Question

The capacitors of capacitance \(4 \mu \mathrm{F}, 6 \mu \mathrm{F}\) and \(12 \mu \mathrm{F}\) are connected first in series and then in parallel. What is the ratio of equivalent capacitance in the two cases? (A) \(2: 3\) (B) \(11: 1\) (C) \(1: 11\) (D) \(1: 3\)

Short answer

The ratio of equivalent capacitance in the two cases is \(1: 11\) (Option C).

Step-by-step solution

To find the equivalent capacitance (\(C_s\)) in a series configuration, we use the following formula: \[\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}\] where \(C_1 = 4 \mu F\), \(C_2 = 6 \mu F\), and \(C_3 = 12 \mu F\). #Step 2: Substitute values and solve for equivalent capacitance in series#

Substitute the values of the capacitors and solve for \(C_s\): \[\frac{1}{C_s} = \frac{1}{4} + \frac{1}{6} + \frac{1}{12}\] #Step 3: Simplify the equation for Cs#

To simplify the equation, we can find the least common denominator of \(4\), \(6\), and \(12\), as follows: \[\frac{1}{C_s} = \frac{3+2+1}{12}\] #Step 4: Solve for Cs#

Solve for \(C_s\): \[\frac{1}{C_s} = \frac{6}{12}\] \[C_s = 2 \mu F\] #Step 5: Calculate the equivalent capacitance in parallel configuration#

To find the equivalent capacitance (\(C_p\)) in a parallel configuration, we use the following formula: \[C_p = C_1 + C_2 + C_3\] #Step 6: Substitute values and solve for equivalent capacitance in parallel#

Substitute the values of the capacitors and solve for \(C_p\): \[C_p = 4 + 6 + 12\] \[C_p = 22 \mu F\] #Step 7: Calculate the ratio of equivalent capacitance in series and parallel configurations#

Now that we have the equivalent capacitance in series and parallel configurations, we can find the ratio between these values: \[\text{Ratio}= \frac{C_s}{C_p}\] #Step 8: Substitute values and solve for ratio#

Substitute the values of \(C_s\) and \(C_p\) and solve for the ratio: \[\text{Ratio}= \frac{2}{22}\] \[\text{Ratio}= \frac{1}{11}\] Therefore, the ratio of equivalent capacitance in the two cases is \(\boldsymbol{1: 11}\) (Option C).

Key concepts

Equivalent Capacitance

Understanding the equivalent capacitance is essential when dealing with multiple capacitors in a circuit. When capacitors are combined, their ability to store electric charge can change depending on their connection, such as in series or parallel. Equivalent capacitance is the potential for these capacitors to exhibit a total capacitance. It's akin to summarizing their effect as a single capacitor in the circuit.

To calculate the equivalent capacitance in a circuit configuration, we use specific formulas for capacitors wired in series or parallel. Identifying whether the capacitors are in series or parallel is pivotal. This determines which formula will be applicable. Once you've deduced the configuration, you apply the relevant equation to find the resultant capacitance value. This simplification is crucial for analyzing complex electrical circuits easily.

Series and Parallel Capacitors

Capacitors can be connected in two primary configurations: series and parallel. In a series configuration, capacitors are linked end to end, and they share the same charge. The equivalent capacitance (\(C_s\)) of capacitors in series is found using:

• The formula: \(\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}\)
• This results in a lower total capacitance than any individual capacitor in the series.

In parallel, capacitors are connected directly across from each other, and the total charge is the sum of the charges on each capacitor. Here, the equivalent capacitance (\(C_p\)) is simply:

• The sum of all capacitances: \(C_p = C_1 + C_2 + C_3\)
• In this case, the equivalent capacitance is larger than the largest individual capacitor.

Grasping how capacitance sums differently in series and parallel configurations helps in designing circuits for desired electrical behaviors.

Capacitance Ratio

Determining the ratio of equivalent capacitance can quickly illustrate how capacitors function differently in series compared to parallel setups. After finding both equivalent capacitances - \(C_s\) for series and \(C_p\) for parallel configurations, the ratio provides us with a simple comparison.

For instance, based on the exercise with capacitances of \(4 \mu F\), \(6 \mu F\), and \(12 \mu F\):

• The series equivalent \(C_s\) was calculated as \(2 \mu F\).
• The parallel equivalent \(C_p\) resulted in \(22 \mu F\).

The ratio becomes \(\frac{1}{11}\), highlighting the significant capacitance difference due to different configurations. Such ratios are helpful in predicting and analyzing circuit performance effectively.

Capacitor Configuration

The arrangement of capacitors, known as capacitor configuration, heavily influences both their collective electrical properties and performance in a circuit. Configurations can be tailored for specific functionalities, like timing circuits, filters, and energy storage systems.

Choosing between series and parallel setups depends on your goal:

• Series configurations are useful for applications needing lower capacitance combined with higher voltage tolerance.
• Parallel arrangements increase total capacitance, suitable for circuits requiring more charge to be stored.

Understanding various capacitor configurations helps in selecting the right design for achieving desired outcomes in electrical and electronic systems. It aids in effectively managing the voltage and charge characteristics in a circuit.