Question

An inductor with inductance \(L=47.0 \mathrm{mH}\) is connected to an AC power source having a peak value of \(12.0 \mathrm{~V}\) and \(f=1000 . \mathrm{Hz} .\) Find the reactance of the inductor and the maximum current in the circuit.

Short answer

Solution: The inductive reactance is approximately 295.167 Ω, and the maximum current in the circuit is approximately 0.0406 A or 40.6 mA.

Step-by-step solution

Find the inductive reactance

Use the formula for inductive reactance, \(X_L = 2\pi fL\). We are given the inductance \(L = 47.0 \mathrm{mH}\) and the frequency \(f = 1000 \mathrm{Hz}\). Plug in these values and find the inductive reactance: \(X_L = 2\pi \cdot (1000 \mathrm{Hz}) \cdot (47.0 \times 10^{-3} \mathrm{H})\) Now calculate the value of \(X_L\): \(X_L = 295.167 \Omega\) The inductive reactance is approximately \(295.167 \Omega\).

Find the maximum current

Use Ohm's law to find the maximum current in the circuit, \(I_{max} = \frac{V_{max}}{X_L}\). We are given the peak voltage \(V_{max} = 12.0 \mathrm{V}\). We just found the inductive reactance is \(295.167 \Omega\): \(I_{max} = \frac{12.0 \mathrm{V}}{295.167 \Omega}\) Now calculate the value of \(I_{max}\): \(I_{max} = 0.0406 \mathrm{A}\) The maximum current in the circuit is approximately \(0.0406 \mathrm{A}\) or \(40.6 \mathrm{mA}\).

Key concepts

AC Circuit Analysis

When studying alternating current (AC) circuits, one must understand how the various components behave as the current change direction periodically. Unlike direct current (DC) circuits where the flow of electric charge is constant, AC circuits carry a current that varies sinusoidally with time. The inductor, a crucial component in our exercise, responds to these changes in current by generating a voltage that opposes the change due to its inductance.

Inductance, denoted by the symbol 'L', is measured in Henry (H), and it quantifies an inductor's ability to store energy in a magnetic field. Inductive reactance, symbolized by \(X_L\), describes how much resistance an inductor offers to the changing current. It is calculated using the formula \(X_L = 2\pi fL\), where \(f\) is the frequency of the current in hertz (Hz). A higher inductive reactance signifies a greater opposition to AC, affecting the circuit's behaviour, especially the current that can flow through it.

In our exercise, with the combination of the AC source's frequency and the inductance of the inductor, we were able to quantify the inductive reactance and understand its impact on the circuit.

Ohm's Law

Ohm’s law is a fundamental principle in electronics that relates the voltage across a component to the current flowing through it and the resistance it presents. The law is simply stated as \(V = IR\), where \(V\) is the voltage, \(I\) is the current, and \(R\) is the resistance. In AC circuits, resistance is replaced with the term 'impedance', representing both the resistive and reactive opposition to current. For the inductor in our exercise, impedance is purely reactive, known as inductive reactance \(X_L\), since ideal inductors do not dissipate energy as a resistor does.

Using Ohm's law, we can also calculate the maximum current \(I_{max}\) in an AC circuit by adjusting the formula to \(I_{max} = \frac{V_{max}}{X_L}\), with \(V_{max}\) being the peak voltage. This shows us the inversely proportional relationship between inductance and current – as inductive reactance increases, the current that the AC source can push through the inductor decreases, which is precisely demonstrated in the step-by-step solution.

Maximum Current

The term 'maximum current' in an AC circuit, such as the one in our textbook exercise, refers to the highest value the current can achieve during its cycle. The importance of knowing the maximum current is crucial for various design criteria, including safety, component selection, and power management. It can be influenced by the source voltage, impedance of the circuit, and in this case, specifically the inductive reactance.

In our textbook example, the maximum current \(I_{max}\) is derived after calculating the inductive reactance \(X_L\). It's important to note that this current will oscillate, reaching this maximum value periodically at the peak voltage (\(V_{max}\)). This value is pivotal when designing a circuit because it determines the potential stress on the components and the safety parameters needed to avoid overloading the circuit. By using \(I_{max} = \frac{V_{max}}{X_L}\), we ensure that students not only learn to compute the required values but also to appreciate the underlying principles dictating the behavior of AC circuits.