Problem 13 - University Physics with Modern Physics

Question

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 \(\mathrm{cm}\) . The coil rotates in a magnetic fleld of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 \(\mathrm{mV} ?\)

Step-by-Step Solution

Answer

The angular speed \( \omega \) is approximately 10.42 rad/s.

Step 1: Understand the Formula for Maximum EMF

The maximum electromotive force (emf) in a rotating coil is given by the formula: \( \varepsilon_{max} = NAB\omega \), where \( N \) is the number of turns, \( A \) is the area of the coil, \( B \) is the magnetic field strength, and \( \omega \) is the angular speed. We need to solve for \( \omega \).

Step 2: Calculate the Area of the Coil

The area \( A \) of the square coil is found by squaring the length of its side: \( A = (0.0160 \, \text{m})^2 = 0.000256 \, \text{m}^2 \).

Step 3: Substitute Known Values into EMF Equation

Substitute the given values into the formula for maximum emf: \( 24.0 \, \mathrm{mV} = (120)(0.000256 \, \text{m}^2)(0.0750 \, \mathrm{T})\omega \). Convert the emf to volts: \( 24.0 \, \mathrm{mV} = 0.0240 \, \mathrm{V} \).

Step 4: Solve for Angular Speed \( \omega \)

Rearrange the equation to solve for \( \omega \): \( \omega = \frac{0.0240}{(120)(0.000256)(0.0750)} \). Calculate \( \omega \): \( \omega \approx 10.42 \, \text{rad/s} \).

Key Concepts

Angular Speed

Angular speed, often symbolized as \( \omega \), is a measure of how fast an object rotates about an axis. It is expressed in radians per second (rad/s). To visualize this concept, imagine a spinning top. The faster it spins, the greater its angular speed. For any rotating object, the angular speed signifies how quickly it completes a rotation.
Angular speed is crucial in many real-world applications, notably in electricity generation. In the context of generators, such as the one in our exercise, angular speed determines how swiftly the coil inside the generator turns.
To calculate angular speed in this exercise, we used the formula for maximum electromotive force (emf):
\[ \varepsilon_{max} = NAB\omega \]
Here:

• \( N \) is the number of turns in the coil.
• \( A \) is the area of the coil.
• \( B \) represents the magnetic field strength.

By solving for \( \omega \), we determined how fast the coil must rotate to achieve a specific emf.

Maximum EMF

The maximum electromotive force (emf) is an important concept in electromagnetic induction. It represents the highest voltage that a generator can produce when operating at optimal conditions. In simple terms, emf is the electrical "push" or voltage generated by a changing magnetic field.
In our exercise, the maximum emf is given to be 24.0 mV, which we converted to volts for calculation as 0.0240 V. To find the maximum emf from a coil, you can use the equation \( \varepsilon_{max} = NAB\omega \). Here:

• \( N \) is the number of turns in the coil.
• \( A \) is the area of the coil, calculated here using \( A = (0.0160 \, \text{m})^2 \).
• \( B \) is the magnetic field strength.
• \( \omega \) is the angular speed of the coil.

This formula highlights the relationship between these factors and how they contribute to the production of emf. Understanding maximum emf is essential as it influences how effective a generator is at converting mechanical energy into electrical energy.

Magnetic Field Strength

Magnetic field strength, denoted by \( B \), quantifies how strong a magnetic field is. In the International System of Units (SI), it is measured in teslas (T).
This physical quantity is essential in electromagnetic applications. In our exercise, we have a magnetic field strength of 0.0750 T. A stronger magnetic field can induce a higher maximum emf in a generator.
When a coil of wire rotates within a magnetic field, it "cuts" through the magnetic lines of force. This movement causes a change in magnetic flux through the coil. Due to electromagnetic induction, this change induces an emf.
The relationship between magnetic field strength and the generator's output is captured by the formula \( \varepsilon_{max} = NAB\omega \). This shows how a strong magnetic field, along with other factors like area and angular speed, impacts emf production. In understanding magnetic field strength, one appreciates its pivotal role in determining the performance of devices like motors and generators.