Question

At what frequency will a \(10.0-\mu \mathrm{F}\) capacitor have reactance \(X_{C}=200 . \Omega ?\)

Short answer

Answer: The frequency at which the 10.0 µF capacitor will have a reactance of 200 Ω is approximately 7957.75 Hz.

Step-by-step solution

Write down the given values and formula

We are given the values for capacitive reactance (X_C) and capacitance (C). Write down these values: Capacitive Reactance, \(X_{C} = 200 \Omega\) Capacitance, \(C = 10.0 \times 10^{-6} \mathrm{F}\) We will use the formula for capacitive reactance to solve for the frequency (f): \(X_{C} = \frac{1}{2 \pi f C}\)

Solve the equation for frequency (f)

Rearrange the formula to solve for frequency (f): \(f = \frac{1}{2 \pi X_{C} C}\)

Substitute the given values and calculate frequency (f)

Plug in the given values of X_C and C into the equation: \(f = \frac{1}{2 \pi (200 \Omega)(10.0 \times 10^{-6} \mathrm{F})}\) Now, calculate the frequency (f): \(f \approx 7957.75 \mathrm{Hz}\)

Write the final answer

The frequency at which the \(10.0-\mu \mathrm{F}\) capacitor will have a reactance of \(200 . \Omega\) is approximately \(7957.75 \mathrm{Hz}\).

Key concepts

Capacitive Reactance

Capacitive reactance is a measure of how much a capacitor resists the flow of alternating current (AC). It is denoted by the symbol \(X_C\) and is measured in ohms (\(\Omega\)). Unlike resistance, which applies to direct current (DC), capacitance reactance only comes into play with AC.
The capacitive reactance decreases as the frequency of the AC increases, because the capacitor will "charge" and "discharge" more quickly.

• High Frequency = Lower Reactance: At higher frequencies, electrons switch direction more often, causing the capacitor to "fill" and "empty" faster.
• Low Frequency = Higher Reactance: Fewer cycles per second mean that the capacitor charges slower, offering more opposition to the flow.

This concept is crucial in designing circuits, as it helps control how much current flows at different frequencies.

Frequency

In the context of electric circuits, frequency refers to how often the current changes direction each second. It is measured in Hertz (Hz), where 1 Hz means one cycle per second.
Frequency plays a big role in determining the performance of AC circuits. Higher frequencies cause reactance to decrease, while lower frequencies increase reactance.
For a capacitor like the one in the exercise, higher frequencies will allow more current to pass through due to lower reactance.

• Frequency is inversely related to reactance: As frequency increases, reactance decreases.
• A balance between frequency and capacitance is essential for controlling the amount of AC that can pass through a circuit.

Formula Derivation

To find out how capacitive reactance relates to frequency and capacitance, the formula derived is \(X_C = \frac{1}{2\pi f C}\). This shows how reactance is inversely proportional to both frequency (\(f\)) and capacitance (\(C\)).
We can reformulate this to solve for frequency when we know the reactance and capacitance:

• Start with the formula: \(X_C = \frac{1}{2\pi f C}\)
• Rearrange to solve for frequency: \(f = \frac{1}{2\pi X_C C}\)

By plugging in the values from the exercise \(X_C = 200 \Omega\) and \(C = 10.0 \times 10^{-6} F\), we calculate the frequency:

• Use the rearranged formula: \(f \approx 7957.75 \text{ Hz}\)

This derived formula helps customize circuits for specific frequencies by adjusting capacitance and measuring reactance, making it a powerful tool in electronics and engineering.