A capacitor is a device that can store electric charge and energy in the form of an electric field. Capacitors can be connected in different ways to form a combination of capacitors, which have an equivalent capacitance that depends on the arrangement and the individual capacitances. There are two common methods of connecting capacitors: in series and in parallel.
Series Combination of Capacitors
Voltage Division: When capacitors are connected in series, the same current flows through each capacitor. However, the voltage across each capacitor can differ. According to Kirchhoff’s voltage law, the total voltage applied across the series combination is divided among the capacitors based on their capacitance values.
Equivalent Capacitance: In a series combination, the equivalent capacitance \(\displaystyle C_{eq}\) is less than any individual capacitance in the combination. The reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances:
\(\displaystyle \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots \)
Energy Storage: Each capacitor in the series combination stores energy in the form of an electric field between its plates. The total energy stored in the combination is the sum of the energies stored in each capacitor.
First, we need to assume that there are n capacitors connected in series, with capacitances C1, C2, C3, …, Cn. The total potential difference across the series combination is V, and the charge on each capacitor is Q.
Definition of capacitance, which states that the ratio of the charge Q to the potential difference V is constant and equal to the capacitance. Therefore, for each capacitor, we have:
\(\displaystyle C_i = \frac{Q}{V_i} \)…………………..(1)
where \(\displaystyle C_i\) is the capacitance and \(\displaystyle V_i\) is the potential difference of the ith capacitor.
We need to use the fact that the total potential difference V is equal to the sum of the potential differences of each capacitor, that is:
\(\displaystyle V = V_1 + V_2 + V_3 + … + V_n\) …………………..(2)
Substituting the expressions for \(\displaystyle V_i\) i.e eq.(2) in eq. (1), we get:
\(\displaystyle V = \frac{Q}{C_1} + \frac{Q}{C_2} + \frac{Q}{C_3} + … + \frac{Q}{C_n}\)…………………..(3)
We divided both sides of this equation (3) by Q, which is the charge on each capacitor, and factored out Q from the right-hand side. This gave us:
\(\displaystyle\frac{V}{Q} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + … + \frac{1}{C_n}\) …………………..(4)
We compared this equation with the definition of capacitance, which states that the ratio of the charge Q to the potential difference V is constant and equal to the capacitance. Therefore, for the series combination, we have:
\(\displaystyle C_{eq} = \frac{Q}{V}\) …………………..(5)
where \(\displaystyle C_{eq}\) is the total capacitance.
Substitute equation (4) in equation (5). The equation for the equivalent capacitance of capacitors in series, we have:
\(\displaystyle C_{eq} = \frac{Q}{\frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + … + \frac{1}{C_n}}\)…………………..(6)
We can divide both the numerator and the denominator by \(\displaystyle\frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + … + \frac{1}{C_n}\), which is the same as multiplying by the reciprocal \(\displaystyle\frac{C_1 C_2 C_3 … C_n}{1}\).
This gives us:
\(\displaystyle C_{eq} = \frac{Q}{\frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + … + \frac{1}{C_n}} \cdot \frac{\frac{C_1 C_2 C_3 … C_n}{1}}{\frac{C_1 C_2 C_3 … C_n}{1}} = \frac{Q C_1 C_2 C_3 … C_n}{1}\)
\(\displaystyle C_{eq} = \frac{1}{\frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + … + \frac{1}{C_n}}\)
\(\displaystyle C_{eq} = \frac{1}{\frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + … + \frac{1}{C_n}}\)
Or,
\(\displaystyle \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots \)
This formula shows that the equivalent capacitance of capacitors in series is the reciprocal of the sum of the reciprocals of the individual capacitances.
Parallel Combination of Capacitors
A parallel combination of capacitors is a circuit configuration where the capacitors are connected in parallel, meaning that both terminals of each capacitor are connected to the same two points in the circuit. The equivalent capacitance of a parallel combination of capacitors is the sum of the individual capacitances, as given by the formula:
\(\displaystyle C_{eq} = C_1 + C_2 + C_3 + …\)
where \(\displaystyle C_{eq}\) is the equivalent capacitance, and \(\displaystyle C_1, C_2, C_3, …\) are the capacitances of the individual capacitors.
