Problem 1 - University Physics with Modern Physics

Question

A flat, rectangular coil consisting of 50 tums measures 25.0 \(\mathrm{cm}\) by 30.0 \(\mathrm{cm}\) . It is in a uniform, \(1.20-\mathrm{T}\) , magnetic field, with the plane of the coil parallel to the field. In 0.222 s, it is rotated so that the plane of the coil is perpendicular to the field. (a) What is the change in the magnetic flux through the coil due to this rotation? (b) Find the magnitude of the average emf induced in the coil during this rotation.

Step-by-Step Solution

Answer

(a) Change in flux: -9.00 \(\mathrm{Wb}\); (b) Average EMF: 2.03 \(\mathrm{V}\).

Step 1: Calculate Initial Magnetic Flux

The initial magnetic flux \( \Phi_i \) is given by \( \Phi_i = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field strength, \( A \) is the area of the coil, and \( \theta \) is the angle between the field and the normal to the coil's surface. Since the plane of the coil is initially parallel to the field, \( \theta = 0^\circ \), so \( \cos(\theta) = 1 \). Thus, \( \Phi_i = 1.20 \times (0.25 \times 0.30) \). Calculate \( \Phi_i \).

Step 2: Calculate Final Magnetic Flux

Once the coil is rotated to be perpendicular to the field, the angle \( \theta \) between the field and the normal to the coil becomes \( 90^\circ \), so \( \cos(\theta) = 0 \). Thus, the final magnetic flux \( \Phi_f = B \cdot A \cdot 0 = 0 \).

Step 3: Calculate Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is given by \( \Delta \Phi = \Phi_f - \Phi_i \). Substitute the values of \( \Phi_f \) and \( \Phi_i \) to find \( \Delta \Phi \).

Step 4: Calculate Area of the Coil

The area \( A \) of the coil is the product of its length and width.\( A = 0.25 \times 0.30 \; \text{m}^2 \). Calculate this value and use it to compute the initial flux in Step 1.

Step 5: Compute Induced EMF

According to Faraday's Law, the magnitude of the average induced EMF is \( |\mathcal{E}| = \frac{N \times \Delta \Phi}{\Delta t} \), where \( N \) is the number of turns, and \( \Delta t \) is the time period. Substitute the given values: \( N = 50 \), \( \Delta t = 0.222 \; \text{s} \), and your computed \( \Delta \Phi \) from Step 3 to find \( |\mathcal{E}| \).

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It basically describes how much magnetic field passes through a particular area.
The formula to calculate magnetic flux \( \Phi \) is given by:

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

where:

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

For a coil that is parallel to the field, \( \theta = 0^\circ \) and \( \cos(\theta) = 1 \), meaning all the magnetic field lines pass through the coil. When it's perpendicular, \( \theta = 90^\circ \) and \( \cos(\theta) = 0 \), thus, no field lines pass through. In the exercise, changing from parallel to perpendicular results in a significant change in magnetic flux, which is essential for inducing an electromotive force (EMF) in a coil.

Faraday's Law of Induction

Faraday's Law of Induction helps explain how electric currents are generated by changing magnetic fields. Essentially, this law states that the induced electromotive force (EMF) in a closed circuit is directly proportional to the rate of change of magnetic flux through the circuit. This relationship can be written as:

• \( \mathcal{E} = -\frac{d\Phi}{dt} \)

The negative sign in the equation is due to Lenz's Law, which tells us that the induced EMF will always work in a direction to oppose the change in flux that produced it.
This principle is fundamental in understanding why rotating the rectangular coil in the textbook exercise induces an EMF. The coil starts parallel to a magnetic field and then becomes perpendicular, creating a change in magnetic flux from maximum to zero, and therefore, according to Faraday's Law, an average EMF is induced over the rotation period.

Rectangular Coil

A rectangular coil, such as the one described in the exercise, is often used in laboratories and experimental setups to demonstrate magnetic effects and calculate induced EMF. The typical rectangular coil consists of several turns of wire wound in a rectangular shape.
Key characteristics are:

• Area (A): Product of the coil's length and width, affecting magnetic flux. Here it's calculated as \( 0.25 \times 0.30 \; \text{m}^2 \), providing a surface for the field to interact with.
• Number of Turns (N): More turns mean higher induced EMF for a given change in flux, as they effectively multiply the effect of a single loop. The exercise specifies 50 turns.

When the coil rotates from being parallel to perpendicular to the magnetic field, the rapid change in field interaction results in a change in the magnetic flux and, therefore, an induced EMF. This setup helps students understand how real-world applications like electric generators and transformers operate by similar mechanisms.