Capacitors in Series and Parallel Class 12 Notes | EduRev

Physics For JEE

Class 12 : Capacitors in Series and Parallel Class 12 Notes | EduRev

The document Capacitors in Series and Parallel Class 12 Notes | EduRev is a part of the Class 12 Course Physics For JEE.
All you need of Class 12 at this link: Class 12

How Capacitors are connected?

Capacitors combination can be made in many ways. The combination is connected to a battery to apply a potential difference (V) and charge the plates (Q). We can define the equivalent capacitance of the combination between two points to be: C = Q/V

Two frequently used methods of combination are: 

  • Parallel combination
  • Series combination

Parallel Combination of Capacitors

When one plate of one capacitor is connected with one plate of the other capacitor, such combination is called parallel combination.

All capacitors have the same potential difference but different charges.

We can say that : Q1 = C1V

Q1 = Charge on capacitor C1

C1 = Capacitance of capacitor C
Capacitors in Series and Parallel Class 12 Notes | EduRev

V = Potential across capacitor C1

The charge on the capacitor is proportional to its capacitance Q ยต C

Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev

Where Q = Q1 + Q2 + Q3 ..............

Key Points:

  1. The maximum charge will flow through the capacitor of the largest value.
  2. Equivalent capacitance of parallel combination, Ceq = C+ C2 + C3
  3. Equivalent capacitance is always greater than the largest capacitor of combination.
  4. Half of the energy supplied by the battery is stored in the form of electrostatic energy and half of the energy is converted into heat through resistance.
  5. Energy stored in the combination :

Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRevCapacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev

Formulae Derivation for Parallel combination:

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 Figure. 

Capacitors in Series and Parallel Class 12 Notes | EduRev

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  ๐ถ๐‘  of the parallel network, we note that the total charge Q stored by the network is the sum of all the individual charges: 

๐‘„ = ๐‘„+ ๐‘„+ ๐‘„3

On the left-hand side of this equation, we use the relation  ๐‘„ = ๐ถ๐‘๐‘‰, which holds for the entire network. On the right-hand side of the equation, we use the relations  

๐‘„= ๐ถ1๐‘‰ , ๐‘„= ๐ถ2๐‘‰ , and  ๐‘„= ๐ถ3๐‘‰  for the three capacitors in the network. 

In this way we obtain ๐ถ๐‘๐‘‰ = ๐ถ1๐‘‰ + ๐ถ2๐‘‰ + ๐ถ3๐‘‰.

This equation, when simplified, is the expression for the equivalent capacitance of the parallel network of three capacitors: ๐ถ๐‘ = ๐ถ+ ๐ถ+ ๐ถ3

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

Series Combination of Capacitors

When initially uncharged capacitors are connected as shown, then the combination is called series combination

Capacitors in Series and Parallel Class 12 Notes | EduRev

All capacitors will have the same charge but different potential difference across them.

We can say that Capacitors in Series and Parallel Class 12 Notes | EduRev

V1 = potential across C1

Q = charge on positive plate of C1

C1 = capacitance of capacitor similarly

Capacitors in Series and Parallel Class 12 Notes | EduRev


V1 : V2 : V3 = Capacitors in Series and Parallel Class 12 Notes | EduRev

We can say that potential difference across capacitor is inversely proportional to its capacitance in series combination. 

Capacitors in Series and Parallel Class 12 Notes | EduRev

Key Points:

  1. In a series combination, the smallest capacitor gets maximum potential.
  2.  Capacitors in Series and Parallel Class 12 Notes | EduRev, Capacitors in Series and Parallel Class 12 Notes | EduRev, Capacitors in Series and Parallel Class 12 Notes | EduRev
    Where V = V1 +   V2 + V3
  3. Equivalent Capacitance: 
    Equivalent capacitance of any combination is that capacitance which when connected in place of the combination stores the same charge and energy as that of the combination
    In series: Capacitors in Series and Parallel Class 12 Notes | EduRev ........................
  4. In series, the combination equivalent is always less than the smallest capacitor of the combination.
  5. Half of the energy supplied by the battery is stored in form of electrostatic energy and half of the energy is converted into heat through resistance.
  6. Energy stored in the combination:
    Ucombination = Capacitors in Series and Parallel Class 12 Notes | EduRev
    Ucombination = Capacitors in Series and Parallel Class 12 Notes | EduRev
    The energy supplied by the battery in charging the combination
    Ubattery = Q ร— V = Q . Capacitors in Series and Parallel Class 12 Notes | EduRev = Capacitors in Series and Parallel Class 12 Notes | EduRev
    Capacitors in Series and Parallel Class 12 Notes | EduRev

Formulae Derivation for Series combination:

Let the capacitance of each capacitor be C1, Cand Cand their equivalent capacitance is Ceq.

As these capacitors are connected in series, thus charge across each capacitor is same as Q. When some electrical components, let say 3, are connected in series with each other, the potential difference of the battery V gets divided across each component as

V1, Vand Vas shown in the figure.

