Question

An inductor with inductance \(L=42.1 \mathrm{mH}\) is connected to an \(\mathrm{AC}\) power source that supplies \(V_{\text {emf }}=19.1 \mathrm{~V}\) at \(f=605 \mathrm{~Hz}\). Find the reactance of the inductor.

Short answer

Answer: The reactance of the inductor is approximately 160.3 Ω.

Step-by-step solution

Identify given values

We are given the inductance L, the voltage V_emf, and the frequency f: - \(L = 42.1 \mathrm{mH} = 42.1 \times 10^{-3} \mathrm{H}\) (converting mH to H) - \(V_{\text{emf}} = 19.1 \mathrm{V}\) - \(f = 605 \mathrm{Hz}\)

Calculate the reactance of the inductor

Now we can find the inductive reactance (\(X_L\)) using the formula \(X_L = 2\pi fL\). Plug in the given values and calculate: \(X_L = 2\pi (605 \mathrm{Hz})(42.1 \times 10^{-3} \mathrm{H})\) \(X_L = 2 \times 3.1416 \times 605 \times 42.1 \times 10^{-3}\) \(X_L ≈ 160.3\) The reactance of the inductor is approximately \(160.3 \Omega\).

Key concepts

Inductance

Inductance is a fundamental property of an electrical conductor which determines how effectively it can store energy in a magnetic field when electric current flows through it. An inductor, the component that exhibits inductance, effectively resists changes in current.

For instance, when an AC current flows through an inductor, the changing magnetic field induced by this alternating current opposes the changes in current, a phenomenon explained by Faraday's law of electromagnetic induction. This opposition is actually a reactive resistance known as inductive reactance. A higher inductance means a greater ability to store magnetic energy and hence, a higher reactance against changing AC currents.

Inductive reactance depends not only on the inductance but also on the frequency of the AC current, as seen in the formula for reactance, which ties in closely with the next core concept, the AC power source.

AC Power Source

An AC power source generates alternating current, which is characterized by the constant change in direction and amplitude of the electric current over time. This differs from direct current (DC), where the flow of electricity is unidirectional.

The AC power source has a specific voltage and frequency. Voltage refers to the electromotive force or potential difference, which drives the current through a circuit. The frequency, measured in Hertz (Hz), indicates how often the current changes direction per second.

The voltage supplied by an AC source can be used to power various components, like resistors, capacitors, and inductors. In the case of the inductor, the AC source’s frequency is a key factor in determining the magnitude of the inductive reactance, as the resistive behavior of an inductor to AC is directly proportional to the frequency, as seen in the exercise.

Frequency

Frequency, in the context of an AC power source, is the number of cycles of the waveform that occur in one second. It is measured in Hertz (Hz). A cycle is a complete oscillation of the current and voltage, from zero to maximum, back to zero, to minimum, and back to zero again.

The frequency of the AC power source greatly influences the behavior of an electrical circuit. The higher the frequency, the greater the inductive reactance of an inductor. This is because inductors oppose any change in current, and with higher frequency, the current changes direction more frequently, thus leading to a higher rate of opposition - or higher inductive reactance - as explained in the solution to the exercise.

Understanding frequency is essential when working with AC circuits, as even a slight change in frequency can significantly alter the inductive reactance and, consequently, the performance and behavior of electrical components within the circuit.