Problem 7 - College Physics

Question

A closely wound rectangular coil of 80 turns has dimensions of 25.0 \(\mathrm{cm}\) by 40.0 \(\mathrm{cm}\) . The plane of the coil is rotated from a position where it makes an angle of \(37.0^{\circ}\) with a magnetic field of 1.10 \(\mathrm{T}\) to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

Step-by-Step Solution

Answer

The average emf induced in the coil is approximately 43.6 mV.

Step 1: Understand the Concept

The problem involves electromagnetic induction. When a coil rotates in a magnetic field, an electromotive force (emf) is induced. The emf can be calculated using Faraday's law, which states that the average emf induced is related to the change in magnetic flux over time.

Step 2: Calculate the Area of the Coil

Calculate the area of the coil. The coil dimensions are 25.0 cm by 40.0 cm, so we find the area:\[A = 25.0 \, \text{cm} \times 40.0 \, \text{cm} = 1000 \, \text{cm}^2 = 0.100 \, \text{m}^2\]

Step 3: Determine the Initial and Final Magnetic Flux

The initial angle between the magnetic field \(B\) and the normal to the coil is \(37.0^{\circ}\). The initial magnetic flux \(\Phi_i\) through the coil is:\[\Phi_i = N \cdot B \cdot A \cdot \cos(\theta_i) = 80 \cdot 1.10 \, \text{T} \cdot 0.100 \, \text{m}^2 \cdot \cos(37.0^{\circ})\]Calculate \(\cos(37.0^{\circ})\) and then \(\Phi_i\).The final position is perpendicular to the field, therefore, \(\theta_f = 0^{\circ}\), and the final magnetic flux \(\Phi_f\) is:\[\Phi_f = N \cdot B \cdot A \cdot \cos(0^{\circ}) = 80 \cdot 1.10 \, \text{T} \cdot 0.100 \, \text{m}^2 \cdot 1\]

Step 4: Calculate the Change in Magnetic Flux

Compute the change in magnetic flux:\[\Delta \Phi = \Phi_f - \Phi_i = (80 \cdot 1.10 \cdot 0.100 \cdot 1) - (80 \cdot 1.10 \cdot 0.100 \cdot \cos(37.0^{\circ}))\]

Step 5: Use Faraday's Law to Find the Average EMF

According to Faraday's law of induction, the average induced emf \(\mathcal{E}_{\text{avg}}\) over a time interval \(\Delta t\) is given by:\[\mathcal{E}_{\text{avg}} = -\frac{\Delta \Phi}{\Delta t}\]Substitute \(\Delta \Phi\) from Step 4 and \(\Delta t = 0.0600 \text{ s}\) into the equation to find \(\mathcal{E}_{\text{avg}}\).

Key Concepts

Faraday's law of induction

Faraday's law of induction is a fundamental principle of electromagnetism, discovered by Michael Faraday. It describes how a change in magnetic flux through a closed circuit induces an electromotive force (emf) in that circuit.
The formula is given by:

• \(\mathcal{E}_{\text{avg}} = -\frac{\Delta \Phi}{\Delta t} \)

Where \(\Delta \Phi\) is the change in magnetic flux and \(\Delta t\) is the time interval over which the change occurs. The negative sign indicates that the direction of the induced emf opposes the change in flux, following Lenz's Law.
This law plays a critical role in the operation of many electrical devices, such as transformers, electric generators, and motors, which utilize the principle of electromagnetic induction.

magnetic flux

Magnetic flux quantifies the number of magnetic field lines passing through a given area. It is a product of the magnetic field strength, the area through which the field lines pass, and the cosine of the angle between the field lines and the perpendicular to the surface.
The equation is:

• \( \Phi = B \cdot A \cdot \cos(\theta) \)

Here, \(B\) is the magnetic field, \(A\) is the area, and \(\theta\) is the angle between the magnetic field and the normal to the surface.
Magnetic flux is measured in webers (Wb), and it provides an insight into how strong and extensive a magnetic field is in a particular area.

induced emf

The induced electromotive force (emf) is a result of changes in magnetic flux through a circuit or coil. According to Faraday's law, the induced emf arises when there is a variation in the magnetic environment around a conductor.
The key factors affecting the induced emf include:

• The rate of change of magnetic flux \(\Delta \Phi/\Delta t\)
• The number of turns in the coil \(N\)

The unit for emf is volts, and it effectively acts as a voltage source, driving current in a closed loop. Induced emf is essential in generating electricity in power plants by converting mechanical energy into electrical energy.

rectangular coil

A rectangular coil is a loop of wire shaped in a rectangle, often used in experiments involving electromagnetic fields.
Key characteristics include:

• Its shape, which makes calculations simpler using rectangular coordinates.
• The area of the coil, which affects the magnitude of the induced emf. For example, the area \(A \) of a coil with dimensions 25.0 cm by 40.0 cm is \(0.100 \, \text{m}^2.\)
• The number of turns (N) in the coil, enhancing the flux change effect, \(N = 80\) in this scenario.

Understanding the coil's properties helps in accurately applying mathematical equations related to electromagnetic induction, making it a common component in laboratory setups and equipment.

magnetic field

A magnetic field is a vector field surrounding magnetic materials and electric currents. It exerts a magnetic force on moving charged particles and magnetic dipoles.
Important characteristics include:

• Its representation by field lines, indicating the direction and strength of the field.
• Magnetic field strength is denoted by \(B\), measured in teslas (T); in this exercise, \(B = 1.10\, \text{T}.\)
• A magnetic field can change over time or space, which is crucial for inducing emf in a coil.

Understanding magnetic fields is vital for various technologies, such as MRI machines in healthcare or MagLev trains in transport, leveraging their properties for innovative applications. A magnetic field's interaction with conductive materials is the cornerstone of electromagnetic induction processes.