The advantage of a parallel combination of capacitors is that it increases the total capacitance and the amount of charge that can be stored in the circuit. The voltage across each capacitor in a parallel combination is the same and equal to the source voltage. The charge on each capacitor is proportional to its capacitance, as given by the formula:
\(\displaystyle Q_i = C_i V\)
where \(\displaystyle Q_i\) is the charge on the \(\displaystyle i-th\) capacitor, \(\displaystyle C_i\) is its capacitance, and V is the source voltage. The total charge in the circuit is the sum of the charges on the individual capacitors, as given by the formula:
\(\displaystyle Q_{total} = Q_1 + Q_2 + Q_3 + …\)
where \(\displaystyle Q_{total}\) is the total charge, and \(\displaystyle Q_1, Q_2, Q_3, …\) are the charges on the individual capacitors.
Derivation of capacitors in parallel
Let us consider two capacitors \(\displaystyle C_1\) and \(\displaystyle C_2\) connected in parallel across a battery of voltage V. The charge on each capacitor will be \(\displaystyle Q_1 = C_1 V\) and \(\displaystyle Q_2 = C_2 V\), respectively. The total charge supplied by the battery will be \(\displaystyle Q = Q_1 + Q_2 = (C_1 + C_2) V\). The equivalent capacitance \(\displaystyle C_{eq}\) of the parallel combination is defined as the ratio of the total charge to the voltage, i.e.,
\(\displaystyle C_{eq} = \frac{Q}{V} = \frac{(C_1 + C_2) V}{V} = C_1 + C_2\)
This result can be generalized for any number of capacitors connected in parallel. If we have n capacitors \(\displaystyle C_1, C_2, …, C_n\) connected in parallel across a battery of voltage V, then the total charge supplied by the battery will be \(\displaystyle Q = (C_1 + C_2 + … + C_n) V\). The equivalent capacitance \(\displaystyle C_{eq}\) of the parallel combination will be
\(\displaystyle C_{eq} = \frac{Q}{V} = \frac{(C_1 + C_2 + … + C_n) V}{V} = C_1 + C_2 + … + C_n\)
Or,
\(\displaystyle C_{eq} = C_1 + C_2 + … + C_n\)
Hence, the equivalent capacitance of capacitors in parallel is the sum of their capacitances.
Key Points
Series Combination Of Capacitors
Some key points to remember in a series combination of capacitors are:
- A series combination of capacitors is a circuit configuration where the capacitors are connected in series, meaning that the same charge flows through each capacitor and the total potential difference is the sum of the individual potential differences.
- The equivalent capacitance of a series combination of capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances, as given by the formula:
\(\displaystyle \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + …\)
- The charge on each capacitor in a series combination is the same and equal to the product of the equivalent capacitance and the source voltage, as given by the formula:
\(\displaystyle Q = C_{eq} V\)
- The potential difference across each capacitor in a series combination is proportional to its capacitance and inversely proportional to the equivalent capacitance, as given by the formula:
\(\displaystyle V_i = \frac{C_{eq}}{C_i} V\)
- The energy stored in a series combination of capacitors is equal to half the product of the equivalent capacitance and the square of the source voltage, as given by the formula:
\(\displaystyle U = \frac{1}{2} C_{eq} V^2\)
- The energy stored in a series combination is the same as that stored in a single capacitor with the equivalent capacitance.
Parallel Combination Of Capacitors
Some key points to remember in a parallel combination of capacitors are:
- In a parallel combination of capacitors, the potential difference across each capacitor is the same and equal to the source voltage.
- The charge on each capacitor is proportional to its capacitance and different for each capacitor.
- The equivalent capacitance of a parallel combination of capacitors is the sum of the individual capacitances, as given by the formula:
\(\displaystyle C_{eq} = C_1 + C_2 + C_3 + …\)
- The total charge in the circuit is the sum of the charges on the individual capacitors, as given by the formula:
\(\displaystyle Q_{total} = Q_1 + Q_2 + Q_3 + …\)
- The advantage of a parallel combination of capacitors is that it increases the total capacitance and the amount of charge that can be stored in the circuit.