Themes > Science > Physics > Electromagnetism > Electrostatics > Capacitors & Dielectrics > Capacitor : Storing Electrical Energy > Capacitor : Storing Electrical Energy

A device that stores electrical energy based on opposite charges is called a capacitor. Storing the various forms of energy is an essential link in transforming energy from one form into another. The capacitor is a device for storing electrical energy. The storage of energy has another, even more vital function, and that is in the amplification of energy. If one stores energy in small amounts and accumulates a large amount over (a long) time, then one can arrange to release the stored energy in a short burst, and in effect generate an enormous amount of power. The capacitor also plays an indispensable role in integrated circuits, and together with semi-conductors, is one of the main devices for controlling and manipulating electrical energy and charge. Recall the energy per unit volume of the electric field in empty space is given by
\begin{displaymath} u_E=\frac{1}{2}\epsilon_0{\bf E}^2 \nonumber \end{displaymath}

To obtain the total energy $U_E$ of the field, one has to integrate (sum over) the contributions of the ${\bf E}$ over all points of space.

 


\fbox{\fbox{\parbox{12cm}{ The total energy stored in an electric field is give... ...gin{equation} U_E=\frac{1}{2}\epsilon_0\int d^3x{\bf E}^2 \end{equation} }}}

 


If the electric field has a constant value, say $E_0$, in a finite cube with volume $vol$, then, from eq.(7.43), the total energy in the electric field is given by

\begin{displaymath} U_E=\frac{1}{2}\epsilon_0E_0^2 \times vol \end{displaymath} (7.35)

 

In other words, the electric field contributes energy for every volume element of space where it is non-zero, and the total energy of the field is found by adding together the contribution to the total energy $U$ from all the volume elements of space. As can be seen from the above derivation, a capacitor is a device that performs an integration of the electric field, and in effect integrates the energy of the electrical field. The term integrated circuit partly derives its terminology from this feature of a capacitor. Electric charge generates electric field, and hence to store electrical energy, we need to collect charges into a storage device. Since most objects found in nature are electrically neutral, to obtain electrical energy, it is logical to try and store positive and negative electric charge separately . However, if we bring together a large collection of positively (or negatively) charged particles, the force of electrical repulsion is very large, the charges would all tend to fly apart, and an enormous amount of energy would be expended to merely keep them in place. Hence, a more efficient arrangement for generating electric field is to bring together equal amounts of positive and negative charges, but separate them so that we are not back to an electrically neutral object. And this in essence is the principle behind the design of all capacitors.

 

Figure 7.12: Capacitor
\begin{figure} \begin{center} \epsfig{file=core/capacitor.eps, width=4cm} \end{center} \end{figure}

 

A parallel plate capacitor is composed out of two conducting plates placed parallel to each other and separated by a distance $d$ filled with an insulator. One can let air fill up the space between the conducting plates, but, as shown in Figure 7.12, a dielectric material is usually placed instead of air in order to increase the capacitance of the capacitor. By charging the conducting plates with opposite charges of amount $Q$ and $-Q$, a potential difference of amount $V$ is created. A measure of how much charge is stored in a capacitor is the change in the potential difference of the capacitor $V$ if charge $Q$ is placed on the conducting plates. The analog of a capacitor is a water storage tank; if a certain volume of water is poured into the tank - the analog of electric charge - the increase in the height of the water is the analog of the increase in potential $V$, and is a measure of the volume of the storage tank, which is the analog of capacitance $C$. Clearly, the simplest case for a capacitor is when the voltage difference $V$ is proportional to the charge $Q$; the proportionality constant is the capacitance of the capacitor, and is denoted by $C$ (not to be confused with the SI unit of charge, namely the coulomb $C$). We hence have
\begin{displaymath} Q=CV \end{displaymath} (7.36)

 

The SI unit of capacitance is the farad, denoted by $F$; we have
$\displaystyle 1\mathrm{farad}$ $\textstyle =$ $\displaystyle \mbox{\rm { coulomb per volt}}$ (7.37)
$\displaystyle \Rightarrow 1F$ $\textstyle =$ $\displaystyle 1 C/V$ (7.38)

