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A capacitor is a device which stores electrical energy by separating two conductors by some distance. It is filled with an insulating material called dielectric. There are certain capacitors in which the plates are separated by vacuum. These kinds of capacitors are called vacuum capacitors. The amount of energy stored in a capacitor is determined by a term called capacitance. Capacitance can be defined as the ratio of the total charge that the capacitor is capable of storing to the total voltage that is being applied onto the capacitor plates. It is expressed mathematically as,
Capacitance (C) = QV
Energy stored in a capacitor
The amount of energy stored in a capacitor is generally denoted by UC. It is the electrostatic potential of the capacitor, which is why it is calculated using the charge and voltage in the circuit. A capacitor stores electrical energy between capacitor plates in which an electrical field is formed when it is being charged. When the capacitor is disconnected from the source of energy, its electrical energy remains the same between the plates.
Let us now assume an empty parallel plate capacitor that is already charged and has a vacuum between its plates. The space between plates consumes some volume, say Ad, which has a uniform electrostatic field E. The total energy in the capacitor is in this space between the plates. This space is called energy density and denoted by uE, which can be calculated as UCAd. If we know the value of the energy density in the space, we can easily find the energy stored in the capacitor.
UC = uE (Ad)
This is possible because energy density in a free space is found by the equation:
uE = 12 0E2
Therefore, the energy stored in the capacitor is:
UC = 012E2Ad
= 12 0VD22Ad
= 12 0VD2A
= 12v2C.
In the previous equations, we have learned that: (C ) = QV. Therefore,
UC = 12v2QV
= 12QV.
This equation is applicable for all kinds of capacitors, even though it was derived for parallel plate capacitors. Now let us suppose there is a capacitor connected to a circuit, which at an instant gives a potential difference of V = q/C, and the charge at that instant is Q = 0. As the capacitor is charged gradually, the charge increases to a final magnitude of Q. If we find the work done by the capacitor in moving a negative charge dq from the negative plate to the positive plate of the capacitor, we get:
W = 0W(0)dW = 0QqCdq = Q2C.
As no parameter in this equation is related to the shape of the capacitor, it is clear that this equation will stand true for all capacitors. We know that the total energy stored in a capacitor is the electric potential between the plates. We also know that this electric potential is equal to the work done by the charge in charging the plates of the capacitor. Thus, UC = W.
Now, we know that UC = Q2/2C and that E = /0and C = QV
We can find uE:
uE = UC Ad
= 12Q2C1Ad
= 12Q201Ad
= 12 Q2/A2 = 220
= 02E2
This proves that the method of using electrical density to find the electric potential or electric energy in a capacitor is correct and is applicable for most capacitors.
Conclusion
It can be concluded that the energy stored in a capacitor in terms of charge and voltage is calculated by the equation UC = 12QV. When the capacitors are in a circuit, be it parallel or series, we need to find the total charge and voltage in the respective cases and use this formula to find the energy stored in it. We also saw and derived the equations for finding the energy in a capacitor and also verified how it stands true for all shapes of capacitors.
