Introduction

This chapter introduces the fundamental concepts of electrostatics, which deals with the study of forces, fields, and potentials arising from static electric charges. We explore everyday phenomena like the crackling of synthetic clothes in dry weather and lightning, attributing them to the accumulation and discharge of static electricity. The concept of static electricity is the foundation for understanding electrical forces and their interactions.

Electric Charge

Historically, the attraction of light objects by rubbed amber, a phenomenon known since ancient times, laid the groundwork for understanding electric charge. The term "electricity" originates from the Greek word for amber. Experiments revealed that rubbing materials could electrify them, causing them to attract or repel other objects. It was observed that like charges repel each other, while unlike charges attract. This led to the conclusion that there are two types of electric charge: positive and negative. When electrified bodies with opposite charges are brought together, their effects neutralize each other. The positive charge is conventionally assigned to the charge on a glass rod rubbed with silk, and the negative charge to the charge on a plastic rod rubbed with cat's fur. An electrically neutral body has a balanced amount of positive and negative charge.

The gold-leaf electroscope is a device used to detect the presence and estimate the amount of electric charge on a body. Materials acquire charge through the transfer of electrons, which are less tightly bound to atoms. A body becomes positively charged by losing electrons and negatively charged by gaining electrons.

Conductors And Insulators

Substances are classified based on their ability to conduct electricity. Conductors readily allow electric charges to move through them due to the presence of free charges (usually electrons). Metals, the human body, and the Earth are good conductors. Insulators, on the other hand, offer high resistance to the flow of electricity because their charges are not free to move. Examples include glass, porcelain, plastic, and wood. When a conductor is charged, the charge distributes itself over its entire surface. In contrast, the charge on an insulator typically remains localized at the point of application.

This difference explains why a nylon or plastic comb gets electrified when used to comb dry hair, while a metal spoon does not. The charges on the metal object leak away through our body to the ground, as both are conductors. For charging a metal object by rubbing, it's important to hold it by an insulating handle to prevent charge dissipation.

A third category, semiconductors, exhibit intermediate resistance to charge movement.

Basic Properties Of Electric Charge

Additivity Of Charges

Electric charges are scalar quantities and can be added algebraically, considering their signs. If a system contains multiple charges $q_1, q_2, \ldots, q_n$, the total charge of the system is simply the sum of these individual charges: $q_{total} = q_1 + q_2 + \ldots + q_n$. This property is similar to how mass is added.

Charge Is Conserved

The principle of conservation of charge states that electric charge can neither be created nor destroyed. When bodies are charged by rubbing, charge is merely transferred from one body to another. In an isolated system, the total charge remains constant, even if charges are redistributed among the bodies within the system due to interactions. While charged particles can be created or destroyed (e.g., in nuclear reactions), they always appear in pairs with equal and opposite charges, ensuring the net charge remains conserved.

Quantisation Of Charge

Electric charge is quantized, meaning that all free charges exist as integral multiples of a fundamental unit of charge, denoted by $e$. This fundamental unit is the magnitude of charge carried by an electron or a proton. Mathematically, the charge $q$ on any body is given by $q = ne$, where $n$ is an integer ($n = 0, \pm 1, \pm 2, \ldots$). The charge on an electron is $-e$, and the charge on a proton is $+e$. The value of $e$ in the SI system is approximately $1.602192 \times 10^{-19}$ C. While this quantization is significant at the microscopic level, it becomes imperceptible at the macroscopic level where the number of charges involved is enormous, leading to the appearance of charge as a continuous quantity.

Example 1.1: If $10^9$ electrons move from one body to another every second, the time required to accumulate a total charge of 1 C is approximately 198 years, highlighting that 1 Coulomb is a very large unit of charge.

Example 1.2: A cup of water (approximately 250g) contains a significant amount of positive and negative charge, approximately $1.34 \times 10^7$ C, due to the presence of electrons and protons in water molecules.

Coulomb’s Law

Coulomb's Law describes the force between two point charges. For two point charges, $q_1$ and $q_2$, separated by a distance $r$ in a vacuum, the magnitude of the electrostatic force $F$ between them is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them:

$F = k \frac{|q_1 q_2|}{r^2}$

where $k$ is Coulomb's constant, approximately $9 \times 10^9 \text{ Nm}^2/\text{C}^2$. The force acts along the line joining the two charges.