A capacitor is charged by connecting it in a circuit with a battery with voltage $V$, as shown in Figure 7.13. Note the capacitor is shown by two parallel lines with a gap to indicate that there is an insulator between the two conducting plates. A insulator in a D.C. circuit breaks the closed circuit, with no current flowing in the circuit. The conducting plate connected to the (+) terminal of the battery will attain the potential $V_+$ by the flow of electrons to the battery, leaving a net charge $+Q$ on the conducting plate. And similarly for the conducting plate connected to the (-) terminal; it will have the potential $V_-$ and charge $-Q$. The charged capacitor will therefore have a voltage difference equal to the battery, namely equal to $V$, with charges $+Q$ and $-Q$ on the two respective conducting plates. After the (transient) process of charging is over, there is no more flow of charges and the current in the circuit is zero.

 

Figure 7.13: Charging a Capacitor
\begin{figure} \begin{center} \epsfig{file=core/circuit1.eps, width=4cm} \end{center} \end{figure}

The energy stored in a capacitor can be found by determining the electric field that has been created due to charging up of the capacitor. The capacitor has a potential difference of $V$ across a distance $d$, and, for now, let air be placed between the conducting plates. Hence the electric field is constant and points from the positively to the negatively charged conductors as shown in Figure 7.12. From eq.(7.18), the constant electric field is given by
\begin{displaymath} E_0=\frac{V}{d} \end{displaymath} (7.39)

The energy stored in the capacitor is, from eq.(7.44), proportional to the volume of the capacitor. For conducting plates with area $A$, the energy stored in the capacitor is consequently given by
$\displaystyle U_{\mathrm{capacitor}}$ $\textstyle =$ $\displaystyle \frac{1}{2}\epsilon_0\times E_0^2\times \mbox{\rm { volume of capacitor}}$ (7.40)
$\displaystyle \Rightarrow U_{\mathrm{capacitor}}$ $\textstyle =$ $\displaystyle \frac{1}{2}\epsilon_0(\frac{V}{d})^2\times (Ad)$ (7.41)
  $\textstyle =$ $\displaystyle \frac{1}{2}\epsilon_0\frac{A}{d}V^2$ (7.42)

It is known from charging up the capacitor that the total energy stored is given by

$\displaystyle U_{\mathrm{capacitor}}$ $\textstyle =$ $\displaystyle \frac{1}{2}CV^2$ (7.43)
  $\textstyle =$ $\displaystyle \frac{Q^2}{2C}$ (7.44)


\fbox{\fbox{\parbox{12cm}{ To find the electrical energy stored in a capacitor,... ...t_0^Q\frac{Q'}{C}dQ'\\ &=&\frac{Q^2}{2C}=\frac{1}{2}CV^2 \end{eqnarray} }}}


Hence, from the above equations, we see that the capacitance of a parallel plate is given by

\begin{displaymath} C=\epsilon_0\frac{A}{d} \end{displaymath} (7.45)


Note that capacitance $C$ depends only on the geometrical shape of the capacitor, as well as the material serving as the insulator. By changing the insulator from air to a dielectric material, one simply changes in $\displaystyle U_{\mathrm{capacitor}}$ the permittivity of the vacuum $\epsilon_0$ to that of the medium, namely $\epsilon$, obtaining the general result
\begin{displaymath} C=\epsilon\frac{A}{d} \end{displaymath} (7.46)

 

As shown in Table 7.2, a properly chosen dielectric can increase the capacitance by a few order of magnitude.

 

Table 7.2: Some Dielectric Materials
Material $\displaystyle \epsilon/\epsilon_0$
Vacuum $1$
Air $1.00054$
Paper 3.5
Silicon 12
Germanium $16$
Water($25^0$) $78.5$
Titania ceramic 130
Strontium titanate $310$

One can replace the insulating dielectric in a capacitor with a semi-conductor, and create a more complex device that stores and discharges electrical energy depending on the state of the semi-conductor.


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