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V = V 1 + V 2 + V 3 .

Potential V is measured across an equivalent capacitor that holds charge Q and has an equivalent capacitance C S . Entering the expressions for V 1 , V 2 , and V 3 , we get

Q C S = Q C 1 + Q C 2 + Q C 3 .

Canceling the charge Q , we obtain an expression containing the equivalent capacitance, C S , of three capacitors connected in series:

1 C S = 1 C 1 + 1 C 2 + 1 C 3 .

This expression can be generalized to any number of capacitors in a series network.

Series combination

For capacitors connected in a series combination    , the reciprocal of the equivalent capacitance is the sum of reciprocals of individual capacitances:

1 C S = 1 C 1 + 1 C 2 + 1 C 3 + .

Equivalent capacitance of a series network

Find the total capacitance for three capacitors connected in series, given their individual capacitances are 1.000 μ F , 5.000 μ F , and 8.000 μ F .

Strategy

Because there are only three capacitors in this network, we can find the equivalent capacitance by using [link] with three terms.

Solution

We enter the given capacitances into [link] :

1 C S = 1 C 1 + 1 C 2 + 1 C 3 = 1 1.000 μ F + 1 5.000 μ F + 1 8.000 μ F 1 C S = 1.325 μ F .

Now we invert this result and obtain C S = μ F 1.325 = 0.755 μ F .

Significance

Note that in a series network of capacitors, the equivalent capacitance is always less than the smallest individual capacitance in the network.

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The parallel combination of capacitors

A parallel combination of three capacitors, with one plate of each capacitor connected to one side of the circuit and the other plate connected to the other side, is illustrated in [link] (a). Since the capacitors are connected in parallel, they all have the same voltage V across their plates . However, each capacitor in the parallel network may store a different charge. To find the equivalent capacitance C P of the parallel network, we note that the total charge Q stored by the network is the sum of all the individual charges:

Q = Q 1 + Q 2 + Q 3 .

On the left-hand side of this equation, we use the relation Q = C P V , which holds for the entire network. On the right-hand side of the equation, we use the relations Q 1 = C 1 V , Q 2 = C 2 V , and Q 3 = C 3 V for the three capacitors in the network. In this way we obtain

C P V = C 1 V + C 2 V + C 3 V .

This equation, when simplified, is the expression for the equivalent capacitance of the parallel network of three capacitors:

C P = C 1 + C 2 + C 3 .

This expression is easily generalized to any number of capacitors connected in parallel in the network.

Parallel combination

For capacitors connected in a parallel combination    , the equivalent (net) capacitance is the sum of all individual capacitances in the network,

C P = C 1 + C 2 + C 3 + .
Figure a shows capacitors C1, C2 and C3 in parallel, with each one connected to a battery. The positive plates of C1, C2 and C3 have charge +Q1, +Q2 and +Q3 respectively and the negative plates have charge –Q1, –Q2 and –Q3 respectively. Figure b shows equivalent capacitor Cp equal to C1 plus C2 plus C3. The charge on the positive plate is equal to +Q equal to Q1 plus Q2 plus Q3. The charge on the negative plate is equal to –Q equal to minus Q1 minus Q2 minus Q3.
(a) Three capacitors are connected in parallel. Each capacitor is connected directly to the battery. (b) The charge on the equivalent capacitor is the sum of the charges on the individual capacitors.

Equivalent capacitance of a parallel network

Find the net capacitance for three capacitors connected in parallel, given their individual capacitances are 1.0 μ F , 5.0 μ F , and 8.0 μ F .

Strategy

Because there are only three capacitors in this network, we can find the equivalent capacitance by using [link] with three terms.

Solution

Entering the given capacitances into [link] yields

C P = C 1 + C 2 + C 3 = 1.0 μ F + 5.0 μ F + 8.0 μ F C P = 14.0 μ F .

Significance

Note that in a parallel network of capacitors, the equivalent capacitance is always larger than any of the individual capacitances in the network.

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Practice Key Terms 2

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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