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Physics Class 12

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The document Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev is a part of the Class 12 Course Physics Class 12.
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3. Energy Stored in a Charged capacitor

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Work has to be done in charging a conductor against the force of repulsion by the already existing charges on it. The work is stored as a potential energy in the electric field of the conductor. Suppose a conductor of capacity C is charged to a potential V0 and let q0 be the charge on the conductor at this instant. The potential of the conductor when (during charging) the charge on it was q (< q0) is,

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Now, work done in bringing a small charge dq at this potential is,

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Therefore, total work done in charging it from 0 to q0 is,

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

This work is stored as the potential energy,

Therefore, Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Further by using q0 = CV0 we can write this expression also as,

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

In general if a conductor of capacity C is charged to a potential V by giving it a charge q, then

 Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

3.1 Energy Density of a Charged Capacitor

This energy is localized on the charges or the plates but is distributed in the field. Since in case of a parallel plate capacitor, the electric field is only between the plates, i.e., in a volume (A × d), the energy density

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

or Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

3.2 Calculation of Capacitance

The method for the calculation of capacitance involves integration of the electric field between two conductors or the plates which are just equipotential surfaces to obtain the potential difference Vab. Thus,

 Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Therefore, Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

3.3 Heat Generated :

(1) Work done by battery

W = QV

Q = charge flow in the battery

V = EMF of battery

(2) W = Ve (When Battery discharging)

W = -Ve (When Battery charging)

(3) Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Q = CV (C = equivalent capacitance)

so W = CV × V = CV2

Now energy on the capacitor Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Therefore, Energy dissipated in form of heat (due to resistance)

H = Work done by battery - {final energy of capacitor - initial energy of capacitor}

Ex.3 At any time S1 switch is opened and S2 is closed then find out heat generated in circuit.

Sol.

 Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev          Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Charge flow through battery = Qf - Qi

= 2CV - CV = CV

H = (CV × 2V) - Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Ex.4 (a) Find the final charge on each capacitor if they are connected as shown in the figure.

Sol. Initially 

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Finally let q charge flows clockwise then

Now applying KVL

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev 
    Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

5q + 50 + 5q - 200 + 2q = 0

12 q - 150 = 0 ⇒ Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

so finally
Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

 

(b) Find heat loss in the above circuit.

ΔH = Energy [initially - finally] on capacitor

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRevDoc: Energy Stored in a Capacitor Class 12 Notes | EduRev

4. Distribution of Charges on Connecting two Charged Capacitors :

When two capacitors C1 and C2 are connected as shown in figure

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Before connecting the capacitors

Parameter

Ist Capacitor

IInd Capacitor

Capacitance

C1

C2

Charge

Q1

Q2

Potential

V1

V2

 

After connecting the capacitors

Parameter

Ist Capacitor

IInd Capacitor

Capacitance

C1

C2

Charge

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Potential

V1

V2


(a) Common potential :

By charge conservation on plates A and C before and after connection.

Q1 +  Q2 = C1V  + C2V

⇒ Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev = Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

(b) Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

(c) Heat loss during redistribution :

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

The loss of energy is in the form of Joule heating in the wire. 

  • When plates of similar charges are connected with each other ( with and - with -) then put all values (Q1, Q2, V1, V2) with positive sign.
  • When plates of opposite polarity are connected with each other ( with -) then take charge and potential of one of the plate to be negative.

 Derivation of above formulae :

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Let potential of B and D is zero and common potential on capacitors is V, then at A and C it will be V.

C1V +  C2V = C1V1 + C2V2

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

H = Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev      Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

when oppositely charged terminals are connected then

Therefore, C1V C2V = C1V1 - C2V2

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Ex.5 Find out the following if A is connected with C and B is connected with D.

(i) How much charge flows in the circuit.

(ii) How much heat is produced in the circuit.

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Sol. Let potential of B and D is zero and common potential on capacitors is V, then at A and C it will be V.

By charge conservation,

3V + 2V = 40 + 30

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev                   

5V = 70 ⇒ V = 14 volt 

Charge flow = 40 - 28 = 12 μC

Now final charges on each plate is shown in the figure.

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

(ii) Heat produced = Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev × 2 × (20)2   Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev × 3 × (10)2Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev × 5 × (14)2

= 400 150 - 490

= 550- 490 = 60 mJ

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

  • When capacitor plates are joined then the charge remains conserved.
  • We can also use direct formula of redistribution as given above.

Ex.6 Repeat above question if A is connected with D and B is connected with C.

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

Sol. Let potential of B and C is zero and common potential

on capacitors is V, then at A and D it will be V

2V +   3V = 10 ⇒ V = 2 volt

Now charge on each plate is shown in the figure.

Heat produced = 400 + 150 - 1

Therefore 2 × 5 × 4

= 550 - 10 = 540 μJ

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

  • Here heat produced is more. Think why ?

Ex.7 Three capacitors as shown of capacitance 1mF, 2mF and 2mF are charged upto potential difference 30 V, 10 V and 15V respectively. If terminal A is connected with D, C is connected with E and F is connected with B. Then find out charge flow in the circuit and find the final charges on capacitors.

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev          Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev                Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev 

Sol. Let charge flow is q.

Now applying Kirchhoff's voltage low

Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev     

     Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

- 2q = - 25

q = 12.5 mC

Final charges on plates

    Doc: Energy Stored in a Capacitor Class 12 Notes | EduRev

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