Chapter 1 — Electric Charges And Fields

Fig 1.1 — Educational Diagram: Electric Charge, Quantization (q = n e) & Conservation

What is electric charge?
Electric charge is a fundamental property of matter that causes it to experience a force in the presence of other charges. There are two kinds of charge: positive and negative. Like charges repel and unlike charges attract.

Quantitative definition and unit
The SI unit of charge is the coulomb (C). Charge is carried by particles — for example, the electron has charge −e and the proton has charge +e, where e = 1.602 × 10⁻¹⁹ C (the elementary charge).

Quantization of charge
Electric charge is quantized: any measurable charge q is an integer multiple of the elementary charge, q = n·e, where n is an integer (positive, negative or zero). This is an experimental fact relevant at atomic and macroscopic scales.

Conservation and additivity
Total electric charge is conserved in isolated systems — it cannot be created or destroyed (only transferred). Charge is additive: the net charge of a system is the algebraic sum of individual charges.

How objects become charged
Common methods of charging include:

• By friction — rubbing two different materials transfers electrons from one to the other (triboelectric effect).
• By conduction (contact) — touching a charged object to a conductor allows charge to flow and redistribute.
• By induction — bringing a charged object near a conductor redistributes charges within the conductor; grounding can remove or supply charge.

Conductors and insulators
In conductors, charges (electrons) are mobile and can move freely; in electrostatic equilibrium, excess charge resides on the surface. In insulators (dielectrics), charges are localized and do not move freely.

Fig 1.2 — Educational Diagram: Methods of Charging (Friction, Conduction, Induction)

Overview: Charging methods are the ways in which neutral bodies acquire net electric charge. In Class 12 Physics the three standard methods are charging by friction (triboelectric charging), charging by conduction (contact), and charging by induction. Each method relies on transfer or redistribution of electrons and obeys charge conservation.

1. Charging by friction (triboelectric charging)

• Mechanism: When two different materials are rubbed together, electrons may be transferred from one surface to the other depending on their tendency to gain or lose electrons (triboelectric series). One body becomes negatively charged (gains electrons) and the other positively charged (loses electrons).
• Key features: Works mainly for insulators and dielectric surfaces; resulting charges tend to remain localized because charges cannot move freely.
• Practical note: Amount of charge depends on material pair, contact area, pressure, rubbing speed, humidity (moist air reduces charging).

2. Charging by conduction (contact)

• Mechanism: When a charged conductor touches another conductor, electrons flow until both reach the same electric potential. Total charge is conserved and redistributes between the conductors.
• Identifying condition: After contact, conductors in electrical contact have equal potential (V1 = V2). For isolated spherical conductors, final charges divide in proportion to their capacitances.
• Special case (identical conductors): If a charged conductor touches an identical uncharged conductor, the initial charge divides equally.

3. Charging by induction

• Mechanism: A charged object brought near a neutral conductor causes redistribution (separation) of charges within the conductor without direct contact. If the conductor is then grounded while the charged object remains nearby, some charge flows to or from Earth; removing the ground and then the external charged object leaves the conductor with net charge of opposite sign to the nearby object.
• Key features: No direct contact needed with the charging object; useful to produce charge of sign opposite to the inducing charge (by grounding during induction).

Other important points:

• Conservation of charge always holds: the algebraic sum of charges before and after any process remains the same (unless charge is exchanged with the environment/earth).
• In conductors, excess charge resides on the outer surface; electric field inside a conductor in electrostatic equilibrium is zero.
• Insulators can be polarized: local bound charges appear but free charge mobility is low, so charges remain localized.

Summary of process steps (conduction vs induction):

• Conduction/contact: Touch charged object to neutral conductor → charges flow until potentials equalize → separate; both are charged (same sign as original if no ground connection).
• Induction (with grounding): Bring charged object near neutral conductor → connect conductor to ground so charges of appropriate sign flow to/from earth → disconnect ground → remove inducing object → conductor retains induced net charge (opposite sign to inducing charge).

Fig 1.3 — Educational Diagram: Coulomb's Law of Electrostatic Force & Vector Formula

Statement: Coulomb's law gives the electrostatic force between two point charges. It states that the magnitude of the force between two point charges q1 and q2 separated by distance r in vacuum is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the separation.

