) \(0.23 \mathrm{~N}-\mathrm{m}\) (d) \(1.36 \mathrm{~N}-\mathrm{m}\)

Step-by-Step Solution

Answer

The magnitude of the torque is 0.96 N-m, answer (a).

Step 1: Identify the Parameters Involved

To solve this problem, we need to first identify and list the relevant parameters: the side of the square coil is 10 cm, which means each side measures 0.1 m. The coil has 20 turns, carries a current of 12 A, is placed in a uniform magnetic field of 0.80 T, and the angle between the normal to the coil and the magnetic field is 30°.

Step 2: Calculate the Area of the Coil

Since the coil is square-shaped, its area can be calculated by squaring its side length. Thus, \( A = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \).

Step 3: Understand the Torque Formula

The torque \( \tau \) experienced by a coil in a magnetic field is given by the formula: \( \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \), where \( n \) is the number of turns, \( I \) the current, \( A \) the area of the coil, \( B \) the magnetic field strength, and \( \theta \) the angle between the magnetic field and the normal to the coil.

Step 4: Plug Values into the Formula

Using the formula, we substitute the given and calculated values: \( \tau = 20 \times 12 \, \text{A} \times 0.01 \, \text{m}^2 \times 0.80 \, \text{T} \times \sin(30^{\circ}) \). Since \( \sin(30^{\circ}) = 0.5 \), we have \( \tau = 20 \times 12 \times 0.01 \times 0.80 \times 0.5 \).

Step 5: Calculate the Torque

Performing the multiplication gives us \( \tau = 20 \times 12 \times 0.01 \times 0.80 \times 0.5 = 0.96 \, \text{N-m} \).

Key Concepts

Magnetic Field

A magnetic field is an invisible force field produced by electrically charged objects and magnets. It's responsible for the magnetic forces that we observe. Understanding magnetic fields is essential when discussing torque effects on currents. In the context of this exercise, a uniform magnetic field with a magnitude of 0.80 T affects the coil. This magnetic field interacts with the electric current within the coil to create torque. Key features of a magnetic field include:

• Magnitude: This is the strength of the magnetic field, measured in teslas (T). Here, it's 0.80 T.
• Direction: Magnetic fields have a specific direction, which can be visualized as field lines going from the north to the south pole of a magnet.
• Interaction with current: When an electric current flows through a conductor in a magnetic field, it experiences a force. This force is perpendicular to both the direction of magnetic field lines and the flow of current.

Understanding these concepts helps in predicting and calculating the effects, such as torque, which occur due to the magnetic field.

Current in Coils

When electricity flows through a coil, it generates a magnetic field around the coil. This phenomenon is pivotal in calculating the torque exerted by the coil in a magnetic field. In our exercise, the square coil consists of 20 turns carrying a current of 12 A. As each turn of the coil contributes to the overall magnetic field, the total effect is amplified by the number of turns. Factors influencing the current's effect include:

• Number of Turns (n): More turns mean a stronger magnetic field and more torque for the same amount of current.
• Current (I): Higher current results in a stronger magnetic field.
• Coil Area (A): Larger coils with more area experience more torque as more area interacts with the magnetic field.

Each of these aspects contributes to the final torque experienced by the coil, as seen in the formula for torque. Therefore, understanding the relationship between current, magnetic fields, and coil geometry is essential in electric and magnetic field analyses.

Trigonometric Functions

In this exercise, the angle plays a crucial role. The normal to the coil makes an angle with the direction of the magnetic field, impacting the torque. Trigonometry comes into play largely due to the sine function, found in the torque formula: \( \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \).Why trigonometry matters:

• Sine of the angle: This function determines the component of the magnetic field interacting with the coil. For example, when \( \theta = 30^{\circ} \), \( \sin(30^{\circ}) = 0.5 \), which reduces the effective component of the field.
• Angle of interaction (\( \theta \)): The angle directly affects torque. When the angle is 90°, the sine function reaches its maximum value of 1, maximizing torque.
• Geometric interpretation: Sine values help represent the effective perpendicularity of forces acting on the plane of the coil.

By understanding trigonometric functions and their implications, you can better grasp how angles influence the behavior of physical systems in magnetic fields. This knowledge directly applies to optimizing the design and function of devices utilizing electromagnets.