**framework: **

**potential difference across capacitor**

When a capacitor is connected to a circuit, there may be a voltage difference across it compared to other capacitors, but this mainly depends on the configuration of the circuit. Therefore, if capacitors are in series configuration and their capacitance is the same, the voltage across each capacitor will be different. Whereas if the capacitors are in parallel configuration then the potential difference across each capacitor will be the same.

**Potential difference across a capacitor in a series combination**

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

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

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

So now substitute the different voltage values on the capacitor into the Kirchhoff voltage equation:

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

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

Now after substituting the value of 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 to find the potential difference across the capacitor would be:

From the above equation, we can say that the potential difference across any capacitor is the product of the voltage and capacitance of the other capacitor divided by the sum of the capacitances. Now if three capacitors are 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 consisting of three capacitors connected in series with a voltage source of 30 volts and find the potential difference across each capacitor if the individual capacitances are 12 F, 6 F and 5 F respectively.

Using the simple formula we explained earlier to calculate the potential difference across a capacitor:

Now, to find the potential difference across the second capacitor:

Now, to find the potential difference across the third capacitor:

**alternative method: **

Another way to find the voltage on a capacitor is to first calculate the total charge and then use the charge equation to find the voltage on each capacitor. First, find the equivalent capacitance:

Now, to find the total charge stored in the capacitor:

Now, finding the voltage across all three capacitors:

From the results of both methods it is clear that both the approaches are correct, however, the first approach is quite easy to follow as one only needs to put the values into the formula. Now we can verify the calculated values of potential difference using Kirchhoff's rule. According to this rule the sum of all voltage differences will be zero:

**Example 2: Find the potential difference across each capacitor in series-parallel combination**

Consider a DC capacitor circuit consisting of three capacitors, two in parallel and the third connected in series 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 only one capacitor in parallel. So, to find the potential difference between capacitors in parallel using the charge equation:

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

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

So, now plug the values into the charge equation to calculate the total charge stored in the capacitors in the circuit:

Now the potential difference between capacitors in parallel will be:

The voltage on both capacitors will be the same, so now for capacitor C_{3} It is in series combination with two parallel connections, the potential difference across it will be:

From the results of both methods it is clear that both the approaches are correct, however, the first approach is quite easy to follow as one only needs to put the values into the formula. Now we can verify the calculated values of potential difference using Kirchhoff's law, according to this rule the sum of all voltage differences will be zero:

**conclusion**

In a series configuration of a circuit, the current in each component is the same, and the voltage is different, whereas in a parallel configuration the voltage or potential across each component is the same. Potential difference is the difference in energy between the components of a circuit, and depending on this potential, the ability to do work varies.

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