Scalar form: F = k |q1 q2| / r^2, where k = 1 / (4 π ε0) ≈ 8.988 × 10^9 N m^2 C^{-2} and ε0 = 8.854 × 10^{-12} F m^{-1}.

Vector form: the force on charge q2 due to q1 is
F_21 = (1 / (4 π ε0)) * (q1 q2 / r^2) r̂, where r̂ is the unit vector pointing from q1 to q2. The sign of q1 q2 determines direction: like charges repel, unlike charges attract.

Superposition principle: For more than two charges, the net force on any charge is the vector sum of forces due to each other charge separately: F_net = Σ F_i (vector addition).

Effect of medium: In a dielectric medium of relative permittivity εr, replace ε0 by ε = εr ε0, or k becomes k = 1/(4 π ε). The force weakens by factor 1/εr.

• Valid for: point charges or spherically symmetric charge distributions (treatable as point charges outside the sphere).
• Limitations: does not include magnetic or relativistic effects; at very short distances quantum effects matter.
• Units: force in newtons (N), charge in coulombs (C), distance in meters (m).

Physical significance: Coulomb's law is the electrostatic analogue of Newton's law of gravitation but with both attractive and repulsive interactions and much stronger coupling per unit source. It underpins the electric field concept: the electric field of a point charge q is E = k q / r^2 r̂.

Fig 1.4 — Educational Diagram: Principle of Superposition for Multiple Charges

Definition: The principle of superposition states that the total electric force (or electric field or potential) at any point due to a system of charges equals the vector sum (or algebraic sum for scalar potential) of the contributions from each charge taken separately, as if the others were absent.

Why it holds (brief): Coulomb's law for point charges is linear and pairwise additive: the force between any two point charges depends only on those two charges. Because the laws are linear, effects from different sources add without producing new types of terms (no products of fields), so the net effect is a sum of individual effects.

Mathematical statement (point charges): For N point charges q_i located at positions r_i, the electric field at point r is

• E(r) = Σ_{i=1}^N E_i(r) = Σ_{i=1}^N k q_i (r − r_i)/|r − r_i|^3, where k = 1/(4πε_0).
• The force on a test charge q at r is F(r) = q E(r) = Σ_{i=1}^N F_i(r) = Σ_{i=1}^N k q q_i (r − r_i)/|r − r_i|^3.

Continuous charge distributions: Replace the sum by an integral: E(r) = k ∫ (ρ(r') (r − r')/|r − r'|^3) dV', and potential V(r) = k ∫ (ρ(r')/|r − r'|) dV'.

Potential superposition: Electric potential is a scalar and also obeys superposition: V_total(r) = Σ_i V_i(r) = k Σ_i q_i/|r − r_i|. Because V is scalar, superposition is an algebraic sum (signs included).

Conditions and limits:

• Valid in classical electrostatics and in linear media where field responses add linearly.
• Breaks down in nonlinear media (materials with nonlinear dielectric response) or in quantum regimes where interactions are not described by classical Coulomb addition.

How to apply (procedure):

1. Compute the field (or force, or potential) produced by each individual charge at the point of interest.
2. Express each contribution as a vector (for field/force) with correct direction and sign.
3. Add all vector contributions component-wise to get the net field or force; add scalars for potential.

Fig 1.5 — Educational Diagram: Continuous Charge Distributions (Linear λ, Surface σ, Volume ρ)

What is a charge distribution?
When many charges are spread over a region (along a line, over a surface or throughout a volume) we describe the arrangement as a charge distribution. If charges are few and separated we treat them as discrete point charges; if they are closely packed we treat the distribution as continuous and use charge densities.

Types of distributions and their densities

• Linear charge density (λ): for charge spread along a line (wire, rod). Definition: λ = dQ/dl. Element: dQ = λ dl. Unit: C m^-1.
• Surface charge density (σ): for charge on a surface (sheet, conductor surface). Definition: σ = dQ/dA. Element: dQ = σ dA. Unit: C m^-2.
• Volume charge density (ρ): for charge distributed in a volume (charged insulating solid, space charge). Definition: ρ = dQ/dV. Element: dQ = ρ dV. Unit: C m^-3.

Integral relations
To get total charge from a density integrate over the appropriate domain:
Q = ∫_line λ(l) dl, Q = ∫_surface σ( r ) dA, Q = ∫_volume ρ( r ) dV.

