Combination of Capacitors Notes | Study Physics For JEE - JEE

JEE: Combination of Capacitors Notes | Study Physics For JEE - JEE

The document Combination of Capacitors Notes | Study Physics For JEE - JEE is a part of the JEE Course Physics For JEE.
All you need of JEE at this link: JEE

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
Combination of Capacitors Notes | Study Physics For JEE - JEE

V = Potential across capacitor C1

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

Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE

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 :

Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEECombination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE

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. 

Combination of Capacitors Notes | Study Physics For JEE - JEE

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  𝐶𝑝  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+ 𝑄+ 𝑄3

On the left-hand side of this equation, we use the relation  𝑄 = 𝐶𝑝𝑉, which holds for the entire network. On the right-hand side of the equation, we use the relations  

𝑄= 𝐶1𝑉 , 𝑄= 𝐶2𝑉 , and  𝑄= 𝐶3𝑉  for the three capacitors in the network. 

In this way we obtain 𝐶𝑝𝑉 = 𝐶1𝑉 + 𝐶2𝑉 + 𝐶3𝑉.

This equation, when simplified, is the expression for the equivalent capacitance of the parallel network of three capacitors: 𝐶𝑝 = 𝐶+ 𝐶+ 𝐶3

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

Combination of Capacitors Notes | Study Physics For JEE - JEE

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

We can say that Combination of Capacitors Notes | Study Physics For JEE - JEE

V1 = potential across C1

Q = charge on positive plate of C1

C1 = capacitance of capacitor similarly

Combination of Capacitors Notes | Study Physics For JEE - JEE


V1 : V2 : V3 = Combination of Capacitors Notes | Study Physics For JEE - JEE

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

Combination of Capacitors Notes | Study Physics For JEE - JEE

Key Points:

  1. In a series combination, the smallest capacitor gets maximum potential.
  2.  Combination of Capacitors Notes | Study Physics For JEE - JEE, Combination of Capacitors Notes | Study Physics For JEE - JEE, Combination of Capacitors Notes | Study Physics For JEE - JEE
    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: Combination of Capacitors Notes | Study Physics For JEE - JEE ........................
  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 = Combination of Capacitors Notes | Study Physics For JEE - JEE
    Ucombination = Combination of Capacitors Notes | Study Physics For JEE - JEE
    The energy supplied by the battery in charging the combination
    Ubattery = Q × V = Q . Combination of Capacitors Notes | Study Physics For JEE - JEE = Combination of Capacitors Notes | Study Physics For JEE - JEE
    Combination of Capacitors Notes | Study Physics For JEE - JEE

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.

Combination of Capacitors Notes | Study Physics For JEE - JEE

∴   V = V+ V+ V3

Using V = Q/C

Combination of Capacitors Notes | Study Physics For JEE - JEE

  Equivalent capacitance for series combination = Combination of Capacitors Notes | Study Physics For JEE - JEE

In general, Combination of Capacitors Notes | Study Physics For JEE - JEE

    

Solved Examples:

Example 1: Find charge on each capacitor.

Combination of Capacitors Notes | Study Physics For JEE - JEE

Sol. Charge on C1 = C1V1 = 2 × (20 - 5)μC

Combination of Capacitors Notes | Study Physics For JEE - JEE

= 30 μC

Charge on C2 = C2V2 = 2 × (20 - (-10))μC

= 60 μC

Charge on C3 = C3V3 = 4 × (20 - 10)μC

= 40 μC


Example 2: Find charge on each capacitor.  

Combination of Capacitors Notes | Study Physics For JEE - JEE

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    

 Combination of Capacitors Notes | Study Physics For JEE - JEE

(x - 10)C1  (x - 0)C2  (x - 20)C3 = 0

⇒ 2x - 20 2x 4x - 80 = 0

⇒ x = 25 Therefore

so Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE


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

Combination of Capacitors Notes | Study Physics For JEE - JEE Combination of Capacitors Notes | Study Physics For JEE - JEE

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

Combination of Capacitors Notes | Study Physics For JEE - JEE

Example 4: 

Combination of Capacitors Notes | Study Physics For JEE - JEE

Find voltage across capacitor C1.

Sol. 

Combination of Capacitors Notes | Study Physics For JEE - JEE

Now from charge conservation at node x and y

for x

(x - 4)C1 + (x - 2)C2 + (x - y)C3 = 0 ⇒    

 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 ⇒ (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 = Combination of Capacitors Notes | Study Physics For JEE - JEE Combination of Capacitors Notes | Study Physics For JEE - JEE


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.
Combination of Capacitors Notes | Study Physics For JEE - JEE

(iii) Utotal in capacitors 

(iv) heat produced in the circuit

Sol. Combination of Capacitors Notes | Study Physics For JEE - JEE

Ceq = 1 μF.

(i) Q = Ceq V = 30 μC

(ii) charge on 3μF capacitor = 30 μC

energy = Combination of Capacitors Notes | Study Physics For JEE - JEE = Combination of Capacitors Notes | Study Physics For JEE - JEE = 150 μJ

(iii) Utotal = Combination of Capacitors Notes | Study Physics For JEE - JEE = 450 μJ

(iv) Heat produced = (30 μC) (30) - 450 μJ = 450 μJ


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.

Combination of Capacitors Notes | Study Physics For JEE - JEE  Combination of Capacitors Notes | Study Physics For JEE - JEE

Combination of Capacitors Notes | Study Physics For JEE - JEE

Now applying kirchhoff voltage law

Combination of Capacitors Notes | Study Physics For JEE - JEE

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

3q = - 10

Charge flow = - Combination of Capacitors Notes | Study Physics For JEE - JEE μC.

Charge on capacitor of capacitance 1μF = 20 q = Combination of Capacitors Notes | Study Physics For JEE - JEE

Charge on capacitor of capacitance 2μF = 30 q = Combination of Capacitors Notes | Study Physics For JEE - JEE

The document Combination of Capacitors Notes | Study Physics For JEE - JEE is a part of the JEE Course Physics For JEE.
All you need of JEE at this link: JEE

Related Searches

Combination of Capacitors Notes | Study Physics For JEE - JEE

,

Sample Paper

,

mock tests for examination

,

video lectures

,

pdf

,

ppt

,

Viva Questions

,

past year papers

,

study material

,

Objective type Questions

,

Combination of Capacitors Notes | Study Physics For JEE - JEE

,

Extra Questions

,

Important questions

,

MCQs

,

Exam

,

Previous Year Questions with Solutions

,

practice quizzes

,

shortcuts and tricks

,

Free

,

Summary

,

Semester Notes

,

Combination of Capacitors Notes | Study Physics For JEE - JEE

;