Problem 2 - University Physics with Modern Physics

Question

In a physics laboratory experiment, a coil with 200 tums enclosing an area of 12 \(\mathrm{cm}^{2}\) 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 \(6.0 \times 0^{-5} \mathrm{T}\) . (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: \(7.2 \times 10^{-7} \text{Tm}^2\); final flux: 0. b) Average emf: \(3.6 \times 10^{-3} \text{V}\).

Step 1: Calculate Initial and Final Magnetic Flux

To calculate the magnetic flux, we need to use the formula \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( \Phi \) is the magnetic flux, \( B \) is the magnetic field strength, \( A \) is the area, and \( \theta \) is the angle between the magnetic field and the normal to the coil's plane. Initially, \( \theta = 0^{\circ} \), so \( \cos(\theta) = 1 \). The initial flux is \( \Phi_i = B \cdot A = 6.0 \times 10^{-5} \times 12 \times 10^{-4} = 7.2 \times 10^{-7} \text{Tesla meters squared (Tm}^2\text{)} \). After rotation, the plane is parallel to the field, \( \theta = 90^{\circ} \), so \( \cos(\theta) = 0 \). The final flux is \( \Phi_f = 0 \).

Step 2: Calculate Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is the difference between the initial and final flux. Thus, \( \Delta \Phi = \Phi_f - \Phi_i = 0 - 7.2 \times 10^{-7} = -7.2 \times 10^{-7} \text{Tm}^2 \).

Step 3: Calculate Average Induced EMF

Using Faraday's law of electromagnetic induction, the average induced emf \( \mathcal{E} \) is given by \( \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \), where \( N \) is the number of turns (200) and \( \Delta t \) is the time interval in seconds (0.040 s). Substitute the values: \( \mathcal{E} = -200 \times \frac{-7.2 \times 10^{-7}}{0.040} = 3.6 \times 10^{-3} \text{V} \). The negative sign from Faraday’s law indicates the direction of the induced emf, but we are interested in the magnitude here.

Key Concepts

Magnetic Flux

Magnetic flux is a core concept in understanding electromagnetic induction. It represents the total magnetic field passing through a given area. Think of it like the number of magnetic field lines penetrating through the surface of a coil. This can be visualized as a density map for magnetic lines in a particular area.

The formula to calculate magnetic flux, \( \Phi \), is given as:

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

Here,

• \( B \) is the magnetic field strength.
• \( A \) is the area through which the field lines pass.
• \( \theta \) is the angle between the magnetic field and the normal to the surface.

In this exercise, calculating the initial magnetic flux involved having the coil perpendicular to the magnetic field. This makes \( \cos(\theta) = 1 \), meaning the maximum number of magnetic field lines pass through.

After rotation, when the coil is parallel to the field, \( \theta = 90^\circ \), leading to \( \cos(\theta) = 0 \), signifying no lines pass through and thus zero magnetic flux. Understanding this relationship helps in grasping how changes in physical alignment can affect magnetic flux values.

Electromagnetic Induction

Electromagnetic induction is the process of generating an electromotive force (EMF) across a conductor when it is exposed to a changing magnetic field. Discovered by Michael Faraday, this phenomenon is foundational in the development of devices like transformers and generators.

Faraday's Law of Electromagnetic Induction states that the induced EMF in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit.

• This can be mathematically represented by the formula: \( \mathcal{E} = - N \frac{\Delta \Phi}{\Delta t} \)

In simple terms, when the magnetic environment of a coil changes, either by moving the coil in a magnetic field or altering the field’s strength or direction, an EMF is induced. The minus sign indicates Lenz’s Law, specifying that the induced EMF will generate a current opposing the change in flux.

Understanding these principles is crucial in analyzing circuits and designing electronic systems that rely on magnetic force.

Coil in Magnetic Field

A coil positioned in a magnetic field experiences certain effects based on its orientation relative to the field. In the experiment, the coil started perpendicular to the earth's magnetic field before rotating to become parallel. This change in position is paramount to understanding how the experiment impacts the flux and the consequent EMF values.

In general:

• A perpendicular coil position allows maximum magnetic field lines to pass through, maximizing flux.
• A parallel position results in no lines passing through, minimizing flux.

Thus, changing the coil’s position relative to the field alters the flux. This change in flux is critical in applications where the movement of the coil or changes in the magnetic environment are utilized to generate electricity, informing the design of sensors and inking mechanisms in devices like microphones and electric guitars.

Induced EMF

The concept of induced EMF is central to Faraday’s experiments and to this problem. It refers to the voltage generated when a coil is exposed to a changing magnetic flux. This change can be caused by moving either the magnetic field or the coil itself.

To compute the average induced EMF, Faraday's Law provides:

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

In our exercise, \( N \) represents the number of turns in the coil, \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the time over which this change occurs. The calculated value, \( 3.6 \times 10^{-3} \) V, represents the EMF's magnitude generated by the coil’s rotation.

The negative sign in the formula is significant, indicating that the induced EMF's direction opposes the change in magnetic flux—an embodiment of Lenz's law. Thus, understanding the dynamics of induced EMF opens doors to harnessing electrical energy conversion systems, pivotal in engineering fields.