Uniform distributions (constant densities)
If the density is uniform, simple relations hold: λ = Q/L, σ = Q/A, ρ = Q/V.

Special notes

• In electrostatic equilibrium in a conductor all excess charge resides on the surface — described by a surface density σ.
• Point charges are represented in continuous language using the Dirac delta: ρ( r ) = q δ( r − r0 ).
• For thin objects you can relate densities approximately: for a thin sheet of thickness t, ρ ≈ σ/t; for a thin wire of radius a, σ ≈ λ/(2πa) (if charge resides on the surface).

Use with Coulomb's law and Gauss's law
Charge densities let you compute fields by integrating Coulomb's law or by using Gauss's law with symmetry. Example results used often in Class 12: field of an infinite line E = λ/(2πε0 r), infinite sheet E = σ/(2ε0) (one-sided conductor: E = σ/ε0), on-axis field of a ring and field inside/outside uniformly charged sphere (see formulas list).

Practical measurement/visualisation
Graphical plots of ρ(x), σ(θ) or λ(x) help visualize where charge is concentrated. Sketching field magnitude versus distance for standard distributions clarifies how geometry affects fields.

Fig 1.6 — Educational Diagram: Electric Field (E = F/q0) & Radial Field of Point Charge

Definition: An electric field at a point in space is a vector quantity that represents the electric force per unit positive test charge placed at that point. If a small positive test charge q0 experiences an electric force F, the electric field E is E = F/q0. The field exists even if no test charge is present.

Nature and direction: E is a vector. The direction of E at a point is the direction of the force on a positive test charge. Field lines show direction: they originate on positive charges and terminate on negative charges; the density of lines indicates field strength.

Point charge (Coulomb's law): For a point charge q located at the origin, the electric field at position r (distance r from the charge) is radial and given by E = (1/4πε0) (q / r^2) r̂. Here ε0 is the permittivity of free space and r̂ is the unit vector from the charge to the point.

Superposition principle: The net electric field due to several charges is the vector sum of fields due to individual charges: E_total = Σ E_i. This also applies to continuous charge distributions (integral form).

Fields of common charge distributions (qualitative):

• Infinite line of charge (linear density λ): E ∝ 1/r, directed radially outward/inward.
• Infinite uniformly charged plane (surface density σ): E is uniform and equals σ/(2ε0) on each side of a single infinite plane; between two oppositely charged parallel plates E = σ/ε0 (approximately uniform).
• Uniformly charged thin spherical shell: outside it behaves like a point charge (E ∝ 1/r^2); inside the shell E = 0.
• Electric dipole (moment p = q·2a): far-field on axis E ∝ 1/r^3; on perpendicular bisector E has opposite sign and magnitude ∝ 1/r^3.

Relation with electric potential: Electric field is the negative gradient of electric potential: E = -∇V. In one dimension, E_x = -dV/dx. This links field lines to equipotential surfaces (E is perpendicular to equipotentials).

Gauss's law (useful for symmetry): The electric flux through a closed surface equals the enclosed charge divided by ε0: ∮ E · dA = Q_enclosed / ε0. This is often used to compute E for symmetric charge distributions (sphere, infinite line, infinite plane).

Energy density: The energy stored per unit volume in an electric field is u = (1/2) ε0 E^2.

Units: Electric field is measured in newtons per coulomb (N/C) or volts per metre (V/m). Both are equivalent.

Important practical points: Field inside a conductor in electrostatic equilibrium is zero; any excess charge resides on the surface. Field direction and magnitude determine forces on charges, motion of charged particles in devices, and electrostatic interactions in everyday technology.

Fig 1.7 — Educational Diagram: Electric Field Lines & Equipotential Surfaces

Overview
Electric field lines and equipotential surfaces are graphical tools used to visualize the electric field (E) and electric potential (V) produced by charges. Field lines show the direction and relative strength of the electric field, while equipotential surfaces connect points having the same electric potential.

Electric field lines (field lines)

• Definition: A field line is an imaginary curve whose tangent at any point gives the direction of the electric field at that point.
• Conventions: Field lines originate on positive charges and terminate on negative charges (or at infinity). The line direction indicates the force on a positive test charge.
• Density and strength: The local density of field lines (number per unit area perpendicular to lines) represents the magnitude of E. Closer lines mean stronger field.
• Properties: Field lines never cross. They are continuous and may start/stop only at charges. For a uniform field (e.g., between parallel plates) lines are parallel, equally spaced, and straight.

