Problem 6 - College Physics

Question

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 \(\mathrm{cm}^{2}\) is rotated from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The rotation takes 0.040 s. The earth's magnetic field at the location of the laboratory is \(6.0 \times 10^{-5} \mathrm{T.}\) (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

Step-by-Step Solution

Answer

(a) Initial flux: \(7.2 \times 10^{-8} \mathrm{Wb}\), final flux: \(0 \mathrm{Wb}\). (b) Average emf: \(3.6 \times 10^{-4} \mathrm{V}\).

Step 1: Understand the Problem

We need to calculate two things: the magnetic flux before and after the coil is rotated, and the average electromotive force (emf) induced in the coil during the rotation.

Step 2: Calculate Initial Magnetic Flux

The magnetic flux \( \Phi \) through the coil is given by \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field, \( A \) is the area, and \( \theta \) is the angle between the field and the normal to the coil. Initially, the coil is perpendicular to the field, so \( \theta = 0 \) and \( \cos(0) = 1 \). Thus, \( \Phi_{initial} = (6.0 \times 10^{-5} \mathrm{T}) \times (12 \times 10^{-4} \mathrm{m}^2) = 7.2 \times 10^{-8} \mathrm{Wb} \).

Step 3: Calculate Final Magnetic Flux

After the rotation, the coil's plane is parallel to the magnetic field, so \( \theta = 90^{\circ} \) and \( \cos(90^{\circ}) = 0 \). Therefore, the final magnetic flux \( \Phi_{final} = 0 \mathrm{Wb} \).

Step 4: Calculate Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is the difference between initial and final flux: \( \Delta \Phi = \Phi_{final} - \Phi_{initial} = 0 - 7.2 \times 10^{-8} \mathrm{Wb} = -7.2 \times 10^{-8} \mathrm{Wb} \).

Step 5: Compute Average Induced EMF

The average induced emf \( \mathcal{E} \) is given by Faraday's Law: \( \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \), where \( N \) is the number of turns and \( \Delta t \) is the time of rotation. Plug in \( N = 200 \), \( \Delta \Phi = -7.2 \times 10^{-8} \mathrm{Wb} \), and \( \Delta t = 0.040 \mathrm{s} \): \( \mathcal{E} = -200 \frac{-7.2 \times 10^{-8}}{0.040} = 3.6 \times 10^{-4} \mathrm{V} \).

Key Concepts

Faraday's Law

Faraday's Law is a fundamental principle that describes how electricity can be generated from a changing magnetic field. It states that an electromotive force (emf) is induced in a circuit when there is a change in magnetic flux. This change can be due to variations in the strength of the magnetic field, the shape of the circuit, or its orientation relative to the field.

The magnitude of the induced emf is proportional to the rate at which the magnetic flux changes. Mathematically, Faraday's Law is expressed as:

• \( \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \)

where \( \mathcal{E} \) is the induced emf, \( N \) is the number of turns in the coil, \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the time period over which the change occurs. The negative sign indicates the direction of the induced emf is such that it opposes the change in flux (Lenz's Law).

Understanding this law helps in recognizing how generators produce electricity and is pivotal in technologies where magnetic fields are used to induce currents.

Induced EMF

Induced emf refers to the voltage generated by changing the magnetic environment of a coil. It arises when a coil rotates in a magnetic field or when the magnetic field around it is time-varying. This rotating or changing magnetic field influences the electrons in the coil, causing them to move, and thus generating an electromotive force.

In the context of the problem, the coil experiences a change in orientation from perpendicular to parallel with the Earth's magnetic field. This changes the magnetic flux through the coil, resulting in an induced emf. The formula \( \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \), derived from Faraday's Law, gives the quantitative measure of this induced emf.

• For a coil with multiple turns, even small changes in magnetic flux can lead to significant emf because the effect is multiplied by the number of turns, \( N \).
• The faster the change occurs (smaller \( \Delta t \)), the larger the induced emf.

Induced emf plays crucial roles in various devices such as electric generators, transformers, and inductors, demonstrating its importance in both theoretical physics and practical electrical engineering.

Coil Rotation

Coil rotation is integral to the concept of electromagnetic induction. When a coil rotates within a magnetic field, it results in a change in the angle between the coil and the field lines, altering the magnetic flux through the coil. This change in flux induces an electromotive force, as described by Faraday's Law.

In the exercise, the coil begins perpendicular to the Earth's magnetic field and rotates to become parallel. Initially, the magnetic flux is maximized because the coil's plane is fully exposed to the magnetic field. As the coil rotates to a parallel position, the flux through it drops to zero, completing a cycle of flux change.

• The speed of rotation affects the rate of flux change and consequently the magnitude of the induced emf.
• Direction and angle of rotation critically influence the path and speed of the change in magnetic flux.

Coil rotation is a crucial mechanism in generators, where mechanical energy is converted into electrical energy through electromagnetic induction. It is the principle behind many machines and devices that harness rotational motion for electricity generation.

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is represented by field lines that emanate from the magnetic source, providing a vector field. The density of these lines indicates the strength of the field; thicker lines show stronger fields.

In this problem, the Earth's magnetic field provides the environment for the coil's rotation. Its strength, denoted as \( B \) in physics, is measured in teslas (T). The Earth's magnetic field is relatively weak, around \( 6.0 \times 10^{-5} \mathrm{T} \), but sufficient to illustrate concepts of electromagnetic induction in laboratory experiments.

• Magnetic fields are fundamental to the operation of motors, magnetic storage media, and beyond.
• The orientation of a magnetic field relative to a coil affects the magnetic flux and thus, any induced emf in the coil.

Understanding magnetic fields is essential in physics and engineering, providing insight into both natural phenomena and man-made technologies. These fields are pivotal in medical imaging, telecommunications, and energy production.