Problem 101 - Physics

Question

Three capacitors \((3.0,7.0\), and \(9.0 \mu \mathrm{F}\) ) are connected in series. What is their equivalent capacitance?

Step-by-Step Solution

Answer

The equivalent capacitance is approximately 1.703 µF.

Step 1: Understand Series Capacitance Formula

For capacitors connected in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals of each individual capacitance. The formula is given by: \[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \] where \( C_1 = 3.0 \mu F \), \( C_2 = 7.0 \mu F \), and \( C_3 = 9.0 \mu F \).

Step 2: Calculate Reciprocal of Each Capacitance

Calculate the reciprocal for each capacitance: - \( \frac{1}{C_1} = \frac{1}{3.0} \approx 0.333 \mu F^{-1} \) - \( \frac{1}{C_2} = \frac{1}{7.0} \approx 0.143 \mu F^{-1} \) - \( \frac{1}{C_3} = \frac{1}{9.0} \approx 0.111 \mu F^{-1} \)

Step 3: Sum the Reciprocals

Sum the reciprocals of all three capacitances: \[ \frac{1}{C_{eq}} = 0.333 + 0.143 + 0.111 = 0.587 \mu F^{-1} \]

Step 4: Calculate Equivalent Capacitance

Finally, take the reciprocal of the total to find the equivalent capacitance: \[ C_{eq} = \frac{1}{0.587} \approx 1.703 \mu F \]

Key Concepts

Series Capacitors

When capacitors are connected one after another like beads on a string, they are said to be in "series." Imagine you have multiple water tanks, each having an inlet and an outlet. Connecting them in series means water must flow from one tank to another in a single line. This is similar to how charges move through series capacitors. They share the same charge, but the voltage across each can differ.

One key characteristic of capacitors in series is that the total capacitance is less than any of the individual capacitances. It's like having several narrow pipes in a row, where the narrowest pipe limits the flow of water the most. Here, the combined capacitance is influenced heavily by the smallest capacitor in the series.

Capacitance Formula

The formula for calculating the equivalent capacitance of capacitors in series is a bit different than when they are in parallel. When in series, you don’t simply add the capacitances.

Instead, you calculate using the reciprocal of the capacitance. The formula for three capacitors in series is:

• \( \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \)

This equation shows that you're adding the reciprocals of individual capacitances to find the reciprocal of the equivalent capacitance.

Keep in mind that the approach comes from the conservation of charge. As charge spreads through the series, the overall impact is a reduced capacitance.

Reciprocal Capacitance

Understanding reciprocal capacitance is crucial when working with series capacitors. Think of reverses in mathematics—like flipping fractions or thinking in terms of 'opposite' operations.

• For a single capacitor: if \( C = 5 \mu F \), then its reciprocal is \( \frac{1}{5} = 0.2 \mu F^{-1} \).

Calculating the reciprocal capacitance lets us appropriately handle multiple series capacitors' inverse relationship to the equivalent capacitance.

This may sound tricky, but remember, after finding the sum of reciprocals, a final reciprocal operation is needed to return to familiar capacitance values.

Capacitors in Series

Capacitors in series present an interesting scenario where they act like one big whole. In this configuration, they're often used in circuits needing lower equivalent capacitance than any single one in the group. This can be useful for fine-tuning capacitance values in sensitive electronic circuits.

The distinct behavior of charge in series capacitors leads to less overall capacitance. Each capacitor holds the same charge despite possibly different voltages across them. This uniform charge storage feature is essential for maintaining stability in series-connected circuits.

Applying these principles can help in designing more efficient electronics and understanding series capacitance effect makes troubleshooting much simpler in practical applications.