Equipotential surfaces

• Definition: An equipotential surface is a surface on which the electric potential V has the same value at every point.
• Work and motion: No work is done by the electric force when a charge moves along an equipotential (ΔV = 0 → W = qΔV = 0).
• Shapes: For a single point charge, equipotentials are concentric spheres (in 3D) or circles (in plane cross-section). For a uniform field they are planes perpendicular to the field lines.
• Properties: Equipotential surfaces are always perpendicular to electric field lines at their intersections.

Mathematical relation between E and V

• Gradient relation: E = -grad V (vector form). The electric field points in the direction of greatest decrease of potential.
• Line integral form: V(b) − V(a) = −∫_a^b E · dl. For uniform E along direction normal to equipotentials: ΔV = −E Δs. Often magnitude: E = −dV/dn, where dn is distance along the normal.

Important consequences

• Because equipotentials are perpendicular to field lines, equipotential surfaces help sketch field patterns by drawing normals to them.
• Inside a conductor in electrostatic equilibrium, E = 0 and the entire conductor is an equipotential. This is the basis for electrostatic shielding (Faraday cage).
• Work required to move charge q between two equipotentials of potentials V1 and V2 is W = q(V1 − V2).

Typical diagrams students should be able to draw and interpret

• Field lines and concentric equipotential circles for a single positive and for a single negative point charge.
• Dipole field lines with corresponding nested equipotential contours (lobe-shaped surfaces around charges).
• Two like charges and two unlike charges: symmetry of lines and equipotential geometry (saddle and null points).
• Uniform field between parallel plates: straight field lines and equally spaced planar equipotentials.

Pedagogical tips
When sketching, mark arrowheads on field lines (direction of force on +q), draw equipotentials as smooth curves/surfaces, and check perpendicularity at intersections. Use denser lines where E is stronger.

Fig 1.8 — Educational Diagram: Electric Dipole Moment (p = q·2a) & Axial vs Equatorial Field

Definition: An electric dipole consists of two equal and opposite point charges +q and −q separated by a small distance 2a. The line joining them is the dipole axis. For most dipole formulas we assume the observation distance r >> a.

Dipole moment: The dipole moment is a vector defined as p = q·2a (magnitude) and its direction is from the negative charge to the positive charge. Its SI unit is C·m.

Electric potential (far-field approximation): At a point given by spherical coordinates (r, θ) measured from the dipole centre (θ is the angle between r and p), the scalar potential for r >> a is

V(r,θ) = (1/4πε₀)·(p cosθ) / r²

Electric field (far-field): From V, the field components in spherical coordinates (r̂, θ̂) are

E_r = (1/4πε₀)·(2p cosθ) / r³
E_θ = (1/4πε₀)·(p sinθ) / r³

Or in vector form:

E(r) = (1/4πε₀)·[3(p·r̂) r̂ − p] / r³

Special lines: Axial line (θ = 0): E_axial = (1/4πε₀)·(2p) / r³ (directed along the axis).
Equatorial plane (θ = 90°): E_equatorial = (1/4πε₀)·(−p) / r³ (directed opposite to p).

Torque and potential energy in a uniform external field E: A dipole in a uniform field experiences a torque

τ = p × E (magnitude τ = pE sinθ)

and potential energy

U = −p·E = −pE cosθ

Distance dependence & validity: The dipole field and potential fall off rapidly with distance: E ∝ 1/r³ and V ∝ 1/r² for r >> a. Exact expressions (without approximation) can be obtained by summing contributions from the two charges, but the above are the standard dipole approximations used in Class 12 when 2a is very small compared to r.

Physical picture & field lines: Field lines originate on +q and terminate on −q; close to the dipole they curve from +q to −q forming the characteristic pattern. At large distances the dipole looks like a single entity with net zero charge but a nonzero dipole moment.

Fig 1.9 — Educational Diagram: Electric Flux (Φ = E · A cos θ) & Area Vector

Definition: Electric flux (Φ) through a surface is a measure of the number of electric field lines crossing that surface. It quantifies how much of the electric field penetrates the surface.

