Electric Charges and Field

Frictional Electricity and Historical Context

Frictional electricity is defined as the electricity developed by rubbing or friction between two substances. When certain bodies, such as a glass rod rubbed with silk cloth or a fountain-pen rubbed against a coat-sleeve, are subjected to friction, they acquire the ability to attract small pieces of paper, straw, lint, or light feathers. This property of attraction is what is called electricity. A body that exhibits this property is said to be electrified or electrically charged. In all these instances, friction is the primary mechanism that causes substances to become charged. For example, a plastic comb passed through dry hair will attract light objects because it has become electrified through friction.

The historical understanding of frictional electricity dates back to $600\,B.C.$ when Thales of Miletus, a founder of Greek science, observed that a piece of amber rubbed with woollen cloth acquired the property of attracting light objects like dust and feathers. In $1600\,A.D.$, William Gilbert, the personal physician to Queen Elizabeth I of England, conducted a systematic study of substances behaving like amber. In his treatise De Magnete (On the Magnet), he introduced the name electrica for such substances. The term electricity itself is derived from the Greek word for amber, which is elektron. This is the historical root of related terms such as electric force, electric charge, and electron. Amber is a yellow resinous substance typically found on the shores of the Baltic Sea.

While electricity and magnetism were once studied separately, we now understand that both phenomena are derived from charged particles. Magnetism specifically arises from charges in motion. Since moving charged particles exert both electric and magnetic forces on one another, the two fields are studied together under the discipline of electromagnetism.

Nature and Classification of Electric Charge

Electric charge is an intrinsic property of elementary particles, such as electrons and protons, which constitute all matter. It is because of these charges that objects exert strong electric forces of attraction or repulsion on one another. Electric charge is a scalar quantity, and its SI unit is the coulomb ($C$). A proton carries a positive charge denoted as $+e$, while an electron carries a negative charge denoted as $-e$. The magnitude of this basic unit of charge is $e = 1.6 \times 10^{-19}\,coulomb$. Large-scale matter is generally electrically neutral because it contains an equal number of electrons and protons. A body becomes negatively charged if there is an excess of electrons and positively charged if there is a deficit of electrons (an excess of protons).

Electrostatics: Concepts and Industrial Applications

Electrostatics is the branch of physics that studies electric charges at rest. It specifically involves the analysis of forces, fields, and potentials associated with static charges. The principles of attraction and repulsion between charged bodies have numerous practical and industrial applications, including the functioning of electrostatic loudspeakers, electrostatic spraying of paints and powder coating, and the collection of fly ash in industrial chimneys. Furthermore, electrostatics is utilized in the design of Xerox copying machines and in cathode-ray tubes used in television and radar systems.

The Two Kinds of Electric Charges and Fundamental Laws

Experimentation conducted roughly 100 years ago by Charles Du Fay and later experiments with pith balls have demonstrated that there are only two kinds of electric charges. The fundamental law of electrostatics states that like charges repel and unlike charges attract each other. For example, two glass rods rubbed with silk will repel each other, while a glass rod rubbed with silk and a plastic rod rubbed with wool will attract each other. Similarly, two pith balls touched by a glass rod rubbed with silk will repel each other, but a pith ball touched by a glass rod will be attracted to one touched by a plastic rod. The property that distinguishes these two kinds of charges is called the polarity of charge.

Historically, Charles Du Fay referred to these charges as vitreous and resinous. Vitreous charge was the kind developed on glass (from the Latin vitrum), while resinous charge was the kind developed on amber. However, these terms became misleading; for instance, ground glass can develop resinous electricity. Benjamin Franklin later introduced the modern convention, replacing vitreous with positive and resinous with negative. Under this convention, the charge on a glass rod rubbed with silk is positive, the charge on a plastic rod rubbed with wool is negative, and thus the charge on an electron is considered negative.

Triboelectric series exist to determine which substance acquires which charge when rubbed. In a series including substances like fur, flannel, sealing wax, glass, cotton, paper, silk, human body, wood, metals, rubber, resin, amber, sulphur, ebonite, and guta percha, the substance appearing earlier in the list will acquire a positive charge when rubbed with a substance appearing later. For example, glass acquires a positive charge when rubbed with silk, but a negative charge when rubbed with flannel.

Electronic Theory of Frictional Electricity

The electronic theory posits that since all matter is made of atoms consisting of a nucleus (protons and neutrons) and orbiting electrons, bulk matter is neutral when these charges cancel out. The electrons in the outer shells are loosely bound. The energy required to remove an electron from a material's surface is known as its "work function." When two bodies are rubbed together, electrons are transferred from the material with the lower work function to the material with the higher work function. Frictional forces themselves have an electric origin because pulling an electron away from an atom requires a strong electric force. During rubbing, only electrons are transferred; protons are never transferred.

