Question

Two capacitors in series are charged through a resistor. Identical capacitors are instead connected in parallel and charged through the same resistor. How do the times required to fully charge the two sets of capacitors compare?

Short answer

Answer: Capacitors connected in series will fully charge faster compared to those connected in parallel. The time required to fully charge capacitors in series is one-fourth the time required for capacitors in parallel when charged through the same resistor.

Step-by-step solution

Determine the equivalent capacitance of capacitors in series and parallel configurations.

When capacitors are connected in series, their equivalent capacitance is given by the formula: \(\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2}\) When capacitors are connected in parallel, the equivalent capacitance is given by the formula: \(C_{eq} = C_1 + C_2\) Assuming both capacitors have the same capacitance \(C\) in each configuration, let's find the equivalent capacitance for both scenarios.

Calculate the equivalent capacitance for capacitors in series.

For capacitors in series: \(\frac{1}{C_{eq}} = \frac{1}{C} + \frac{1}{C}\) \(\frac{1}{C_{eq}} = \frac{2}{C}\) \(C_{eq(series)} = \frac{C}{2}\)

Calculate the equivalent capacitance for capacitors in parallel.

For capacitors in parallel: \(C_{eq(parallel)} = C + C\) \(C_{eq(parallel)} = 2C\)

Determine the time constant for each configuration.

The time constant for charging a capacitor through a resistor is given by the formula: \(\tau = R C_{eq}\) Here, \(R\) is the resistance through which capacitors are charged. The time required to fully charge the capacitors (to about 99%) is around 5 times the time constant, i.e. \(t_{charge} \approx 5\tau\). Let's calculate the time constant for each configuration.

Calculate the charging time for capacitors in series.

For capacitors in series: \(\tau_{series} = R C_{eq(series)} = R \cdot \frac{C}{2}\) \(t_{charge(series)} \approx 5 \tau_{series} = 5R \cdot \frac{C}{2}\)

Calculate the charging time for capacitors in parallel.

For capacitors in parallel: \(\tau_{parallel} = R C_{eq(parallel)} = R \cdot 2C\) \(t_{charge(parallel)} \approx 5 \tau_{parallel} = 5R \cdot 2C\)

Compare the times required to fully charge the capacitors.

From Steps 5 and 6, we found that: \(t_{charge(series)} = 5R \cdot \frac{C}{2}\) \(t_{charge(parallel)} = 5R \cdot 2C\) The ratio of charging time between the two configurations is: \(\frac{t_{charge(series)}}{t_{charge(parallel)}} = \frac{5R \cdot \frac{C}{2}}{5R \cdot 2C} = \frac{1}{4}\) Thus, the time required to fully charge capacitors in series is one-fourth the time required for capacitors in parallel when charged through the same resistor.