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When capacitors are connected in series, they share the same charge, but the voltage is divided among them. This concept is crucial to understand when determining the equivalent capacitance of capacitors in series.
To find the equivalent capacitance for capacitors connected in series, use the formula:

• \( \frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} \)

Here, \( C_1 \) and \( C_2 \) are the capacitances of the individual capacitors. The equivalent capacitance, \( C_s \), is always less than any of the individual capacitances involved. This is because you're essentially lengthening the path the charge must travel, which acts like stretching out a spring to store less energy, making it harder to store the same charge.
For example, a \(3.00\, \mu\mathrm{F}\) capacitor in series with a \(5.00\, \mu\mathrm{F}\) capacitor results in an equivalent capacitance that can be calculated using this method.

Capacitors in parallel provide a different function than those in series. When you connect capacitors in parallel, the voltage across each capacitor is the same, but the total capacitance increases. This set-up is useful for storing more charge.
To find the equivalent capacitance of parallel capacitors, simply add up their capacitances:

• \( C_p = C_1 + C_2 \)

For the given problem, however, we don't need to calculate the equivalent capacitance directly; we need to understand the impact of parallel connection. Once the \(7.00\, \mu\mathrm{F}\) capacitor was connected in parallel with the \(3.00\, \mu\mathrm{F}\) capacitor, the voltage across them remained the same. This principle allows us to know that the voltage across any capacitor in that parallel section will equal the voltage across the \(3.00\, \mu\mathrm{F}\) capacitor, making calculations straightforward.

The voltage across capacitors is a measure of the electrical potential difference. In a series connection, it's divided between capacitors, while in parallel, all capacitors have the same voltage across them.
For series capacitors, since they hold the same charge, the voltage can be found using:

• \( V = \frac{Q}{C} \)

where \( V \) is the voltage, \( Q \) is the charge, and \( C \) is the capacitance of each capacitor.
After calculating the charge in the series section, you can use it to find the voltage across any single capacitor in the series. For instance, for the \(3.00\, \mu\mathrm{F}\) capacitor, the voltage across it was found by dividing the charge (\(56.25\, \mu C\)) by its capacitance, resulting in \(18.75\, V\). Given the series nature, this voltage matched that found across the parallel-connected \(7.00\, \mu\mathrm{F}\) capacitor.

Equivalent capacitance is a concept that helps simplify complex circuits by replacing multiple capacitors with a single capacitor that stores the same total charge and voltage.
For series capacitors, the equivalent capacitance is less than any contributing capacitance, reflecting their ability to "stretch" the storage capability like a less effective spring.
Calculating equivalent capacitance for parallel capacitors is generally easier because it simply involves summing their capacitances, which directly impacts the circuit's ability to store more charge.
This theoretical equivalence provides an intuitive way to think about how capacitors work together, whether in storing more charge in parallel or balancing charge over a lengthened path in series. Understanding this helps in analyzing the expected behavior of real-world electronic circuits.