Name: capacitor charging and stored electric energy
Uploaded: 2023-04-30T12:38:06.000Z
Duration: 1 min 12 s
Channel: JoVE

K12 Standards

Tennessee Academic Standards

High School

Physics

Energy

PHYS.PS3.13

Energy Stored in a Capacitor

Consider a parallel plate capacitor connected to a battery. Work is done to move the electrons such that a potential difference is developed across the plates.
Suppose, at time t, the plates have acquired charge q, the potential difference across the plates is expressed as the ratio of the acquired charge to the capacitance of the capacitor.
Now, to increase the charge on the plates by a small amount, additional work done is expressed as the product of the potential difference between the plates and the additional charge acquired.
Integrating the expression for the additional work done within the limits of zero to Q, the total work done to acquire a final charge Q can be obtained.
Now, the potential energy gained by the capacitor equals the total work done to acquire charge Q, which can be expressed in terms of potential difference.
Substituting for capacitance and potential difference in terms of electric field, the potential energy per unit volume of the capacitor gives the energy density between the charged capacitor plates.

energy stored in a capacitor

When an archer pulls the string in a bow, he saves the work done in the form of elastic potential energy. When he releases the string, the potential energy is released as kinetic energy of the arrow. A capacitor works on the same principle in which the work done is saved as electric potential energy. The potential energy (UC) could be calculated by measuring the work done (W) to charge the capacitor.
<img alt="static equilibrium equation W=Uc; formula; physics analysis" src="/files/ftp_upload/13738/13738_Equation_1.svg" style="text-align: center; height: 12px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
Let us consider the case of a parallel plate capacitor. When the capacitor is connected to a battery, the plate attached to the battery's negative side gets more electrons, repelling more electrons in the other plate. Hence the second plate gets an equal positive charge. At any instant of time when the capacitor is getting charged, if q and V are the charge and potential difference across the plates, respectively, then they are related by the following equation:
<img alt="Electrical potential formula V'=q/C; diagram with capacitance concepts." src="/files/ftp_upload/13738/13738_Equation_2.svg" style="text-align: center; height: 25px;" data-altupdatedat="1761830439000" data-alt="Equation2" />
In equation (2), C is the capacitance of the parallel plate capacitor. As the capacitor is being charged, the charge gradually builds upon its plates, and after some time, it reaches the final value Q. The amount of work done (dW) to move a charge element dq is Vdq. We get the potential energy stored in the capacitor using the equations (1) and (2). Thus,
<img alt="Capacitor energy equation: \(U_C=W=\int_0^W dW=\frac{1}{C}\int_0^Q qdq=\frac{Q^2}{2C}=\frac{1}{2}CV^2=\frac{1}{2}QV\)" src="/files/ftp_upload/13738/13738_Equation_3.svg" style="text-align: center; height: 50px;" data-altupdatedat="1761830439000" data-alt="Equation3" />
We can now find the energy density stored in vacuum between the plates of a charged parallel-plate capacitor from the potential energy stored in a capacitor. The energy density is then defined as the potential energy per unit volume. If A and d are the area and distance between the plates, then from the expressions for electric field and capacitance, that is E = &#963;/&#949;o and C = &#949;o A/d, the energy density is obtained as:
<img alt="Formula: Electrostatic Energy Density derivation, showing equations with variables and constants." src="/files/ftp_upload/13738/13738_Equation_4.svg" style="text-align: center; height: 46px;" data-altupdatedat="1761830439000" data-alt="Equation4" />

When an archer pulls the string in a bow, he saves the work done in the form of elastic potential energy. When he releases the string, the potential ...

Education

JoVE Core

Unit 1

Capacitance

KEY TERMS AND CONCEPTS

capacitor charging and stored electric energy

A charged capacitor stores electric potential energy by doing work during charging. This idea is similar to an archer pulling back a bowstring and storing elastic potential energy. When the arrow is released, that stored energy becomes motion. In the same way, the work put into a capacitor is stored as electric potential energy.

For a parallel plate capacitor, the process starts when it is connected to a battery. The plate attached to the negative side of the battery gains extra electrons. Those electrons repel electrons on the other plate. As a result, the second plate gets an equal positive charge.

At any moment while the capacitor is charging, the charge q and potential difference V across the plates are related by q = CV, where C is the capacitance of the parallel plate capacitor. As charge builds up, it increases gradually on the plates. After some time, the capacitor reaches its final charge Q. The work done to move a small charge element dq is Vdq.

The electric potential energy stored in the capacitor is found by using the charging relations and the work done. The same ideas can also be used to find the energy density in the vacuum between the plates of a charged parallel-plate capacitor. Energy density means potential energy per unit volume. If A is the plate area and d is the distance between the plates, then the expressions for electric field and capacitance, E = σ/ε0 and C = ε0A/d, lead to the energy density formula.

Capacitor Charging and Stored Electric Energy

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A charged capacitor stores electric potential energy by doing work during charging. This idea is similar to an archer pulling back a bowstring and storing elastic potential energy. When the arrow is released, that stored energy becomes motion. In the same way, the work put into a capacitor is stored as electric potential energy. For a parallel plate capacitor, the process starts when it is connected to a battery. The plate attached to the negative side of the battery gains extra electrons.

energy stored in a capacitor: problem solving

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key terms and concepts

Learn how a parallel plate capacitor stores electric potential energy during charging, how q = CV applies, and how energy density in the plate gap is found.

Capacitor Charging and Stored Electric Energy | JoVE K12

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Virginia  Standards of Learning

HIgh School

Physical Science

PS.9

PS.9.a

Next Generation Science Standards

HS-PS3-5

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PHYS.PS3.11

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PHYS.PS3.12

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PHYS.PS3.15

Capacitor Charging and Stored Electric Energy

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