Question

A circular coil of wire with 20 turns and a radius of \(40.0 \mathrm{~cm}\) is laying flat on a horizontal table as shown in the figure. There is a uniform magnetic field extending over the entire table with a magnitude of \(5.00 \mathrm{~T}\) and directed to the north and downward, making an angle of \(25.8^{\circ}\) with the horizontal. What is the magnitude of the magnetic flux through the coil?

Short answer

Question: Calculate the magnitude of the magnetic flux through a circular coil of 20 turns, radius 40 cm, and placed in a uniform magnetic field of 5.00 T, directed at an angle of 25.8° downward from the horizontal. Answer: First, calculate the area of the circular coil: \(A = \pi (0.4)^2\) \(A \approx 0.5027 \, \mathrm{m}^2\) Next, find the angle between the magnetic field and the normal to the coil: \(\theta = 90^{\circ} - 25.8^{\circ}\) \(\theta = 64.2^{\circ}\) Finally, calculate the magnetic flux through the coil: \(\Phi_B = (20)(5.00)(0.5027)(\cos(64.2^{\circ}))\) \(\Phi_B \approx 50.27 \, \mathrm{Wb}\) The magnitude of the magnetic flux through the circular coil is approximately 50.27 Wb.

Step-by-step solution

Calculate the Area of the Circular Coil

The area of a circle can be calculated using the formula: \(A = \pi r^2\) Where, A = Area r = radius Given, radius r = 40.0 cm = 0.4 m (as we need to convert it to meters) Therefore, the area A of the circular coil will be: \(A = \pi (0.4)^2\)

Calculate the angle between the magnetic field and the normal to the coil

The magnetic field is making an angle of \(25.8^{\circ}\) downward from the horizontal. Thus, we need to find the angle between the magnetic field and the vertical axis (normal to the coil) which is: \(\theta = 90^{\circ} - 25.8^{\circ}\)

Calculate the Magnetic Flux through the coil

Now, we have all the required values to calculate the magnetic flux through the circular coil. Using the formula: \(\Phi_B = NBA\cos\theta\) Where, N = 20 turns B = 5.00 T A = \(\pi (0.4)^2 \,\mathrm{m}^2\) \(\theta = (90^{\circ} - 25.8^{\circ})\) We can now plug in all the values and calculate the magnetic flux: \(\Phi_B = (20)(5.00)(\pi (0.4)^2)(\cos(90^{\circ} - 25.8^{\circ}))\) Calculate the value of magnetic flux to get the magnitude of the magnetic flux through the coil.

Key concepts

Understanding the Circular Coil

A circular coil is a coil of wire that is shaped into a circle. These coils are commonly used in physics and engineering applications because of their ability to generate magnetic fields or to detect changes in magnetic flux.

In this context, the circular coil is made from wire and has 20 turns. Coil's properties such as its radius significantly impact how it interacts with any magnetic fields it encounters.

The radius of the coil given in this problem is 40 cm, which is converted to meters (0.4 m) for calculation purposes. Using the formula for the area of a circle, the area of the coil can be calculated as

• Area = \( \pi r^2 \)
• Where r = 0.4 m

This area is crucial for calculating how much magnetic flux the coil experiences when subjected to an external magnetic field.

The Role of Magnetic Field

A magnetic field is an invisible force field that exerts a force on particles that are sensitive to magnetism. Magnetic fields are vector fields, and they have direction and magnitude.

In this exercise, the magnetic field through which the coil is placed has a strength of 5.0 T (teslas), representing a strong magnetic field.

The direction of the field is also critical as it influences how the magnetic field interacts with the coil, affecting the flux through it. The problem describes the magnetic field as directed to the north and downward, making it vital to understand its interaction with the coil's orientation.

Determining this interaction is one key step towards calculating the magnetic flux, where you consider how the field angle relative to the coil impacts the resultant magnetic flux.

Angle of Inclination and its Influence

The angle of inclination, in the context of this problem, is the angle between the direction of the magnetic field and the horizontal plane. Here, this angle is given as 25.8 degrees.

However, to fully determine the magnetic flux through the coil, we require the angle between the magnetic field and the perpendicular (normal) to the surface of the coil.

This results from using cosine rule in the flux calculation. Transforming the angle of inclination involves subtracting it from 90 degrees, yielding the necessary

• \(\theta = 90^{\circ} - 25.8^{\circ}\)

By correctly determining this angle, we ensure that our calculation accurately reflects the physical scenario and how it influences the magnetic flux through the coil.

Significance of Number of Turns

The number of turns (N) in a coil determines how intensively the coil can interact with a magnetic field. More turns mean the coil can capture more of the magnetic field's energy, increasing the resultant magnetic flux.

In this exercise, the coil has 20 turns, and this value is directly employed in the formula to compute the magnetic flux. This is because the flux through one loop multiplies across the multiple turns of the coil.

When calculating magnetic flux, the formula

• \(\Phi_B = NBA\cos\theta\)

shows how the number of turns amplifies the flux. Thus, a coil with more turns generally increases the total magnetic flux the coil experiences under the same conditions, a crucial consideration in the practical design of inductive components.