There is always a change in mass during the charging process because electrons have a finite mass ($m_e = 9.1 \times 10^{-31}\,kg$). A positively charged body loses electrons, resulting in a slight decrease in mass, while a negatively charged body gains electrons, resulting in a slight increase in mass. Furthermore, electric charge is always conserved during the rubbing process; the total charge of the two-body system remains zero.

Conductors, Insulators, and Earthing

Substances are classified based on their ability to allow the flow of electric charges. Conductors, such as metals, human and animal bodies, graphite, acids, and alkalies, contain a large number of free electrons, allowing charges to flow easily. In contrast, insulators (or dielectrics), such as glass, diamond, porcelain, plastic, nylon, wood, and mica, have electrons tightly bound to the nucleus and lack free charge carriers, offering high resistance to electricity. A key distinction is that charge transferred to a conductor distributes itself over the entire surface, whereas charge placed on an insulator stays at the point of contact.

A metal rod cannot be electrified by rubbing while held in a bare hand because the human body is a conductor, and the charge will be transferred to the earth. To charge it, one must use an insulating handle. This leads to the concept of grounding or earthing, which is the process of sharing charges with the earth. In household circuits, a three-core wiring system is used: a live wire (red), a neutral wire (black), and an earth wire (green). The earth wire is connected to a metal plate buried in the ground. If a fault occurs and the live wire touches the metallic body of an appliance, the charge flows to the earth, protecting the user from electric shock.

Electrostatic Induction

Electrostatic induction is the phenomenon of temporary electrification of a conductor where opposite charges appear at the closer end and similar charges appear at the farther end in the presence of a nearby charged body. These charges are called induced charges, and the external charge identified for the effect is the inducing charge.

For example, to charge two metal spheres oppositely by induction:

1. Place two spheres in contact on insulating stands.
2. Bring a positively charged glass rod near the left sphere; electrons will be attracted to the left side, leaving the right side of the second sphere positive.
3. While the rod is still present, separate the spheres.
4. Remove the rod; the charges will redistribute, and the spheres will now be oppositely charged.

A single sphere can also be charged by induction. To charge it positively, a negatively charged plastic rod is brought near. The far end of the sphere is earthed, allowing electrons to flow to the ground. When the earth connection is removed and then the plastic rod is removed, the remaining positive charge spreads uniformly over the sphere. A device used to detect charge and its polarity is the gold-leaf electroscope, where the divergence of two gold leaves indicates the presence and amount of charge.

Fundamental Properties of Electric Charge

There are three basic properties of electric charge: additivity, quantization, and conservation.

Additivity of charge means the total charge of an extended system is the algebraic sum of all individual charges. For a system with charges $q_1, q_2, ..., q_n$, the total charge is: $q = q_1 + q_2 + ... + q_n$

Quantization of electric charge is the principle that the total charge ($q$) of a body is always an integral multiple of a basic quantum of charge ($e$): $q = ne$ where $n = 0, \pm 1, \pm 2, \pm 3, ...$. The cause of quantization is that only integral numbers of electrons can be transferred. This was experimentally verified by Faraday’s laws of electrolysis and Millikan’s oil drop experiment. At a macroscopic level, quantization can often be ignored because the charge $e$ is so small that the flow of charge appears continuous. However, at a microscopic level, it is essential. Modern physics suggests protons and neutrons are made of quarks with charges of $\frac{2}{3}e$ and $-\frac{1}{3}e$, but as quarks do not exist in isolation, the quantum of charge remains $e$.

Conservation of charge states that the total charge of an isolated system remains constant. Charges can neither be created nor destroyed; they can only be transferred. Examples include the ionization of NaCl ($NaCl \rightarrow Na^{+} + Cl^{-}$), pair production where a gamma-ray photon materializes into an electron-positron pair, and nuclear fission. Even in high-energy physics where mass and energy convert, the law of conservation of charge holds strictly.

Comparison Between Charge and Mass

1. Electric charge can be positive, negative, or zero, whereas mass is always positive.
2. Charge is quantized ($q = ne$), while the quantization of mass is not yet established.
3. The charge on a body does not depend on its speed (invariant), but the mass of a body increases with speed according to the theory of relativity: $m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$
4. Charge is strictly conserved, whereas mass can be converted into energy ($E = mc^2$).
5. Electrostatic forces can be attractive or repulsive, but gravitational forces are always attractive.

Coulomb’s Law of Electric Force

Formulated by Charles Augustin Coulomb in $1785$, Coulomb's law states that the force of attraction or repulsion between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The force acts along the line joining the charges: $F = k \frac{q_1 q_2}{r^2}$ In SI units, for charges in vacuum: $k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9\,N\,m^2\,C^{-2}$ where $\epsilon_0$ is the permittivity of free space ($\epsilon_0 = 8.854 \times 10^{-12}\,C^2\,N^{-1}\,m^{-2}$).

