Capacitors and Dielectrics Lecture Notes

The Limitations of Single Spherical Isolated Conductors

The question arises whether a spherical-shaped single insulated conductor can be effectively used to store electric charge. Scientifically, it is rarely employed for this purpose because it can only store a limited amount of electric charge. Adding more charges ($Q$) to such a conductor inevitably leads to an increase in the electric potential ($V$) at a specific distance ($r$) from the center of the charge. This relationship is mathematically defined by the following equations:

$V = \frac{1}{4\pi \epsilon_0} \times \frac{Q}{r}$

or

$V = K \times \frac{Q}{r}$

As the electric potential increases, the electric potential difference between the conductor and any surrounding medium, such as air, also increases. This subsequently raises the intensity of the electric field ($E$) to a critical threshold where electric discharge occurs through the surrounding air. Consequently, any attempt to continue adding charges to a charged spherical-shaped single insulated conductor results in immediate discharge because the increased potential and potential difference eventually reach a limit dictated by the surrounding environment's dielectric capabilities.

The Evolution and Definition of the Capacitor

To overcome the limitations of single conductors, it is possible to build a system or device designed to store large amounts of electrical charges and electrical energy. This is achieved by utilizing a system of two conductors of any shape. These two conductors are separated by an insulator. Such a system can store positive charges on one conductor and negative charges on the other, and it is formally known as a capacitor ($\text{المآسعة}$).

A capacitor is defined as a device consisting of a pair (or more) of conductive parallel plates separated by an insulator, used explicitly for storing electrical charges and electrical energy. Capacitors are manufactured in various physical shapes and sizes depending on their intended technological application. The three most common shapes include:

1. Two parallel plates capacitor.

2. Two concentric cylinders capacitor.

3. Two concentric spheres capacitor.

The simplest and most prevalent form used in practical applications is the parallel plate capacitor, where two conductors separated by a dielectric are usually plane and parallel.

Anatomy and Mechanics of the Parallel Plate Capacitor

A parallel plate capacitor consists of two identical, isolated, and parallel conductive plates. Each plate possess a surface area ($A$) and the plates are separated by a distance ($d$). These plates are charged with equal quantities of charge but of opposite types.

To charge a parallel plate capacitor, one plate is connected to the positive terminal of a battery, causing it to display a positive charge ($+Q$). The other plate is connected to the negative terminal of the battery, displaying a negative charge ($-Q$). Both of these charges reside on the opposite inner surfaces of the plates due to the attraction force between them. Because the plates carry charges that are equal in magnitude but opposite in type, the net charge on the plates of a charged capacitor is always zero ($Q_{\text{total}} = +Q - Q = 0$).

Electric Field Dynamics and Potential in Capacitors

The electric field ($E$) generated between the plates of a parallel plate capacitor is considered uniform (regular) under specific conditions. It is treated as regular if the distance ($d$) between the plates is very short compared to the physical dimensions of the plates, allowing the researcher to ignore the irregularity of electric field lines at the edges.

Every point on a single plate of a charged capacitor exists at the same electric potential because the plates are made of conductive material and are isolated. However, an electric potential difference ($\Delta V$) is generated between the two plates. This occurs because the capacitor stores positive charge ($+Q$) on one plate, giving it a high potential, and negative charge ($-Q$) on the other plate, giving it a low potential. The relative difference between these high and low potentials constitutes the voltage or potential difference across the capacitor.

The Concept and Quantification of Capacitance

Capacitance ($C$) is defined as the ratio of the charge ($Q$) stored on either of the plates to the potential difference ($\Delta V$) between the plates. It is a measure of the charge that needs to be placed on the plates to generate a specific electric potential difference. A capacitor with a larger capacitance is capable of holding a larger amount of charge. The mathematical relationship is expressed as:

$C = \frac{Q}{\Delta V}$

The standard unit of measurement for capacitance is the Farad, symbolized by ($F$). One Farad is the capacitance of a capacitor that stores a charge of one Coulomb ($1\,C$) resulting in a potential difference of one Volt ($1\,V$) between its plates. The Farad is quite a large unit for most practical purposes, so sub-units are frequently used:

1. Microfarad: $1\,\mu F = 10^{-6}\,F$

2. Nanofarad: $1\,nF = 10^{-9}\,F$

3. Picofarad: $1\,pF = 10^{-12}\,F$

There are three units equivalent to the Farad:

$\text{Farad} = \frac{\text{Coulomb}}{\text{Volt}}$

$\text{Farad} = \frac{\text{Coulomb}^2}{\text{Joule}}$

$\text{Farad} = \frac{\text{Joule}}{\text{Volt}^2}$

If a capacitor is said to have a capacitance of $2\,\mu F$, it means that a charge of $2\,\mu C$ is required to raise the potential difference between the two ends of the capacitor by $1\,V$.

