The Formula For Energy Stored In A Capacitor

Firstly, let us learn briefly about what a capacitor is and discuss its terminologies.

What is a capacitor?

A capacitor is a device that stores electrical energy within an electric field.

Structure: A capacitor consists of two conductors placed closely but insulated from each other. If one of the conductors has a charge of +Q on it, the other conductor has a charge of -Q on it.

Types of capacitors:

The various types of capacitors available include 

Electrolytic Capacitor, Mica Capacitor, Film Capacitor, Non-Polarized Capacitor, Ceramic Capacitor, etc. But, the Parallel Plate Capacitor is the most used capacitor. In this article, too, we will be studying parallel plate capacitors.

Applications:

The various applications of capacitors are:

1. Used for storing energy.

2. Used for power conditioning

3. Used for signal decoupling

4. Used for signal coupling

5. Used for electronic noise filtering

6. Used for remote sensing

Basic terminologies:

The terminologies associated with a capacitor are:

1. Charge (Q)- it is the value of charge stored in the conductors of a capacitor.

2. Potential difference(V)- the voltage difference between the ends of the conductors is called the potential difference.

3. Capacitance: The capacitance of a capacitor is defined as the ratio of the electric charge ‘Q’ stored in the capacitor and the potential difference ‘V’ present in the capacitor. It is denoted by ‘C’.

Therefore,  (C = Q/V)

Standard unit:

The SI unit of the capacitance of a capacitor is Farad(F). 

The formula for energy stored in a capacitor:

A capacitor containing a charge Q has energy stored between its conductors. This energy is electrostatic potential energy and is denoted by UC.

To derive the formula for energy stored in the capacitor, let us take the example of a charged, empty parallel plate capacitor, a capacitor with a vacuum between its plates or its conductors.

Consider the volume between the two conductors to be Ad and the electrostatic field present between them to be E.

The electric density in this region between the given conductors can be given as,

                                                        UE=(½) ε0E2

If this energy density is multiplied by the volume between the two plates, Ad,

We obtain,

UC= UEAd

 =(½) ε0E2Ad

=(½) ε0 V2/d2Ad [E=V/d]

=(½) V2ε0A/d

=(½)V2C [C= ε0A/d]

This equation can be further simplified as follows,

UC= (½) V2C

=(½)*((Q*Q)/C)

=(½)QV

Thus, from the above equations, we concluded that the formula for energy stored in the capacitor is,

 UC = (½) V2C

Combinations of capacitors:

We find either a series or parallel combination of capacitors in many circuits. In that case, the total energy stored in the capacitors is given by the formula:

U = U1 + U2 + U3 + …+ UN,

 where U1, U2 are the energies stored in one capacitor.

For both series and parallel combinations of capacitors, the energy stored in them is additive in nature.

Numericals:

Let’s check out some problems to find the energy stored in a capacitor.

1. If a capacitor has a charge of 10C and a potential difference of 20 volts across its ends, find the energy stored in the capacitor.

Ans: Using the formula,

       U=(½)QV, the energy stored in this capacitor is

  =(½)*10*20

=100J

2. If a capacitor has a charge of 20C and a potential difference of 200 volts across its ends, find the energy stored in the capacitor.

Ans: Using the formula,

 U=(½)QV, the energy stored in this capacitor is

  =(½)*20*200

=2000J

3. If a capacitor has a charge of 10C and capacitance of 20F across its ends, find the energy stored in the capacitor.

Ans: Using the formula,

U=(½)*((Q*Q)/C)

=10*10/(2*20)

=2.5J

4. If a capacitor has a charge of 0C and a potential difference of 90 volts across its ends, find the energy stored in the capacitor.

Ans: Using the formula,

       U=(½)*(QV), the energy stored in this capacitor is

  =(½)*0*90

=0J

5. If a capacitor has a charge of 10C and a potential difference of 200 volts across its ends, find the energy stored in the capacitor.

Ans: Using the formula,

       U=(½)QV, the energy stored in this capacitor is

  =(½)*10*200

=1000J

Practice time:

Put your acquired knowledge to the test by solving the given problems:

1. If the energy stored in a capacitor is 100J and the capacitor has a charge of 10C, find the capacitance of the capacitor.

2. If the energy stored in a capacitor is 500J and the capacitor has a charge of 10C, find the capacitance of the capacitor.

3. If the energy stored in a capacitor is 1000 J and the capacitor has a charge of 20C, find the potential difference across its ends.

4. If the energy stored in a capacitor is 100J and the capacitor has a charge of 10C, find the potential difference across its ends.

5. Derive the formula for energy stored in a capacitor.

Conclusion:

In this entire material, we focused mainly on the derivation of the formula for energy stored in a capacitor. We derived two different formulas to find the energy in terms of charge and potential difference and charge and capacitance. To sharpen our prerequisites, we also learned what a capacitor is, its symbol, its circuit diagram, etc. We revised the basic terminologies associated with capacitors and had a quick recap about capacitance and its units. As we derived the formula, we went for its implementation and solved a few questions. Hope this material will ease the topic.