One coulomb is defined as that amount of charge that repels an equal and similar charge with a force of $9 \times 10^9\,N$ when placed in vacuum at a distance of one metre. Other units include the statcoulomb ($1\,C = 3 \times 10^9\,statC$) and the abcoulomb ($1\,C = \frac{1}{10}\,abcoulomb$). Coulomb's law is only valid for stationary point charges and for distances greater than nuclear dimensions ($10^{-15}\,m$).

Dielectric Constant and Relative Permittivity

Permittivity is the property of a medium that determines the electric force between two charges in that medium. Relative permittivity ($\epsilon_r$) or dielectric constant ($K$) is the ratio of the force between two charges in vacuum ($F_{vac}$) to the force in a medium ($F_{med}$): $K = \epsilon_r = \frac{\epsilon}{\epsilon_0} = \frac{F_{vac}}{F_{med}}$ For vacuum, $K = 1$; for air, $K = 1.00054$; and for water, $K = 80$. When a medium of dielectric constant $K$ is introduced, the force becomes $\frac{1}{K}$ times the force in vacuum.

Numerical Examples of Quantization and Forces

Example 1: How many electronic charges form one coulomb? $n = \frac{q}{e} = \frac{1\,C}{1.6 \times 10^{-19}\,C} = 6.25 \times 10^{18}$.

Example 2: Time required to collect $1\,C$ if a body emits $10^9$ electrons per second. Charge per second: $10^9 \times 1.6 \times 10^{-19} = 1.6 \times 10^{-10}\,C/s$. Time: $\frac{1}{1.6 \times 10^{-10}} = 6.25 \times 10^9\,seconds \approx 198\,years$.

Example 3: Force comparison for an electron and a proton separated by distance $r$. The ratio of electrostatic force ($F_e$) to gravitational force ($F_g$) is approximately $2.27 \times 10^{39}$, proving that electrostatic forces are enormously stronger than gravity. Richard Feynman noted that if two people stood at arm's length and had just a $1\%$ excess of electrons, the repulsive force would be enough to lift the entire Earth.

Principle of Superposition

While Coulomb's law describes the force between two point charges, the principle of superposition allows for the calculation of the force on a single charge due to a group of charges. It states that the total force on a given charge is the vector sum of the forces exerted on it by all other charges individually, with each individual force being unaffected by the presence of other charges: $\mathbf{F}_1 = \mathbf{F}_{12} + \mathbf{F}_{13} + ... + \mathbf{F}_{1N}$ For a charge $q_a$ located at $\mathbf{r}_a$ due to other charges $q_b$: $\mathbf{F}_a = \frac{q_a}{4\pi\epsilon_0} \sum_{b=1, b \neq a}^{N} \frac{q_b}{|\mathbf{r}_a - \mathbf{r}_b|^3} (\mathbf{r}_a - \mathbf{r}_b)$

The Electric Field

An electric field exists at a point if a force of electrical origin is exerted on a stationary charged body at that point. Quantitatively, the electric field intensity ($\mathbf{E}$) is the force experienced by a unit positive test charge ($q_0$): $\mathbf{E} = \lim_{q_0 \to 0} \frac{\mathbf{F}}{q_0}$ The test charge must be vanishingly small so as not to disturb the source charge distribution. The SI unit is Newton per coulomb ($NC^{-1}$) or volt per metre ($V\,m^{-1}$). Its dimensions are $[MLT^{-3}A^{-1}]$. The electric field is a vector field, meaning every point in space is associated with a specific vector. The force on any charge $q$ in a field is given by $\mathbf{F} = q\mathbf{E}$.

Electric Field of a Point Charge and Systems

The electric field of a point charge $q$ at distance $r$ is: $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ This field is spherically symmetric or radial. For a system of charges, the resulting electric field is the vector sum of the individual fields (Principle of Superposition for Fields): $\mathbf{E} = \mathbf{E}_1 + \mathbf{E}_2 + ... + \mathbf{E}_n$

Continuous Charge Distributions

When dealing with macroscopic charges, the discrete nature of charge is ignored in favor of a continuous distribution. There are three types:

1. Line Charge Distribution: Charge per unit length $\lambda$ ($C\,m^{-1}$).    $dq = \lambda\,dL$
2. Surface Charge Distribution: Charge per unit area $\sigma$ ($C\,m^{-2}$).    $dq = \sigma\,dS$
3. Volume Charge Distribution: Charge per unit volume $\rho$ ($C\,m^{-3}$).    $dq = \rho\,dV$

The total field is found by integrating over the distribution: $\mathbf{E} = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{r^2} \mathbf{\hat{r}}$ For an infinitely long thin wire of density $\lambda$, the field at distance $r$ is: $E = \frac{\lambda}{2\pi\epsilon_0 r}$ For a charged ring of radius $a$ at a point $x$ on its axis: $E = \frac{1}{4\pi\epsilon_0} \frac{qx}{(x^2 + a^2)^{3/2}}$

Electric Dipoles, Fields, and Torques

An electric dipole consists of a pair of equal and opposite charges ($+q$ and $-q$) separated by a small distance $2a$. The dipole moment ($\mathbf{p}$) is a vector with magnitude $p = q \times 2a$, directed from the negative to the positive charge.