Proportion, Proportionality, and the Hazards of Charged Capacitors

The relationship between the amount of charge ($Q$) on the plates and the electrical potential difference ($\Delta V$) is directly proportional. An increase in charge results in a corresponding increase in potential difference. If the potential difference across a capacitor with constant capacitance is doubled ($\Delta V_2 = 2\Delta V_1$), the stored charge will also double ($Q_2 = 2Q_1$).

Regarding safety, a charged capacitor where the potential difference is very high can be extremely dangerous even if it is disconnected from its voltage source. If the plates are touched by hand, the high amount of stored charge will discharge quickly through the hand. The human hand acts as a conductive material between the plates, leading to a dangerous electrical shock. To handle such a capacitor safely, it must first be discharged using a wire made of conductive material covered with insulation to connect the two ends, or by using electric discharging tongs or a screwdriver.

There is a nuance regarding the term "charge." If a friend claims a charged capacitor stores a specific value of charge and you claim the net charge is zero, both are correct according to physical theory. The "charge of the capacitor" refers to the magnitude of charge on one plate ($Q$), whereas the "net charge" refers to the sum of positive and negative charges ($+Q + (-Q) = 0$).

Dielectric Materials: Polar vs. Non-Polar Classifications

Dielectric materials are non-conducting substances under normal conditions (standard temperature and pressure) that serve to decrease the amount of the electric field in which they are placed. They are classified into two categories:

1. Polar Dielectrics: Examples include pure water. The molecules of these materials possess permanent electric bipolar moments (dipoles). In these molecules, the distance between the centers of negative and positive charges is constant.

2. Non-Polar Dielectrics: Examples include glass and polyethylene. The molecules of these materials do not have permanent moments; instead, they gain temporary bipolar electric moments through electric induction when exposed to an external electric field. The distance between the centers of their positive and negative charges is not constant.

The Influence of Dielectrics on Electric Fields and Potentials

When a polar dielectric is inserted between the plates of a charged capacitor, the external electric field ($E$) affects the dipoles, aligning them along the field. This generates an internal electric field ($E_d$) inside the dielectric which directed opposite to the external field. Consequently, the net electric field ($E_k$) is reduced according to:

$E_k = E - E_d$

When a non-polar dielectric is inserted, the external electric field minimally displaces the centers of the positive and negative charges in the molecules, causing them to gain temporary dipole moments through induction. The molecules turn into dipoles aligned opposite to the external field, and surface charges appear on the sides of the dielectric. The dielectric thus becomes polarized. These surface charges generate an internal field ($E_d$) that opposes the external field, weakening it.

In both cases, the net electric field is decreased by the ratio of the dielectric constant ($K$):

$E_k = \frac{E}{K}$

Since the electric field is directly proportional to the potential difference ($E = \frac{\Delta V}{d}$), the potential difference also decreases by the ratio of ($K$) when the capacitor is isolated:

$\Delta V_K = \frac{\Delta V}{K}$

Technical Definitions: Dielectric Constant and Dielectric Strength

The electric field between two capacitor plates is defined as the ratio of the potential difference to the distance ($d$) between them, measured in ($\text{Volt/meter}$):

$E = \frac{\Delta V}{d}$

Dielectric Strength is defined as the maximum amount of electric field that a material can withstand before electrical breakdown occurs. It serves as a measure of a material's ability to resist the applied electric field and is measured in ($\text{Volt/m}$).

Dielectric Constant ($K$), or relative permittivity, is the ratio between the capacitance of the capacitor after inserting the dielectric material ($C_k$) and its capacitance in a vacuum or air ($C$):

$K = \frac{C_k}{C}$

This value is dimensionless and depends entirely on the type of dielectric material used. Inserting a dielectric material instead of air provides two practical benefits:

1. It increases the capacitance of the capacitor ($C_k = K \times C$).

2. It prevents early electric breakdown of the dielectric between the plates when a large potential difference is applied.