SI unit: newton·meter²/coulomb (N·m²/C) or volt·meter (V·m).

Flux for a uniform field and flat surface: If a uniform electric field E passes through a flat surface of area A, and the angle between the field vector and the outward normal to the surface is θ, the electric flux is

Φ = E A cosθ

This formula implies maximum flux when the field is normal (θ = 0) and zero flux when the field is parallel to the surface (θ = 90°).

General definition (for non-uniform fields or curved surfaces): For any surface S, divide it into infinitesimal area elements dA with local outward normal. The net flux is the surface integral

Φ = ∮_S E · dA = ∮_S E cosθ dA

Closed surfaces and Gauss's law: For a closed surface (a surface that fully encloses a volume) the net electric flux equals the net charge enclosed (q_enc) divided by the permittivity of free space (ε₀):

Φ_closed = ∮_closed E · dA = q_enc / ε₀

This is Gauss's law. It is especially useful for highly symmetric charge distributions (spherical, cylindrical, planar) to find electric fields.

Properties and useful points:

• Flux can be positive, negative or zero depending on field direction relative to the outward normal.
• For a point charge q at the center of a spherical surface, flux through the sphere is q/ε₀, independent of the sphere's radius.
• If no charge is enclosed by a closed surface, net flux through it is zero (field lines entering equal those leaving).
• Local form of Gauss's law: ∇·E = ρ/ε₀, where ρ is charge density.

How to apply: For problems, identify whether surface is open or closed, check symmetry (can simplify E), choose suitable Gaussian surface if using Gauss's law, or evaluate the surface integral directly for non-uniform cases.

Fig 1.10 — Educational Diagram: Gauss's Law of Electrostatics (Φ = ∮ E·dA = Q/ε0)

Definition: Gauss's law relates the electric flux through a closed surface to the total charge enclosed by that surface. In integral form: Φ = ∮ E · dA = Q_enc / ε0, where Φ is electric flux, E is electric field, dA is an outward area element, Q_enc is total enclosed charge and ε0 is the permittivity of free space.

Electric flux: Electric flux through a surface measures how many field lines pass through that surface. For a uniform field and flat area, flux Φ = E A cosθ, where θ is the angle between E and the outward normal.

Gaussian surface: A mathematical closed surface chosen to exploit symmetry (sphere, cylinder, plane slab) so that the dot product E·dA is easy to evaluate. Gauss's law is always true, but it is most useful when the field has high symmetry so E is constant over parts of the surface or perpendicular/parallel to them.

Using Gauss's law (steps):

• Choose a Gaussian surface that matches the symmetry of the charge distribution (spherical for point/spherical charges, cylindrical for infinite line, pillbox for infinite plane).
• Write Φ = ∮ E·dA and simplify using symmetry so E can be taken outside the integral on appropriate surface portions.
• Compute Q_enc inside the surface (include volume or surface charge as needed).
• Set Φ = Q_enc/ε0 and solve for E.

Common results obtained from Gauss's law:

• Point charge (or uniformly charged sphere, outside): E = Q / (4πε0 r²) (radial, inverse-square).
• Infinite line charge (linear density λ): E = λ / (2πε0 r) (radial, falls as 1/r).
• Infinite plane sheet (surface density σ): E = σ / (2ε0) (constant, independent of distance).
• Conductor in electrostatic equilibrium: electric field inside is zero; any net charge resides on the surface; just outside surface E = σ / ε0 (normal to surface).

Differential form: Using the divergence theorem, Gauss's law becomes ∇·E = ρ/ε0, where ρ is volume charge density. This form is useful in field pointwise analysis and in Maxwell's equations.

Limitations and remarks: Gauss's law always holds, but it yields simple algebraic results only when symmetry allows E to be constant or zero on chosen surface parts. For arbitrary charge distributions, direct use gives an integral equation that may be hard to solve.

Fig 1.11 — Educational Diagram: Applications of Gauss's Law (Line, Sheet & Spherical Shell)

Gauss's law (statement): The net electric flux through any closed surface (Gaussian surface) equals the enclosed charge divided by the permittivity of free space: ∮E · dA = Q_enclosed/ε₀. Gauss's law is especially powerful for calculating electric fields when the charge distribution has high symmetry (spherical, cylindrical, planar).

