(i) Capacitor in series (ii) Capacitance in parallel - Explanation with Solved Example Problems

**Capacitor
in series and**** ****parallel**

Consider three
capacitors of capacitance C_{1},â€†C_{2} and C_{3}
connected in series with a battery of voltage V as shown in the Figure 1.60
(a).

As soon as the battery is connected to the capacitors in series, the electrons of charge â€“Q are transferred from
negative terminal to the right plate of C_{3} which pushes the
electrons of same amount -Q from left plate of C_{3} to the right plate
of C_{2} due to electrostatic induction. Similarly, the left plate of C_{2}
pushes the charges of â€“Q to the right plate of C_{1} which induces the
positive charge +Q on the left plate of C_{1}. At the same time,
electrons of charge â€“Q are transferred from left plate of C_{1} to
positive terminal of the battery.

By these processes, each
capacitor stores the same amount of charge Q. The capacitances of the
capacitors are in general different, so that the voltage across each capacitor
is also different and are denoted as V_{1}, V_{2} and V_{3}
respectively.

The total voltage across
each capacitor must be equal to the voltage of the battery.

If three capacitors in
series are considered to form an equivalent single capacitor C_{s}
shown in Figure 1.60(b), then we have V=Q/C_{s}. Substituting this expression into
equation (1.104), we get

Thus, the inverse of the
equivalent capacitance C_{S} of three capacitors connected in series is
equal to the sum of the inverses of each capacitance. This equivalent
capacitance C_{S} is always less than the smallest individual
capacitance in the series.

Consider three
capacitors of capacitance C_{1}, C_{2} and C_{3}
connected in parallel with a battery of voltage V as shown in Figure 1.61 (a).

Since corresponding
sides of the capacitors are connected to the same positive and negative
terminals of the battery, the voltage across each capacitor is equal to the
batteryâ€™s voltage. Since capacitance of the capacitors is different, the charge
stored in each capacitor is not the same. Let the charge stored in the three
capacitors be Q_{1}, Q_{2,} and Q_{3} respectively.
According to the law of conservation of total charge, the sum of these three
charges is equal to the charge Q transferred by the battery,

If these three
capacitors are considered to form a single capacitance C_{P} which
stores the total charge Q as shown in the Figure 1.61(b), then we can write Q =
C_{P}V. Substituting this in equation (1.107), we get

Thus, the equivalent
capacitance of capacitors connected in parallel is equal to the sum of the
individual capacitances.

The equivalent
capacitance C_{P} in a parallel connection is always greater than the
largest individual capacitance. In a parallel connection, it is equivalent as
area of each capacitance adds to give more effective area such that total
capacitance increases.

**EXAMPLE 1.22**

Find the equivalent capacitance
between P and Q for the configuration shown below in the figure (a).

**Solution**

The capacitors 1 ÂµF and 3ÂµF are connected in parallel and 6ÂµF and 2 ÂµF are also separately connected in parallel. So these parallel combinations reduced to equivalent single capacitances in their respective positions, as shown in the figure (b).

C_{eq} = 1ÂµF + 3ÂµF = 4ÂµF

C_{eq} = 6ÂµF + 2ÂµF = 8ÂµF

From the figure (b), we infer that
the two 4 ÂµF capacitors are connected in series and the two 8 ÂµF capacitors are
connected in series. By using formula for the series, we can reduce to their
equivalent capacitances as shown in figure (c).

From the figure (c), we infer that
2ÂµF and 4ÂµF are connected in parallel. So the equivalent capacitance is given
in the figure (d).

C_{eq} = 2ÂµF + 4ÂµF = 6ÂµF

Thus the combination of capacitances
in figure (a) can be replaced by a single capacitance 6 ÂµF.

Tags : Explanation with Solved Example Problems , 12th Physics : Electrostatics

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12th Physics : Electrostatics : Capacitor in series and parallel | Explanation with Solved Example Problems

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