Problem 10 - University Physics with Modern 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 \(1.70 \mathrm{~T}\) to a position perpendicular to the field. The rotation takes \(0.0600 \mathrm{~s}\). What is the average emf induced in the coil?

Step-by-Step Solution

Answer

The average emf induced in the coil is approximately 0.34 V.

Step 1: Define the formula for average induced emf

The formula for the average induced electromotive force (emf) is given by Faraday's law of electromagnetic induction: \(\text{emf}_{\text{avg}} = -N \frac{\Delta \Phi}{\Delta t}\), where \(N\) is the number of turns, \(\Delta \Phi\) is the change in magnetic flux, and \(\Delta t\) is the change in time.

Step 2: Calculate the initial magnetic flux

The initial magnetic flux \(\Phi_i\) is calculated by the formula: \(\Phi_i = B \times A \times \cos \theta\), where \(B = 1.70 \mathrm{~T}\) is the magnetic field, \(A\) is the area of the coil, and \(\theta = 37.0^{\circ}\) is the angle. First, calculate the area \(A = 0.25 \times 0.40 = 0.10 \mathrm{~m^2}\), and then \(\Phi_i = 1.70 \times 0.10 \times \cos 37.0^{\circ}\).

Step 3: Calculate the final magnetic flux

When the coil is perpendicular to the magnetic field, the angle \(\theta\) is \(0^{\circ}\). Therefore, the final magnetic flux \(\Phi_f = B \times A \times \cos 0^{\circ} = 1.70 \times 0.10 = 0.17 \mathrm{~Wb}\).

Step 4: Determine the change in magnetic flux

Calculate \(\Delta \Phi = \Phi_f - \Phi_i\). Substitute \(\Phi_i = 1.70 \times 0.10 \times \cos 37.0^{\circ}\) and \(\Phi_f = 0.17\). Use \(\cos 37.0^{\circ} \approx 0.7986\). So, \(\Delta \Phi = 0.17 - (1.70 \times 0.10 \times 0.7986)\).

Step 5: Calculate the average induced emf

Now, use \(\Delta t = 0.0600 \mathrm{~s}\) and \(N = 80 \) in the formula for average induced emf. Substitute \(\Delta \Phi\) from Step 4 into \(\text{emf}_{\text{avg}} = -N \frac{\Delta \Phi}{\Delta t}\).

Key Concepts

Faraday's law

Faraday's law of electromagnetic induction is a fundamental principle that describes how an electromotive force (emf) is induced in a coil due to a changing magnetic field. According to this law, the induced emf is directly proportional to the rate of change of magnetic flux through the coil. It can be mathematically expressed as:

• \( \text{emf} = -N \frac{d\Phi}{dt} \)

where:

• \( N \) is the number of turns in the coil
• \( \Phi \) is the magnetic flux
• \( t \) is time

The negative sign indicates Lenz's law, meaning that the induced emf produces a current that opposes the change in magnetic flux. This principle explains the working of generators and transformers in converting mechanical energy into electrical energy. Faraday's law is essential for understanding how electricity can be generated from motion or changing fields.

magnetic flux

Magnetic flux (\( \Phi \) ) quantitatively describes the strength and extent of a magnetic field passing through an area. It's similar to the idea of water flow through a surface. When analyzing a coil in a magnetic field, the magnetic flux can be described with the formula:

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

where:

• \( B \) is the magnetic field strength (in Teslas)
• \( A \) is the area through which the field lines pass (in square meters)
• \( \theta \) is the angle between the field lines and the normal (perpendicular) to the surface

The unit of magnetic flux is the Weber (\( \mathrm{Wb} \)).
In a rotational context, as with the coil problem, the angle changes with time, which influences the magnetic flux and subsequently the induced emf. Understanding magnetic flux is key to determining how systems will react under changes in magnetic conditions.

average induced emf

Average induced emf is the mean value of the emf induced in a coil over a specific time period. In systems where the magnetic flux changes, calculating this average helps understand the effective electromotive force influencing the circuit during a complete motion or change. The formula given by Faraday's law for average induced emf is:

• \( \text{emf}_{\text{avg}} = -N \frac{\Delta \Phi}{\Delta t} \)

Here:

• \( \Delta \Phi \) represents the change in magnetic flux
• \( \Delta t \) is the time interval over which the change happens
• \( N \) is the number of turns of the coil

The negative sign accounts for the direction of induced emf as per Lenz's law, indicating the system’s tendency to counteract changes in magnetic flux. Calculating the average induced emf provides insight into the behavior of the circuit when subjected to temporal changes in magnetic conditions.

coil rotation

Coil rotation in a magnetic field dynamically changes the orientation of the coil relative to the field. This movement modifies the angle \( \theta \), thereby affecting the magnetic flux (\( \Phi = B \times A \times \cos \theta \)).
In scenarios like our exercise, the coil's initial position forms an angle of \( 37.0^{\circ} \) with the magnetic field. By moving to a perpendicular position, the coil ensures maximum interaction with the field, leading to maximum final flux. As the coil rotates:

• The change in angle affects \( \cos \theta \) , altering the magnetic flux
• The constant rotation speed provides uniform variation rate

This rotational motion, and the resulting change in flux over time, is pivotal in the operation of many electrical devices, such as alternators and dynamos, which rely on converting mechanical energy to electrical through electromagnetic induction.