Electric Field of a Dipole:

1. On the Axial Line: At distance $r$ from the center:    $E_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2pr}{(r^2 - a^2)^2} \approx \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3}$ (for $r \gg a$)
2. On the Equatorial Line: At distance $r$ from the center:    $E_{equa} = \frac{1}{4\pi\epsilon_0} \frac{p}{(r^2 + a^2)^{3/2}} \approx \frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$ (for $r \gg a$) The axial field is twice the strength of the equatorial field at the same distance. The dipole field falls off as $\frac{1}{r^3}$, faster than the $\frac{1}{r^2}$ of a point charge.

Dipole in a Uniform Electric Field: While the net translational force is zero, the dipole experiences a torque (couple) that tends to align it with the field: $\tau = pE \sin(\theta)$ In vector form: $\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$ The torque is maximum when the dipole is perpendicular to the field ($\theta = 90^{\circ}$) and zero when aligned ($\theta = 0^{\circ}$).

Electric Field Lines

Proposed by Michael Faraday, electric field lines are imaginary curves used to visualize electric fields. The tangent at any point gives the direction of the field, and the density of lines represents the field strength.

Properties of Field Lines:

1. They are continuous curves starting at positive charges and ending at negative charges. They do not form closed loops.
2. Tangents give the direction of $\mathbf{E}$.
3. No two lines can intersect; otherwise, there would be two directions of $\mathbf{E}$ at one point.
4. Lines are always normal to the surface of a conductor in equilibrium.
5. They contract lengthwise (explaining attraction) and expand laterally (explaining repulsion).
6. They do not pass through the interior of a conductor.

Electric Flux and Gauss's Theorem

Electric flux ($\Phi_E$) is a measure of the total number of electric field lines passing normally through a given area. For a surface $S$ in a field $\mathbf{E}$: $\Phi_E = \int_S \mathbf{E} \cdot d\mathbf{S}$ The SI unit is $N\,m^2\,C^{-1}$ or $V\,m$. The area vector $d\mathbf{S}$ is always taken along the outward normal for a closed surface.

Gauss's Theorem states that the total electric flux through any closed surface (a Gaussian surface) is $\frac{1}{\epsilon_0}$ times the net charge enclosed by the surface: $\Phi_E = \oint \mathbf{E} \cdot d\mathbf{S} = \frac{q}{\epsilon_0}$

Applications of Gauss's Theorem:

1. Infinitely Long Wire: $E = \frac{\lambda}{2\pi\epsilon_0 r}$.
2. Infinite Plane Sheet: For a sheet with surface density $\sigma$:    $E = \frac{\sigma}{2\epsilon_0}$    Note: This field is independent of the distance $r$ from the sheet.
3. Oppositely Charged Parallel Plates: The field between them is $E = \frac{\sigma}{\epsilon_0}$, and outside it is zero.
4. Thin Spherical Shell:
   • Outside ($r > R$): $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$. (Behaves like a point charge at the center).
   • At the Surface ($r = R$): $E = \frac{\sigma}{\epsilon_0}$.
   • Inside ($r < R$): $E = 0$. This is because no charge is enclosed by the internal Gaussian surface.

Questions & Discussion

Question: Why can we not electrify a metal rod by rubbing it while holding it in our hand? Response: The human body is a conductor. Any charge developed on the metal rod is immediately transferred to the earth through the body. To charge it, the rod must be equipped with an insulating handle (plastic or rubber) and rubbed without touching the metal.

Question: What does $q_1 + q_2 = 0$ signify? Response: It signifies that the two charges are equal in magnitude and opposite in sign, meaning the system as a whole is electrically neutral.

Question: Why are aircraft tires made slightly conducting? Response: During landing, friction with the airstrip can generate significant static electricity. Making tires slightly conducting allows this charge to drain safely to the ground, preventing sparks that could lead to fire.

Question: Can two like charges attract each other? Response: Yes, if one charge is very large compared to the other. The large charge can induce a significant opposite charge on the closer end of the smaller body, overcoming the force of repulsion from the original small charge.

Question: Why do field lines not form closed loops? Response: Electrostatic fields are conservative. The work done in moving a charge in a closed loop must be zero, which is inconsistent with the existence of closed field lines. Lines must start on a positive charge and terminate on a negative one.