The constant $k$ can be expressed as $k = \frac{1}{4\pi\epsilon_0}$, where $\epsilon_0$ is the permittivity of free space, with a value of approximately $8.854 \times 10^{-12} \text{ C}^2\text{ N}^{-1}\text{m}^{-2}$.

In vector form, the force on charge $q_1$ due to charge $q_2$ is given by:

$\vec{F}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{21}^2} \hat{r}_{21}$

where $\vec{r}_{21}$ is the vector from $q_1$ to $q_2$ and $\hat{r}_{21}$ is the unit vector in that direction. Coulomb's law obeys Newton's third law, meaning the force on $q_2$ due to $q_1$ is equal in magnitude and opposite in direction to the force on $q_1$ due to $q_2$. This law is valid for any sign of charges, correctly predicting repulsion for like charges and attraction for unlike charges.

Example 1.3: Compares the electrostatic force and gravitational force between charged particles. It highlights that electrical forces are significantly stronger than gravitational forces. For instance, the ratio of the electric force to the gravitational force between an electron and a proton is approximately $2.4 \times 10^{39}$. The example also calculates the acceleration of an electron and a proton due to their mutual electrostatic attraction, showing very large accelerations.

Example 1.4: Demonstrates the application of Coulomb's law and the principle of charge redistribution when spheres are touched. It shows that when charged spheres touch identical uncharged spheres, the charge is halved, and subsequent force calculations can be made.

Forces Between Multiple Charges

When a system contains multiple charges, the total force on any one charge is the vector sum of the forces exerted by each of the other charges individually. This is known as the principle of superposition. The force between any pair of charges is unaffected by the presence of other charges. For a system of charges $q_1, q_2, \ldots, q_n$, the total force $\vec{F}_1$ on charge $q_1$ is given by:

$\vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + \ldots + \vec{F}_{1n} = \sum_{i=2}^{n} \vec{F}_{1i}$

where $\vec{F}_{1i}$ is the force on $q_1$ due to $q_i$, calculated using Coulomb's Law.

Example 1.5: Calculates the force on a charge placed at the centroid of an equilateral triangle with equal charges at its vertices. Due to symmetry, the forces from the three charges cancel each other, resulting in zero net force on the central charge.

Example 1.6: Analyzes the forces on charges placed at the vertices of an equilateral triangle when one charge is negative. It demonstrates how to find the resultant force on each charge by vector addition and notes that the sum of forces on all charges is zero, consistent with Newton's third law.

Electric Field

The concept of an electric field is introduced to describe the influence of a source charge on the space around it. An electric field exists at every point in space around a charged object. When a test charge is placed in this field, it experiences a force. The electric field $\vec{E}$ at a point is defined as the force $\vec{F}$ experienced by a unit positive test charge placed at that point:

$\vec{E} = \frac{\vec{F}}{q_0}$

where $q_0$ is the test charge. For a point charge $Q$, the electric field at a distance $r$ is given by:

$\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \hat{r}$

The electric field is a vector quantity, with its direction determined by the sign of the source charge: radially outward for positive charges and radially inward for negative charges. The magnitude of the electric field decreases with the square of the distance from the source charge.

The physical significance of the electric field lies in its ability to characterize the electrical environment of a charge configuration, independent of any test charge. It provides a convenient way to describe forces in time-dependent electromagnetic phenomena, accounting for the finite speed of signal propagation.

Electric Field Due To A System Of Charges

For a system of discrete charges, the electric field at any point is the vector sum of the electric fields produced by each individual charge at that point, as per the superposition principle:

$\vec{E} = \sum_{i=1}^{n} \vec{E}_i$

where $\vec{E}_i$ is the electric field due to the $i$-th charge.

Example 1.7: Calculates the time of fall for an electron and a proton in a uniform electric field, comparing it to free fall under gravity. It shows that the acceleration in an electric field is significantly larger than due to gravity, and that heavier particles accelerate less.

Example 1.8: Calculates the electric field at different points due to a system of two point charges. It demonstrates how to find the resultant electric field by vector addition, considering the contributions from each charge.

Physical Significance Of Electric Field

The concept of the electric field is introduced as a useful tool for describing the electrical environment around a system of charges. It represents the force that a unit positive test charge would experience at a point, independently of the test charge itself. While not strictly necessary for electrostatics, the electric field becomes crucial in understanding time-dependent electromagnetic phenomena, where it explains the propagation of forces and energy through space.

