Problem 2 - University Physics with Modern Physics

Question

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm\(^2\) is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is 6.0 \(\times\) 10\(^{-5}\) 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 is 7.2 x 10^-7 T·m^2; final flux is 0. (b) Average emf is 3.6 x 10^-3 V.

Step 1: Understand the Problem Variables

We need to calculate two main things: the magnetic flux before and after the rotation, and the average electromotive force (emf) induced in the coil. We have the following values given:- Number of turns, \(N = 200\)- Area of the coil, \(A = 12\, \text{cm}^2 = 12 \times 10^{-4} \, \text{m}^2\)- Earth's magnetic field, \(B = 6.0 \times 10^{-5} \, \text{T}\)- Rotation time, \(t = 0.040 \, \text{s}\).

Step 2: Calculate Initial Magnetic Flux

The magnetic flux (\(\Phi\)) through the coil when the plane is perpendicular to the magnetic field is calculated using the formula \(\Phi = B \cdot A \cdot \cos(\theta)\). Initially, the plane is perpendicular, thus \(\theta = 0^\circ\) and \(\cos(0^\circ) = 1\). So, \[\Phi_{\text{initial}} = B \cdot A = 6.0 \times 10^{-5} \, \text{T} \times 12 \times 10^{-4} \, \text{m}^2 = 7.2 \times 10^{-7} \, \text{T}\cdot\text{m}^2.\]

Step 3: Calculate Final Magnetic Flux

After rotation, the plane of the coil is parallel to the magnetic field, meaning \(\theta = 90^\circ\). Here, \(\cos(90^\circ) = 0\). Therefore, the magnetic flux becomes:\[\Phi_{\text{final}} = B \cdot A \cdot \cos(90^\circ) = 0.\]

Step 4: Calculate Average Induced EMF

The average emf induced in the coil can be found using Faraday's law of electromagnetic induction, which states \(\epsilon = -N \frac{\Delta\Phi}{\Delta t}\). The change in magnetic flux \(\Delta\Phi\) is \(\Phi_{\text{final}} - \Phi_{\text{initial}} = 0 - 7.2 \times 10^{-7} \, \text{T}\cdot\text{m}^2 = -7.2 \times 10^{-7} \, \text{T}\cdot\text{m}^2\). Thus,\[\epsilon = -200 \times \frac{-7.2 \times 10^{-7}}{0.040} = 3.6 \times 10^{-3} \, \text{V}.\]

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental principle in physics where a change in magnetic environment induces an electromotive force (emf) in a conductor. This phenomenon is the foundation for the working of generators and transformers.
It occurs when there is a change in magnetic flux through a surface. According to electromagnetic induction:

• The motion of a magnetic field relative to a coil or conductor can create an emf.
• The strength of the induced emf depends on the rate at which the magnetic field changes.
• This process can generate electrical currents in loops if they are part of a closed circuit.

In the context of our exercise, electromagnetic induction explains how the rotation of the coil in Earth's magnetic field generates an emf. As the coil rotates from perpendicular to parallel, the change in orientation causes a variation in magnetic flux through the coil, which results in the production of emf.

Faraday's Law

Faraday's Law of Electromagnetic Induction quantitatively describes how an electric field is induced by changing magnetic conditions. According to Faraday's law:
"The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit."

• The law is mathematically expressed as: \(\epsilon = -N \frac{\Delta\Phi}{\Delta t}\), where \(\epsilon\) is emf, \(N\) is the number of turns, \(\Delta\Phi\) is the change in magnetic flux, and \(\Delta t\) is the time over which this change occurs.
• The negative sign in Faraday's law is due to Lenz's Law, which states that the direction of induced emf will oppose the change in flux.

In our example, we used Faraday's Law to calculate the average emf induced in the coil due to its rotation in the Earth's magnetic field.

Magnetic Field

A magnetic field is a vector field that exerts force on particles that are magnetic (like iron) or electrically charged.
The magnetic field direction is defined by the direction a north pole of a magnet points at. It can be visualized using magnetic field lines.

• The strength of a magnetic field is measured in Tesla (T) or Gauss (G).
• The Earth's magnetic field at the location given in the exercise is \(6.0 \times 10^{-5}\, \text{T}\).
• Magnetic fields are central to the induction process; they vary either through motion or changes within the field itself.

The importance of the magnetic field in this exercise is evident, as it is the interaction between the magnetic field and the coil that causes changes in flux, ultimately leading to electromagnetic induction.

Electric Coil

An electric coil is a series of loops that can conduct electrical current. Coils are integral components in many electronic devices.
They are crucial in processes involving electromagnetic induction.

• Coils are characterized by their number of turns (\(N\)), which increase their ability to induce emf.
• In our exercise, the coil has 200 turns, which amplifies the induced emf through each turn contributing to the total voltage.
• The area enclosed by the coil loops also impacts the magnetic flux and, thus, the induced emf.

Electric coils can be designed in various shapes and sizes depending on their application, impacting the efficiency and output of electromagnetic induction.