Formula for energy stored in the capacitor – Derive

We know that a capacitor can store electric charge and can release them whenever required. So, a capacitor has the capacity to store electric charges. Again, if we apply a voltage to a capacitor an electric field develops inside it. The application of voltage produces electric field and this electric field forces the charges to move from one plate to another plate. This rises a voltage across the capacitor which is different from the voltage of the battery. Electrons will continue to move from one plate to another plate until the voltage across the capacitor becomes equal to the voltage of the battery. As the voltage across the capacitor develops an energy starts to be stored in the capacitor. In this article, we are going to discuss the formula and derivation of the energy stored in the capacitor. This will be helpful for the students of class 12.

Capacitor
Capacitor

Contents in this article:

  1. Which type of energy stored in a capacitor?
  2. Formula for energy stored in the capacitor.
  3. Derivation of energy stored in the capacitor.
  4. How to increase the energy stored in a capacitor?
  5. How to find the energy stored in a capacitor?
  6. Maximum energy stored in a capacitor.
  7. Problems related to energy of a capacitor.

Which type of energy stored in a capacitor?

The energy stored by the capacitor is the electrostatic potential energy. Such type of energy appears due to the storage of electric charges in the electric field. All type of capacitors like parallel plate capacitor, spherical capacitor, cylindrical capacitor, etc. stores same type of energy inside them. The unit of the energy stored in the capacitor is same as the unit of energy we know. It is Joule in SI system and erg in CGS.

Formula for energy stored in the capacitor

If Q, V and C be the charge, voltage and capacitance of a capacitor, then the formula for the energy stored in the capacitor is, \small {\color{Blue} U=\frac{1}{2}CV^{2}}. ……………….(1)

Again, Q=CV. So, we can re-write the equation in two different ways as, \small {\color{Blue} U=\frac{1}{2}QV} …………(2)

And, \small {\color{Blue} U=\frac{1}{2}\frac{Q^{2}}{C}} ……………..(3)

The above three equations give the formula for the energy stored by a capacitor.

Derivation of the energy stored in the capacitor

As the charges shifted from one plate to another plate of a capacitor, a voltage develops in the capacitor. This voltage opposes further shifting of electric charges. Now, to give more charges to the capacitor a work is to be done against the voltage drop. This work stores as the electrostatic potential energy in the capacitor.

Let at any instant the electric charge on the capacitor is q and the voltage is v. Now, to give another dq amount charge to the capacitor, the work done against the developed voltage is, \small {\color{Blue} dW=v.dq}

Now, q = C.v or, v = q/C.

Then, \small {\color{Blue} dW=\frac{q}{C}dq}

Now, let we want to charge the capacitor up to Q amount from zero value. Then total work done in charging the capacitor by Q is, \small {\color{Blue} W=\int_{0}^{Q}\frac{q.dq}{C}}

or, \small {\color{Blue} W=\frac{1}{2}CV^{2}} …………………..(4)

This work done is stored as the electrostatic potential energy (U) of the capacitor.

Now, using Q=CV formula one can re-write this equation in other two forms.

How to find the energy stored in a capacitor?

One can easily determine the energy stored in a capacitor by using above formulae. We have to know the values of any two quantities among C, V and Q. One can measure the value of voltage (V) by a voltmeter or multimeter and capacitance of the capacitor can be determined from the formula for respective capacitors. (See the formula for different types of capacitors). Again, we can use the energy stored calculator to find the energy of the capacitor.

How to increase the potential energy of a capacitor?

Parallel plate Capacitor with circular plate
Parallel plate Capacitor with circular plate

There are two ways to increase the energy stored in a capacitor.

  1. By increasing the voltage of the capacitor without changing the parameters and properties of the capacitor.
  2. In the above formulae, one can see that electrostatic potential energy of the capacitor will increase if the capacitance increases when voltage remains same. So, one can increase the energy stored in a parallel plate capacitor by inserting dielectric medium or slab between the plates at the time of charging the capacitor.

Numerical problems related to the energy of capacitor

  1. A parallel plate capacitor is fully charged by 2 coulomb and thereby its potential increases to 3 volts. What will be the maximum energy stored in the parallel plate capacitor?

Answer: Here, the maximum charge of the parallel plate capacitor is 2 C and the corresponding voltage is 3 volts. Then using equation-2 we get,

Energy stored = 1/2 (QV) = (2×3)÷2 = 3 Joule.

2. A parallel plate capacitor has its capacitance of 2 micro-farad. Now, if you place a dielectric medium (K=2) between the plates keeping a battery of 10 voltage on. What will be the ratio of potential energy of the capacitor before and after placing the dielectric medium?

Answer: Here, the battery is always on. So, the voltage is constant all time. given, V=10 volts, C= 2 micro-farad. After placing the dielectric medium or slab, the capacitance becomes C2=KC=4 micro-farad.

Now, from equation-1, \small U=\frac{1}{2}CV^{2}.

So, \small \frac{U_{1}}{U_{2}}=\frac{\frac{CV^{2}}{2}}{\frac{4CV^{2}}{2}}

or, \small \frac{U_{1}}{U_{2}}=\frac{1}{4}

or, U1 : U2 = 1 : 4

Thus, the final energy increases and becomes four times the initial value of energy.

Some popular questions on this topic:

  1. Where the energy stores in a capacitor?
  2. Derive the expression for the energy stored in a capacitor.
  3. Deduce the expression for the electrostatic energy stored in a capacitor.
  4. If we insert a dielectric medium in the capacitor keeping the voltage constant, will its potential energy increase or decrease? Explain.

This is all from this article on the electrostatic energy of a capacitor. If you have any question on this topic you can ask me in the comment section.

Thank you!

Related posts:

  1. Different type of capacitors and formula for capacitance
  2. Parallel plate capacitor with dielectric medium
  3. Electrostatic potential energy

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