When there are two charge carriers in an area that have different charges, this means that their ability to do work differs. So, the difference in their potential to do work is often referred to as the potential difference. From a border perspective, we can say that it is the difference in energy stored in two or more devices.

Outline:

• The Potential Difference Across the Capacitor
• The potential difference across a capacitor in a series combination
• Example 1
• Example 2
• Conclusion

The Potential Difference Across the Capacitor

When a capacitor is connected to a circuit, there can be a voltage difference around it as compared to other capacitors, but it primarily depends on the configuration of the circuit. So, if the capacitors are in series configuration and have equal capacitance, then the voltage of each capacitor will be different. Whereas the potential difference of each capacitor will be the same if the capacitors are in parallel configuration.

The potential difference across a capacitor in a series combination

In any series circuit, the current flowing through all the components is equal, however, the voltage across each of the components is different. So, in the case of capacitors, the voltage around the capacitors will be different. Consider a simple capacitor circuit having two capacitors connected in a series configuration:

So according to the Kirchhoff voltage law, the total voltage will be:

Now as we know the charge on a capacitor is proportional to the capacitance and voltage, so from this equation, we can find the voltage if the capacitance and charge values are known.

So now place the values of individual voltages across the capacitor in the Kirchhoff voltage equation:

If the charge on all the capacitors is the same, then the above equation can be written as:

Now to find the potential difference across the first capacitor, the above equation can be:

Now after placing the value of the charge and further simplifying the equation, we get:

Since we are finding the potential difference across the first capacitor, the equation can be rewritten as:

So now for the second capacitor, the equation for finding the potential difference across the capacitor will be:

From the above equation, we can say that the potential difference across any capacitor is the product of voltage and the capacitance of the other capacitor divided by the sum of capacitance. Now in case if there are three capacitors connected in series then the voltage across all three will be:

Example 1: Find the Potential Difference Across each Capacitor in Series

Consider a circuit having three capacitors connected in series with a voltage source of 30 Volts and find the potential difference across each capacitor if the individual capacitance is 12 F, 6 F, and 5 F respectively.

Using the simple formula, we explained earlier to calculate the difference of potential across the first capacitor:

Now, finding the difference of potential across the second capacitor:

Now, finding the difference of potential across the third capacitor:

Alternate Way:

Another way to find the voltage across the capacitor is by first calculating the total charge and then using the charge equation to find the voltage across each capacitor. First, find the equivalent capacitance:

Now, finding the total charge stored in the capacitors:

Now, finding the voltage across all three capacitors:

It is obvious from the results of both methods that both approaches are correct, however, the first approach is quite easy to follow as one needs to just place the values in the formula. Now we can verify the calculated values of potential difference by using Kirchhoff law. According to this law sum of all the voltage differences will be zero:

Example 2: Find the Potential Difference Across each Capacitor in a Series-parallel Combination

Consider a DC capacitor circuit having three capacitors, two are in parallel and the third one in series is connected to a 9 Volt supply. Find the potential difference across each capacitor.

As in parallel combination, the voltage difference remains the same, so we have to find the potential difference for just one capacitor in parallel. So, to find the potential difference between the capacitors in parallel using the charge equation:

Now in the case of parallel capacitors, the equivalent capacitance of parallel capacitance will be:

Now both capacitors are in series so, the equivalent capacitance of capacitors will be:

So, now Just place the values in the charge equation for calculating the total charge stored in the capacitors in circuits:

Now the potential difference across the capacitors in parallel will be:

The voltage across both the capacitors will be the same, so now for the capacitor C₃ that is in series combination with the two parallel ones, the potential difference across it will be:

It is obvious from the results of both methods that both approaches are correct, however, the first approach is quite easy to follow as one needs to just place the values in the formula. Now we can verify the calculated values of potential difference by using Kirchhoff’s law, according to this law the sum of all the voltage differences will be zero:

Conclusion

In the series configuration of a circuit, the current across each of the components is equal, and the voltage is different, whereas in parallel configuration the voltage or the potential across each component is the same. The potential difference is the difference in energies between the components of a circuit, and based on this potential, the ability to do work differs.

The same is true in the case of capacitors, so there are two approaches to finding the potential difference across one is by using the charge equation and the other is through the potential difference equation.