Question

(a) What is the reactance of a 3.00-H inductor at a frequency of 80.0 Hz? (b) What is the inductance of an inductor whose reactance is 120\(\Omega\) at 80.0 Hz? (c) What is the reactance of a 4.00-\(\mu\)F capacitor at a frequency of 80.0 Hz? (d) What is the capacitance of a capacitor whose reactance is 120 \(\Omega\) at 80.0 Hz?

Short answer

(a) 1507.96 Ω, (b) 0.238 H, (c) 498.75 Ω, (d) 16.58 μF.

Step-by-step solution

Understand Reactance in Inductors

The reactance of an inductor, known as inductive reactance, is given by the formula \( X_L = 2\pi f L \), where \( f \) is the frequency and \( L \) is the inductance.

Calculate Reactance of the Inductor

For part (a), substitute \( f = 80.0 \text{ Hz} \) and \( L = 3.00 \text{ H} \) into the formula for inductive reactance: \( X_L = 2\pi \times 80.0 \times 3.00 \). Calculate \( X_L \).

Calculate Inductance from Reactance

For part (b), rearrange the inductive reactance formula to find inductance: \( L = \frac{X_L}{2\pi f} \), where \( X_L = 120 \Omega \) and \( f = 80.0 \text{ Hz} \). Solve for \( L \).

Understand Reactance in Capacitors

The reactance of a capacitor, called capacitive reactance, is given by the formula \( X_C = \frac{1}{2\pi f C} \), where \( C \) is the capacitance.

Calculate Reactance of the Capacitor

For part (c), substitute \( f = 80.0 \text{ Hz} \) and \( C = 4.00 \mu\text{F} \) (converted to Farads as \( 4.00 \times 10^{-6} \text{ F} \)) into the capacitive reactance formula. Calculate \( X_C \).

Calculate Capacitance from Reactance

For part (d), rearrange the capacitive reactance formula to find capacitance: \( C = \frac{1}{2\pi f X_C} \), where \( X_C = 120 \Omega \) and \( f = 80.0 \text{ Hz} \). Solve for \( C \).

Key concepts

Inductive Reactance

Inductive reactance is a property that describes how much an inductor opposes the flow of alternating current (AC) through it. It's measured in ohms (\(\Omega\)) and is symbolized by \(X_L\). Think of inductive reactance as a kind of resistance, but unlike regular resistance, it depends on the frequency of the current and the inductance of the coil.

• Formula: \(X_L = 2\pi f L\)
• \(f\) is the frequency in hertz (Hz)
• \(L\) is the inductance in henrys (H)

When the frequency \(f\) or the inductance \(L\) increases, so does the inductive reactance. This means the inductor will block more current as the frequency rises. Inductive reactance plays a crucial role in tuning circuits, such as those in radios, by allowing only desired frequencies to pass.

Capacitive Reactance

Capacitive reactance is the measure of how much a capacitor resists the flow of AC through it. Just like inductive reactance, it's also measured in ohms (\(\Omega\)) but is denoted by \(X_C\). The interesting part about capacitive reactance is that it actually decreases with an increase in frequency.

• Formula: \(X_C = \frac{1}{2\pi f C}\)
• \(f\) is the frequency in hertz (Hz)
• \(C\) is the capacitance in farads (F)

So, if you increase the frequency or capacitance, the capacitive reactance decreases. Capacitors are often used in AC circuits to control the flow of electrons, and a lower reactance means the capacitor will pass more current at higher frequencies.

Frequency

Frequency, in the context of electronics and AC circuits, refers to how often an electrical signal repeats its pattern over time. It's measured in hertz (Hz), which indicates the number of cycles per second. In a circuit:

• A higher frequency means that the electrical signal oscillates more frequently.
• The response of both inductive and capacitive reactance is highly dependent on frequency.

For example, in the formula for inductive reactance \(X_L = 2\pi f L\), as frequency increases, so does \(X_L\). Conversely, in the formula for capacitive reactance \(X_C = \frac{1}{2\pi f C}\), an increase in frequency results in a decrease in \(X_C\). This frequency dependence helps engineers design circuits that filter or amplify specific frequency ranges.

Inductance

Inductance is a property of an inductor that determines its ability to store energy in a magnetic field when electricity flows through it. Measured in henrys (H), inductance influences inductive reactance, especially in varying AC circuits.

• More inductance means a greater ability to oppose changes in current.
• The formula \(X_L = 2\pi f L\) shows that increasing \(L\) increases \(X_L\).

Inductors are key components in filters, transformers, and inductance-capacitance circuits, which rely on their ability to block or allow AC at specific frequencies. Understanding inductance helps in the design of stable circuits that respond correctly to electrical signals.

Capacitance

Capacitance is the ability of a capacitor to store an electric charge. It's measured in farads (F) and impacts how capacitors behave in AC circuits. Higher capacitance allows a capacitor to store more energy, giving it a larger role in filtering and timing circuits.

• In the formula \(X_C = \frac{1}{2\pi f C}\), increasing \(C\) lowers \(X_C\).
• Capacitors are designed to pass AC while blocking DC.

In many electronic applications, capacitors help stabilize voltage and filter out noise, thanks to their ability to manage AC frequencies. Choosing the right capacitance is crucial for effective signal processing and energy storage in electronic systems.