Problem 37 - Physics for Scientists and Engineers with Modern Physics

Question

(I) A simple generator is used to generate a peak output voltage of 24.0 \(\mathrm{V}\) . The square armature consists of windings that are 5.15 \(\mathrm{cm}\) on a side and rotates in a field of 0.420 \(\mathrm{T}\) at a rate of 60.0 \(\mathrm{rev} / \mathrm{s} .\) How many loops of wire should be wound on the square armature?

Step-by-Step Solution

Answer

58 loops.

Step 1: Determine the Expression for EMF

The peak output voltage, \(V_0\), of a generator can be determined using the formula: \( V_0 = NAB\omega \sin(\theta) \). Here, \(A\) is the area of the armature, \(N\) is the number of loops, \(B\) is the magnetic field strength, and \(\omega\) is the angular velocity. We want the maximum EMF, which is \( NAB\omega \) since \( \sin(\theta) = 1 \) at peak.

Step 2: Calculate the Area of the Loop

The armature is square-shaped with each side measuring 5.15 cm. Convert this into meters to maintain SI units: \(5.15 \text{ cm} = 0.0515 \text{ m}\). Thus, the area of the armature is \( A = (0.0515 \text{ m})^2 \). Calculate this to find \( A = 0.00265225 \text{ m}^2 \).

Step 3: Find the Angular Velocity

The armature rotates at a rate of 60.0 revolutions per second. Angular velocity, \( \omega \), is given by \( \omega = 2\pi \times \text{frequency} \). Thus, \( \omega = 2\pi \times 60.0 \) radians per second. Calculate this to find \( \omega = 376.99 \text{ rad/s} \).

Step 4: Use the Peak Voltage Equation

Substitute the known values into the peak output voltage formula: \( 24 = N \times 0.00265225 \times 0.420 \times 376.99 \). Simplify to solve for \( N \): \( N = \frac{24} {0.00265225 \times 0.420 \times 376.99 } \).

Step 5: Calculate the Number of Loops

Compute \( N = \frac{24}{0.420 \times 0.37699 \times 0.00265225} \approx 57.97 \). Since the number of loops must be an integer, round \( N \) to the nearest whole number, which is 58.

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is at the core of understanding how generators and transformers work. It describes how a changing magnetic field within a closed loop induces an electromotive force (EMF), often referred to as voltage, in the loop. This principle is crucial because it allows us to generate electricity in a practical and efficient manner.

In simple terms, if a coil of wire is exposed to an environment where the magnetic flux is changing, an electric current is generated within that wire. This is the basic working principle of devices like the simple generator described in the exercise. Faraday's Law quantifies this process with the equation:

\[ \varepsilon = -N \frac{d\Phi_B}{dt} \]

Where:

• \(\varepsilon\) is the induced EMF.
• \(N\) represents the number of loops or turns in the coil.
• \(\frac{d\Phi_B}{dt}\) is the rate of change of the magnetic flux.

The negative sign denotes Lenz's Law, which states that the induced EMF will always work to oppose the change in flux. In the context of the generator problem, finding the correct number of loops (\(N\)) helps ensure that the desired voltage is achieved.

Magnetic Flux

Magnetic flux gives us a measure of the total magnetic field passing through a given area. It's particularly relevant in Faraday's Law, as it plays a critical role in determining the magnitude of the induced EMF.

Magnetic flux through a surface is defined by the equation:

\[ \Phi_B = B \times A \times \cos(\theta) \]

Where:

• \(\Phi_B\) is the magnetic flux.
• \(B\) is the magnetic field strength.
• \(A\) represents the area the field is passing through.
• \(\theta\) is the angle between the magnetic field and the normal (perpendicular) to the surface.

In the problem's context, the armature (which is the loop of wire) rotates in a magnetic field. The magnetic flux is at its peak when \(\theta = 0\), meaning the field lines are perpendicular to the loop's surface. Calculating changes in flux as the loop rotates provides the necessary inputs to determine the resulting EMF using Faraday's Law.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates, which is central to understanding devices like generators. In this case, the armature rotates at a given speed, and converting this rotational speed into angular velocity helps us apply the formulas effectively.

Angular velocity \(\omega\) is found using the equation:

\[ \omega = 2\pi \times \text{frequency} \]

Where:

• \(\omega\) is the angular velocity in radians per second.
• The frequency refers to how many complete revolutions occur per second.

Angular velocity is pivotal in calculating the induced EMF in this exercise. It is directly proportional to the maximum potential difference generated, as shown in the equation \(EMF = NAB\omega\). Thus, understanding and accurately computing \(\omega\) allows us to correctly determine the number of loops (\(N\)) needed for the generator to achieve a specific output voltage.