Electric Field Lines

Electric field lines are a visual tool for mapping the electric field in space. They are drawn such that the tangent to a line at any point represents the direction of the electric field at that point. The density of the field lines (number of lines per unit area perpendicular to the lines) is proportional to the magnitude of the electric field; closer lines indicate a stronger field, and more spaced-out lines indicate a weaker field.

Key properties of electric field lines include:

• They originate from positive charges and terminate on negative charges.
• They are continuous curves without any breaks in charge-free regions.
• Two field lines never intersect each other, as the electric field at a point has a unique direction.
• Electrostatic field lines do not form closed loops, indicating that the electrostatic field is conservative.

Figure 1.12 shows the field lines of a point charge, illustrating the radial outward direction and the decreasing density with distance. Figure 1.13 demonstrates how the density of field lines represents the field strength, which varies inversely with the square of the distance ($1/r^2$). Figure 1.14 illustrates field lines for various charge configurations, such as single charges, dipoles, and systems of like charges.

Electric Flux

Electric flux ($\Phi_E$) is a measure of the electric field passing through a given surface. It is defined as the dot product of the electric field vector ($\vec{E}$) and the area vector ($\vec{DS}$):

$\Phi_E = \vec{E} \cdot \vec{DS} = E DS \cos\theta$

where $\theta$ is the angle between $\vec{E}$ and $\vec{DS}$. The area vector $\vec{DS}$ has a magnitude equal to the area of the surface element and a direction perpendicular to the surface. For a closed surface, the outward normal is conventionally chosen as the direction of the area vector.

The total electric flux through any surface S is obtained by integrating the flux over all area elements:

$\Phi_E = \oint_S \vec{E} \cdot d\vec{S}$

The unit of electric flux is N m$^2$/C.

Figure 1.15 illustrates how the flux depends on the angle between the electric field and the area element. Figure 1.16 defines the area vector and the convention for closed surfaces.

Electric Dipole

An electric dipole consists of two equal and opposite point charges, $+q$ and $-q$, separated by a distance $2a$. The line joining the two charges defines the dipole axis, and the direction from $-q$ to $+q$ is conventionally taken as the direction of the dipole moment.

The Field Of An Electric Dipole

The electric field produced by a dipole falls off faster than that of a single point charge, specifically as $1/r^3$ at large distances ($r \gg a$). The electric field has different magnitudes and directions at points on the dipole axis and in the equatorial plane (perpendicular to the axis through the center).

• On the dipole axis: $\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3} \hat{p}$, where $\hat{p}$ is the unit vector along the dipole axis.
• On the equatorial plane: $\vec{E} = -\frac{1}{4\pi\epsilon_0} \frac{p}{r^3} \hat{p}$. The negative sign indicates that the field is opposite to the dipole moment.

The dipole moment ($\vec{p}$) is defined as the product of one of the charges and the vector separation between them: $\vec{p} = q (2a) \hat{p}$.

Example 1.9: Calculates the electric field at a point on the axis and on the equatorial plane of a given electric dipole, verifying the $1/r^3$ dependence at large distances.

Physical Significance Of Dipoles

Many molecules possess a permanent electric dipole moment if the centers of their positive and negative charges do not coincide (polar molecules, e.g., H$_2$O). Other molecules develop a dipole moment when placed in an external electric field (non-polar molecules, e.g., CO$_2$, CH$_4$). The presence of electric dipoles in materials gives rise to various important properties and applications.

Dipole In A Uniform External Field

When an electric dipole is placed in a uniform external electric field $\vec{E}$, the forces on the two charges $+q$ and $-q$ are equal and opposite ($q\vec{E}$ and $-q\vec{E}$). Since these forces act at different points, they produce a torque ($\vec{\tau}$) on the dipole, tending to align it with the external field. The torque is given by:

$\vec{\tau} = \vec{p} \times \vec{E}$

The magnitude of the torque is $\tau = pE\sin\theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$. If the net force on the dipole is zero (as in a uniform field), the torque is independent of the point of application of forces.

In a non-uniform electric field, the dipole experiences both a net force and a torque. The net force tends to move the dipole towards regions of stronger electric field if the dipole moment is aligned with the field, and towards regions of weaker field if it is antiparallel.

