Advertisements
Advertisements
प्रश्न
Derive the expression for resultant capacitance, when the capacitor is connected in series.
उत्तर
Consider three capacitors of capacitance C1, C2 and C3 connected in series with a battery of voltage V as shown in figure (a).
As soon as the battery is connected to the capacitors in series, the electrons of charge -Q are transferred from negative terminal to the right plate of C3which pushes the electrons of same amount -Q from left plate of C3 to the right plate of C2 due to electrostatic induction. Similarly, the left plate of C2 pushes the charges of Q to the right plate of which induces the positive charge +Q on the left plate of C1 At the same time, electrons of charge -Q are transferred from left plate of C1 to positive terminal of the battery.
(a)
Capacitors connected in series
(b)
Equivalent capacitors CS
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 V1, V2 and V3 respectively.
The total voltage across each capacitor must be equal to the voltage of the battery.
V = V1 + V2 + V3 ….. (1)
Since Q = CV, we have V = `"Q"/"C"_1 + "Q"/"C"_2 + "Q"/"C"_3`
Q = `(1/"C"_1 + 1/"C"_2 + 1/"C"_3)` .....(2)
If three capacitors in series are considered to form an equivalent single capacitor Cs shown in figure (b), then we have V = `"Q"/"C"_"s"`
Substituting this expression into equation (2) we get
V = `"Q"/"C"_"s" = "Q"(1/"C"_1 + 1/"C"_2 + 1/"C"_3)`
`1/"C"_"s" = 1/"C"_1 + 1/"C"_2 + 1/"C"_3` ...(3)
Thus, the inverse of the equivalent capacitance Cs of three capacitors connected in series is equal to the sum of the inverses of each capacitance. This equivalent capacitance Cs is always less than the smallest individual capacitance in the series.
APPEARS IN
संबंधित प्रश्न
A capacitor of capacitance ‘C’ is charged to ‘V’ volts by a battery. After some time the battery is disconnected and the distance between the plates is doubled. Now a slab of dielectric constant, 1 < k < 2, is introduced to fill the space between the plates. How will the following be affected? (a) The electric field between the plates of the capacitor Justify your answer by writing the necessary expressions.
(i) Find equivalent capacitance between A and B in the combination given below. Each capacitor is of 2 µF capacitance.
(ii) If a dc source of 7 V is connected across AB, how much charge is drawn from the source and what is the energy stored in the network?
The equivalent capacitance of the combination shown in the figure is _________ .
Following operations can be performed on a capacitor:
X − connect the capacitor to a battery of emf ε.
Y − disconnect the battery.
Z − reconnect the battery with polarity reversed.
W − insert a dielectric slab in the capacitor.
(a) In XYZ (perform X, then Y, then Z) the stored electric energy remains unchanged and no thermal energy is developed.
(b) The charge appearing on the capacitor is greater after the action XWY than after the action XYZ.
(c) The electric energy stored in the capacitor is greater after the action WXY than after the action XYW.
(d) The electric field in the capacitor after the action XW is the same as that after WX.
Three capacitors having capacitances 20 µF, 30 µF and 40 µF are connected in series with a 12 V battery. Find the charge on each of the capacitors. How much work has been done by the battery in charging the capacitors?
Obtain an expression for equivalent capacitance when three capacitors C1, C2 and C3 are connected in series.
A parallel plate capacitor stores a charge Q at a voltage V. Suppose the area of the parallel plate capacitor and the distance between the plates are each doubled then which is the quantity that will change?
Three capacitors are connected in a triangle as shown in the figure. The equivalent capacitance between points A and C is ______.
For the given capacitor configuration
- Find the charges on each capacitor
- potential difference across them
- energy stored in each capacitor.
A capacitor of 4 µ F is connected as shown in the circuit (Figure). The internal resistance of the battery is 0.5 Ω. The amount of charge on the capacitor plates will be ______.