Factors Determining Capacitance and Empirical Demonstrations

Three primary factors affect the capacitance of a parallel plate capacitor:

1. Surface Area ($A$): Capacitance is directly proportional to the facing surface area ($C \propto A$).

2. Distance ($d$): Capacitance is inversely proportional to the distance between the plates ($C \propto \frac{1}{d}$).

3. Dielectric Medium: The presence of a dielectric constant ($K$) increases capacitance. The general formula is:

$C = \epsilon_0 \times \frac{K \times A}{d}$

Through practical activities, these can be verified. For area: by reducing the surface area to half ($A \rightarrow \frac{1}{2} A$) while keeping charge constant, the voltmeter reading doubles ($\Delta V \rightarrow 2\Delta V$), meaning capacitance decreases. For distance: by halving the distance ($d \rightarrow \frac{1}{2} d$), the voltmeter reading halves ($\Delta V \rightarrow \frac{1}{2} \Delta V$), meaning capacitance increases.

Faraday’s experiment demonstrates that inserting a dielectric into a disconnected charged capacitor causes a drop in the voltmeter reading ($\Delta V_k = \frac{\Delta V}{K}$). Since $C = \frac{Q}{\Delta V}$, a decrease in voltage leads to an increase in capacitance ($C_k = K \times C$).

To maximize capacitance during manufacturing, engineers control these factors by using very thin, wide ribbons of metal plates wrapped into cylindrical shapes, separated by thin ribbons of insulators with high dielectric constants.

Energy and Power in the Capacitor's Electric Field

Moving electric charges between positions of different potential requires work. According to the law of conservation of energy, this work is stored as potential energy ($PE$) in the electric field of the capacitor. The amount of stored energy can be calculated by determining the area of the triangle on a graph of charge ($Q$) versus potential difference ($\Delta V$). The following formulas apply:

$PE = \frac{1}{2} \times Q \times \Delta V$

$PE = \frac{1}{2} \times C \times (\Delta V)^2$

$PE = \frac{1}{2} \times \frac{Q^2}{C}$

Energy is measured in Joules ($J$). Power ($P$) is the rate of energy transfer, measured in Watts ($W$) or Joules per second ($J/s$):

$P = \frac{PE}{t}$

Functional variations in energy based on doubling parameters include:

• If capacitance is constant and $\Delta V$ is doubled, energy increases fourfold ($PE_2 = 4PE_1$) because $PE \propto (\Delta V)^2$.

• If charge is constant and $\Delta V$ is doubled, energy increases to double the amount ($PE_2 = 2PE_1$) because $PE \propto \Delta V$.

• If $\Delta V$ is constant and charge is doubled, energy increases to double the amount ($PE_2 = 2PE_1$).

• If capacitance is constant and charge is doubled, energy increases fourfold ($PE_2 = 4PE_1$) because $PE \propto Q^2$.

Comparative Analysis of Physical Property Changes under Varied Constraints

Consider a parallel plate capacitor (air as insulator) charged by a battery and then disconnected. If a dielectric with $K = 2$ is inserted:

1. Charge ($Q$): Remains constant because it is disconnected.

2. Capacitance ($C$): Increases to double ($C_k = 2C$).

3. Potential Difference ($\Delta V$): Decreases to half ($\Delta V_k = \frac{\Delta V}{2}$).

4. Electric Field ($E$): Decreases to half ($E_k = \frac{E}{2}$).

5. Stored Energy ($PE$): Decreases to half ($PE_k = \frac{PE}{2}$).

Consider the same capacitor remaining connected to the source while a dielectric with $K = 6$ is inserted:

1. Charge ($Q$): Increases by six times ($Q_k = 6Q$).

2. Capacitance ($C$): Increases by six times ($C_k = 6C$).

3. Potential Difference ($\Delta V$): Remains constant because it is still connected to the battery.

4. Electric Field ($E$): Remains constant since $\Delta V$ and $d$ are constant.

5. Stored Energy ($PE$): Increases by six times ($PE_k = 6PE$).