This phenomenon explains how a charged comb attracts uncharged pieces of paper. The charged comb creates a non-uniform electric field, inducing a dipole moment in the paper, and the net force pulls the paper towards the comb.

Continuous Charge Distribution

For many practical situations, it is more convenient to consider charges as being distributed continuously over a region rather than as discrete point charges. This is analogous to treating mass as a continuous distribution. We define charge densities:

• Linear charge density ($\lambda$): Charge per unit length, $\lambda = \frac{dQ}{dl}$. Units: C/m.
• Surface charge density ($\sigma$): Charge per unit area, $\sigma = \frac{dQ}{dS}$. Units: C/m$^2$.
• Volume charge density ($\rho$): Charge per unit volume, $\rho = \frac{dQ}{dV}$. Units: C/m$^3$.

These densities are macroscopic averages, smoothing out the discrete nature of charges. The electric field due to a continuous charge distribution can be calculated by integrating the contributions from infinitesimal charge elements ($dQ = \lambda dl$, $dQ = \sigma dS$, or $dQ = \rho dV$) over the distribution, using Coulomb's Law and the superposition principle.

$d\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{dQ}{r^2} \hat{r}$

$\vec{E} = \int d\vec{E}$

Gauss’s Law

Gauss's Law provides a powerful method for calculating electric fields, particularly for charge distributions with high symmetry. It relates the total electric flux ($\Phi_E$) through any closed surface (Gaussian surface) to the total electric charge ($q_{enc}$) enclosed by that surface:

$\Phi_E = \oint_S \vec{E} \cdot d\vec{S} = \frac{q_{enc}}{\epsilon_0}$

Key points about Gauss's Law:

• It is valid for any closed surface, regardless of its shape or size.
• $q_{enc}$ includes all charges enclosed by the surface.
• The electric field $\vec{E}$ on the left side is due to all charges (both inside and outside the surface).
• A Gaussian surface cannot pass through a discrete point charge.
• It simplifies the calculation of electric fields for symmetric charge distributions.

The law is a direct consequence of Coulomb's inverse-square law and the superposition principle. If the net flux through a closed surface is zero, it implies that the net charge enclosed by the surface is zero.

Example 1.10: Calculates the electric flux through a cube with a non-uniform electric field and then uses Gauss's Law to find the enclosed charge.

Example 1.11: Calculates the flux through the faces and sides of a cylinder in a uniform electric field and determines the net charge inside.

Applications Of Gauss’s Law

Gauss's Law is particularly useful for finding electric fields in situations with high symmetry.

Field Due To An Infinitely Long Straight Uniformly Charged Wire

For an infinitely long thin wire with uniform linear charge density $\lambda$, the electric field at a radial distance $r$ is:

$\vec{E} = \frac{1}{2\pi\epsilon_0} \frac{\lambda}{r} \hat{n}$

where $\hat{n}$ is the unit vector pointing radially outward from the wire. The field is directed radially outward for positive $\lambda$ and inward for negative $\lambda$. This result is an approximation valid for points close to the central portion of a very long wire.

Field Due To A Uniformly Charged Infinite Plane Sheet

For an infinite plane sheet with uniform surface charge density $\sigma$, the electric field at any point is:

$\vec{E} = \frac{\sigma}{2\epsilon_0} \hat{n}$

where $\hat{n}$ is a unit vector normal to the plane, pointing away from it. The field is uniform and directed away from the sheet if $\sigma > 0$, and towards the sheet if $\sigma < 0$. This equation is also an approximation for large plane sheets, valid in the central regions away from the edges.

Field Due To A Uniformly Charged Thin Spherical Shell

For a thin spherical shell of radius $R$ and uniform surface charge density $\sigma$ (total charge $q = 4\pi R^2 \sigma$):

• Outside the shell ($r > R$): The electric field is identical to that of a point charge $q$ located at the center: $\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}$
• Inside the shell ($r < R$): The electric field is zero: $\vec{E} = 0$.

This result for the field inside a uniformly charged shell is zero, which is a direct consequence of Gauss's Law and the symmetry of the charge distribution.

Example 1.12: Applies Gauss's Law to a model of an atom with a positively charged nucleus and a uniform distribution of negative charge. It calculates the electric field both inside and outside the region of negative charge.

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