Capacitors in Series and Parallel Class 12 Notes | EduRev

โˆด   V = V+ V+ V3

Using V = Q/C

Capacitors in Series and Parallel Class 12 Notes | EduRev

  Equivalent capacitance for series combination = Capacitors in Series and Parallel Class 12 Notes | EduRev

In general, Capacitors in Series and Parallel Class 12 Notes | EduRev

    

Solved Examples:

Example 1: Find charge on each capacitor.

Capacitors in Series and Parallel Class 12 Notes | EduRev

Sol. Charge on C1 = C1V1 = 2 ร— (20 - 5)ฮผC

Capacitors in Series and Parallel Class 12 Notes | EduRev

= 30 ฮผC

Charge on C2 = C2V2 = 2 ร— (20 - (-10))ฮผC

= 60 ฮผC

Charge on C3 = C3V3 = 4 ร— (20 - 10)ฮผC

= 40 ฮผC


Example 2: Find charge on each capacitor.  

Capacitors in Series and Parallel Class 12 Notes | EduRev

Sol. Charge on C1 = (x - 10) C1

Charge on C2 = (x - 0) C2

Charge on C3 = (x - 20) C3

Now from charge conservation at node x    

 Capacitors in Series and Parallel Class 12 Notes | EduRev

(x - 10)C1  (x - 0)C2  (x - 20)C3 = 0

โ‡’ 2x - 20 2x 4x - 80 = 0

โ‡’ x = 25 Therefore

so Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev


Example 3: In the given circuit find out the charge on each capacitor. (Initially they are uncharged)

Capacitors in Series and Parallel Class 12 Notes | EduRev Capacitors in Series and Parallel Class 12 Notes | EduRev

Sol. Let potential at A is 0, so at D it is 30 V, at F it is 10 V and at point G potential is -25V. Now apply Kirchhoff's Ist law at point E. (total charge of all the plates connected to 'E' must be same as before i.e. 0)

Therefore, (x - 10) +  (x - 30) 2 +(x 25) 2 = 0

5x = 20

x = 4

Final charges :

Q2mF = (30 - 4) 2 = 52 mC

Q1mF = (10 - 4) = 6 mC

Q2mF = (4 - (-25)) 2 = 58 mC

Capacitors in Series and Parallel Class 12 Notes | EduRev

Example 4: 

Capacitors in Series and Parallel Class 12 Notes | EduRev

Find voltage across capacitor C1.

Sol. 

Capacitors in Series and Parallel Class 12 Notes | EduRev

Now from charge conservation at node x and y

for x

(x - 4)C1 + (x - 2)C2 + (x - y)C3 = 0 โ‡’    

 2(x - 4) + 2(x - 2) (x - y) 2 = 0

6x - 2y - 12 = 0 .....(1)

For y

(y - x)C3 +  [y -(-4)]C4  (y - 0)C5 = 0 โ‡’ (y - x)2 (y 4) 2 y 2 = 0

= 6y - 2x 8 = 0 .....(2)

eq. (1) & (2)

y = - 3 Therefore 

 x = 7 Therefore 

So potential difference = x - y = Capacitors in Series and Parallel Class 12 Notes | EduRev Capacitors in Series and Parallel Class 12 Notes | EduRev


Example 5: Three initially uncharged capacitors are connected in series as shown in circuit with a battery of emf 30V. Find out following :

(i) charge flow through the battery,

(ii) potential energy in 3 mF capacitor.
Capacitors in Series and Parallel Class 12 Notes | EduRev

(iii) Utotal in capacitors 

(iv) heat produced in the circuit

Sol. Capacitors in Series and Parallel Class 12 Notes | EduRev

Ceq = 1 ฮผF.

(i) Q = Ceq V = 30 ฮผC

(ii) charge on 3ฮผF capacitor = 30 ฮผC

energy = Capacitors in Series and Parallel Class 12 Notes | EduRev = Capacitors in Series and Parallel Class 12 Notes | EduRev = 150 ฮผJ

(iii) Utotal = Capacitors in Series and Parallel Class 12 Notes | EduRev = 450 ฮผJ

(iv) Heat produced = (30 ฮผC) (30) - 450 ฮผJ = 450 ฮผJ


Example 6: Two capacitors of capacitance 1 mF and 2mF are charged to potential difference 20 V and 15 V as shown in figure. If now terminal B and C are connected together terminal A with positive of battery and D with negative terminal of battery then find out final charges on both the capacitor.

Capacitors in Series and Parallel Class 12 Notes | EduRev  Capacitors in Series and Parallel Class 12 Notes | EduRev

Capacitors in Series and Parallel Class 12 Notes | EduRev

Now applying kirchhoff voltage law

Capacitors in Series and Parallel Class 12 Notes | EduRev

- 40 - 2q - 30 - q = - 60

3q = - 10

Charge flow = - Capacitors in Series and Parallel Class 12 Notes | EduRev ฮผC.

Charge on capacitor of capacitance 1ฮผF = 20 q = Capacitors in Series and Parallel Class 12 Notes | EduRev

Charge on capacitor of capacitance 2ฮผF = 30 q = Capacitors in Series and Parallel Class 12 Notes | EduRev


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