Energy Stored in a Capacitor

You already know that capacitors can store electric charges. But, do you know how is the energy stored in a capacitor? And how much energy a capacitor can hold? Here we will study about the energy stored in a capacitor. We will see how much heat we can get out of a combination of capacitors.

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Energy Stored in a Capacitor

Work has to be done to transfer charges onto a conductor, against the force of repulsion from the already existing charges on it. This work is stored as a potential energy of the electric field of the conductor.

 

Suppose a conductor of capacity C is at 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,

V ∝ q or V = Cq; where ‘C’ is a constant of proportionality that depends on the nature of the material of the conductor. This constant is known as the capacitance.

If we wish to transfer more charge to this conductor, work has to be done against the repulsive forces of the charges already present on the conductor. Let us say that we have to transfer a small charge ‘dq’ which takes a small amount of work ‘dW’. Then work done in bringing a small charge dq at this potential (V) is =

The total work done in charging it from 0 to q₀ is now easy to calculate. All we have to do is to take an integral of the above equation between the relevant limits as shown below:

This work is stored as the potential energy and we have:

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

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

Browse more Topics Under Electrostatic Potential And Capacitance

• Electric Potential Energy and Electric Potential
• Capacitors and Capacitance
• Electrostatics of Conductors
• The Parallel Plate Capacitor
• Combination of Capacitors
• Dielectrics and Polarisation
• Effect of Dielectric on Capacitance
• Van De Graaff Generator

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 =

U_E = U/Volume; using the formula C = ε₀A/d, we can write it as:

Browse more Topics under Electrostatic Potential And Capacitance

• Electric Potential Energy and Electric Potential
• Capacitors and Capacitance
• Electrostatics of Conductors
• The Parallel Plate Capacitor
• Combination of Capacitors
• Dielectrics and Polarisation
• Effect of Dielectric on Capacitance
• Van De Graaff Generator

Heat Generated

Since, Q = CV (C = equivalent capacitance)

So, W = (1/2) (CV)² / C = 1/2 CV²

Now the energy stored in a capacitor, U = W =

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

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

Distribution of Charges on Connecting two Charged Capacitors

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

 

We can sum it up in the following table:

(a) Common potential:

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

Q₁ +  Q₂ = C₁V  + C₂V

⇒ 

or we can say that the common potential will be = (Total charge on the capacitors)/(Total capacitance of the system)

(b) To find the values of final charge on either of the capacitors, we use the following:

here ‘V’ is the common potential. Similarly, we can write that:

The heat lost during redistribution:

The following points are to be noted:

• When plates of similar charges are connected with each other ( + with + and – with -) then put all values (Q₁, Q₂, V₁, V₂) with a 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.

Solved Question for You

Question 1: Write a note on conductors and insulators.

Answer: Conductor contains a large number of free charge carriers to conduct electricity while insulator does not contain any free charge carriers to conduct electricity.

• Examples of conductors are metals and graphite.
• Examples of insulators are plastic rod and nylon.