Problem 12 - Physics

Question

An ac generator has a frequency of \(7.5 \mathrm{kHz}\) and a voltage of \(39 \mathrm{~V}\). When an inductor is connected between the terminals of this generator, the current in the inductor is \(42 \mathrm{~mA}\). What is the inductance of the inductor?

Step-by-Step Solution

Answer

The inductance of the inductor is approximately 19.7 mH.

Step 1: Understanding the Problem

We are given a generator with a frequency of \(7.5 \text{kHz}\) and a voltage of \(39 \text{~V}\). An inductor is connected which allows a current of \(42 \text{~mA}\) to flow. We need to find the inductance \(L\) of the inductor.

Step 2: Formula for Inductive Reactance

The inductive reactance \(X_L\) is given by the formula \(X_L = 2\pi f L\), where \(f\) is the frequency and \(L\) is the inductance. The reactance is also related to voltage and current by \(X_L = \frac{V}{I}\).

Step 3: Convert to Standard Units

Convert the given current from milliamperes to amperes. Thus, \(42 \text{~mA} = 0.042 \text{~A}\). Also, convert the frequency from kilohertz to hertz: \(7.5 \text{kHz} = 7500 \text{~Hz}\).

Step 4: Calculate Inductive Reactance

Use the formula \(X_L = \frac{V}{I}\) to calculate the inductive reactance.\[ X_L = \frac{39}{0.042} = 928.57 \text{~Ohms} \]

Step 5: Calculate Inductance

Rearrange the inductive reactance formula to solve for \(L\): \(L = \frac{X_L}{2\pi f}\).Substitute known values:\[ L = \frac{928.57}{2\pi \times 7500} \approx 0.0197 \text{~H} \]Thus, the inductance \(L\) is approximately \(19.7 \text{~mH}\).

Key Concepts

AC Generator

An AC generator, also known as an alternator, is a device that converts mechanical energy into electrical energy in the form of alternating current (AC). It functions by rotating a coil of wire in a magnetic field, which induces an electric current within the wire. This process involves key components such as a rotor (the moving part) and a stator (the stationary part). The generator relies on Faraday's law of electromagnetic induction, which states that a change in magnetic environment of a coil of wire will induce an electromotive force (EMF) in the coil.

• AC generators are commonly used in power plants to produce electricity for the grid.
• The frequency of the generated AC can vary, with common standards being 50 Hz or 60 Hz, depending on the country.
• For this problem, the generator has a frequency of 7.5 kHz, which is much higher than standard household frequencies.

Understanding AC generators is crucial as they play a pivotal role in powering various devices and systems by supplying alternating current efficiently over long distances. This basic operation helps in maintaining a constant supply of current and voltage.

Inductive Reactance

Inductive reactance is a concept used to describe how an inductor opposes the change in current in a circuit. It's a key component of AC circuits that can be visualized as the resistance offered by an inductor to alternating current. Unlike resistance in DC circuits, inductive reactance depends on the frequency of the AC signal and the inductance of the coil.

• It is calculated using the formula: \( X_L = 2\pi f L \), where \( f \) is frequency and \( L \) is inductance.
• In our exercise, the inductive reactance \( X_L \) can also be determined by \( X_L = \frac{V}{I} \), where \( V \) is voltage and \( I \) is current.
• Inductive reactance affects the phase relationship between voltage and current, leading the current to lag behind the voltage in a purely inductive circuit.

As frequency or inductance increases, the inductive reactance rises, thereby limiting the current flow in the circuit. This property is vital for variety of applications such as tuning circuits in radios.

Frequency

Frequency is a fundamental concept that refers to the number of cycles a wave completes in one second. Measured in hertz (Hz), it is a crucial parameter in AC circuits because it directly influences how the circuit components, such as capacitors and inductors, behave and interact with AC signals.

• In the formula for inductive reactance \(X_L = 2\pi f L\), frequency \(f\) plays a critical role in determining how much the inductor will oppose the current.
• A higher frequency results in higher inductive reactance, which reduces current through an inductor.
• In our problem, the frequency is given as 7.5 kHz, which means each second has 7500 cycles of AC.

Understanding frequency is essential for designing and analyzing AC circuits. It affects not just the reactance but also the overall power delivery and efficiency in an AC power system. Thus, having a clear grasp of frequency and its implications loop throughout the electrical behavior in inductors and other components.

Voltage and Current Relationship

In AC circuits, the relationship between voltage and current is essential for determining how the circuit functions. Unlike DC circuits where the relationship is straightforward, AC circuits require understanding of phase differences, especially when reactance is involved.

• The key formula \( X_L = \frac{V}{I} \) relates the voltage across and the current through an inductor in AC circuits.
• In a purely inductive circuit, the voltage leads the current by 90 degrees, which indicates a phase difference caused by the energy-storing nature of inductors.
• This phase difference can affect how power is applied in the circuit, especially in applications like transformers and power supply units.

When visualizing the voltage and current relationship on a graph, they form sinusoidal waves that are out of phase in inductive circuits. The reactive elements, such as inductive loads, store and release energy, which shifts these waves relative to one another. This concept is pivotal for the proper design and functioning of electrical systems, ensuring that devices receive the right amount of power without premature wear or inefficiency.