Problem 8 - Applied Physics

Question

Find the capacitive reactance of a \(6.00-\mathrm{mF}\) capacitor in a circuit of frequency \(10 \overline{0} \mathrm{~Hz}\).

Step-by-Step Solution

Answer

The capacitive reactance is approximately 2.65 Ω.

Step 1: Understand Capacitive Reactance

Capacitive reactance is a measure of a capacitor's opposition to alternating current (AC) and is denoted by the symbol \(X_c\). It is calculated using the formula \(X_c = \frac{1}{2 \pi f C}\), where \(f\) is the frequency of the AC signal in hertz and \(C\) is the capacitance in farads.

Step 2: Convert Units if Necessary

The capacitance is given as \(6.00\, \text{mF}\). We convert this to farads: \(6.00\, \text{mF} = 6.00 \times 10^{-3}\, \text{F}\). The frequency \(f\) is given as \(100\, \text{Hz}\). These values should be used in the formula without further conversion.

Step 3: Substitute Values into the Formula

Use the formula for capacitive reactance: \(X_c = \frac{1}{2 \pi f C}\). Substitute \(f = 100\, \text{Hz}\) and \(C = 6.00 \times 10^{-3}\, \text{F}\) into the formula:\[ X_c = \frac{1}{2 \pi \times 100 \times 6.00 \times 10^{-3}} \]

Step 4: Calculate Capacitive Reactance

Calculate \(X_c\):\[ X_c = \frac{1}{2 \pi \times 100 \times 6.00 \times 10^{-3}} \approx 2.65 \Omega \]This calculation involves dividing 1 by the product of \(2\pi\), \(100\, \text{Hz}\), and \(6.00 \times 10^{-3}\, \text{F}\).

Key Concepts

Capacitive Reactance

Capacitive reactance (\(X_c\)) is a term used to describe how a capacitor resists the flow of alternating current (AC) in an electrical circuit. Unlike resistance, which remains constant for all current types, capacitive reactance is frequency-dependent. It means that its resistance to the current changes based on the frequency of the AC signal.

To find capacitive reactance, we use the formula:\[ X_c = \frac{1}{2 \pi f C} \]where:

• \(f\) is the frequency in hertz (Hz)
• \(C\) is the capacitance in farads (F)

This formula makes sense because, as the frequency increases, the reactance (\(X_c\)) decreases, indicating that the capacitor allows more current to flow through. Conversely, at lower frequencies, the capacitive reactance increases, meaning the capacitor blocks more of the AC current.

Alternating Current (AC)

Alternating Current, abbreviated as AC, is a type of electrical current in which the flow of electric charge periodically reverses direction. It is the predominant form of electricity used in homes and industries around the world. The main advantage of AC over direct current (DC) is that it can be easily transformed to different voltages, which allows for long-distance transmission with minimal energy loss.

AC electricity is typically used to power household appliances and equipment. Every country has a standard AC supply frequency, most commonly either 50 or 60 Hz.
In AC circuits, devices like capacitors and inductors exhibit reactive properties, which influence how the current and voltage relate to each other. Capacitors, for example, tend to "resist" changes in voltage in these circuits, which is where the concept of capacitive reactance comes into play.

Frequency in Circuits

Frequency, in electrical and electronic circuits, refers to the number of cycles a periodic signal completes in one second, and is measured in hertz (Hz). It plays a crucial role in determining how various components in a circuit behave.

In an AC circuit, each component will react differently to changes in frequency:

• Resistors are generally unaffected by changes in frequency.
• Inductors and capacitors, however, are highly sensitive to frequency changes.

Particularly for capacitors, the capacitive reactance (\(X_c\)) is inversely proportional to the frequency. This means at higher frequencies, capacitors allow more current to pass through and vice versa. This property is utilized in many practical applications, such as in filters and tuners.
Knowing the frequency of a circuit is essential in designing and troubleshooting electronic systems, as it affects signal timings and the impedance of reactive components.