General procedure to apply Gauss's law:

• Identify symmetry (spherical, cylindrical, or planar).
• Choose a Gaussian surface that matches the symmetry so that E has constant magnitude on parts of the surface and contributions simplify.
• Compute flux by evaluating ∮E·dA (often reduces to E·A times symmetry factor).
• Compute the enclosed charge Q_enclosed.
• Use E = Q_enclosed/(ε₀·A) or the equivalent result from the flux equation to get E.

Key applications and results (with the typical Gaussian surface used):

• Uniformly charged thin spherical shell (spherical symmetry):
  • Gaussian surface: concentric sphere of radius r.
  • Outside (r > R): field behaves like a point charge: E = (1/(4πε₀))·Q/r².
  • Inside (r < R): E = 0 (no enclosed charge).
• Uniformly charged solid (non-conducting) sphere (spherical symmetry):
  • Outside (r > R): same as point charge: E = (1/(4πε₀))·Q/r².
  • Inside (r < R): field increases linearly with r: E = (1/(4πε₀))·(Q·r/R³) = (ρ·r)/(3ε₀), where ρ is volume charge density.
• Infinite line of charge (cylindrical symmetry):
  • Gaussian surface: coaxial cylinder of radius r and length L.
  • Field: E = λ/(2π ε₀ r), directed radially outward (λ = linear charge density).
• Infinite charged plane (planar symmetry):
  • Gaussian surface: pillbox straddling the plane.
  • Field (single infinite sheet, both sides): E = σ/(2ε₀), constant and independent of distance; σ = surface charge density.
  • Two oppositely charged infinite plates (parallel-plate capacitor): inside E = σ/ε₀ (if plates are infinite), outside fields cancel.
• Infinite cylindrical shell (conducting):
  • Inside hollow region (r < a): E = 0 (for conducting shell).
  • Outside (r > a): E = λ/(2π ε₀ r) as if all charge were on axis.
• Conducting solid sphere:
  • All excess charge resides on the outer surface; inside conductor E = 0.
  • Outside field: E = (1/(4πε₀))·Q/r².
• Coaxial cable (application of cylindrical symmetry):
  • Between inner radius a and outer radius b, if inner conductor carries +λ and outer carries -λ, then E(r) = λ/(2π ε₀ r) for a < r < b.
  • Inside conductors the field is zero; useful for shielding and transmission-line design.
• Uniformly charged infinite slab (thickness 2a):
  • Field inside is linear with distance from center: E = ρ x/ε₀ (for |x| < a), where x is distance from mid-plane; outside it becomes constant E = ρ a/ε₀ = σ/2ε₀ (matches infinite sheet limit).

Important physical consequences and uses:

• Electrostatic shielding (Faraday cage): conductor interior has E = 0, so sensitive electronics can be protected from external fields.
• Capacitor design: field between parallel plates derived using Gauss's law gives capacitance per area, energy density (1/2)ε₀E².
• Coaxial cable and transmission lines: field distribution and shielding derived from cylindrical Gauss law results.
• Electrostatic precipitators and droplet/particle manipulation exploit field near plates and edges; lightning rods use field enhancement at sharp points (not directly from Gauss but consistent with surface charge concentration).

Limitations: Gauss's law always holds, but finding E easily from it requires high symmetry (spherical, cylindrical, planar). For arbitrary charge distributions, other methods (superposition, Coulomb's law, numerical methods) are needed.

Fig 1.12 — Educational Diagram: Electrostatic Shielding (E=0 in Conductor Cavity & Faraday Cage)

Definition: A conductor in electrostatic equilibrium is a conducting body in which charges are at rest and the electric field inside the conducting material is zero.

Key properties and reasoning:

• Electric field inside is zero: Free charges in a conductor move until the internal electric field cancels. At equilibrium, E = 0 everywhere inside the conducting material.
• All excess charge resides on the surface: Using Gauss's law for a Gaussian surface drawn entirely inside the conductor: flux = 0 ⇒ net enclosed charge = 0, so any excess charge must be on the conductor's surface.
• Surface field is perpendicular: The electric field at the conductor surface is normal to the surface. If there were a tangential component, free charges would move along the surface and equilibrium would not exist.
• Boundary condition: Just outside the surface the normal component of the electric field E_n satisfies E_n = σ/ε0, where σ is the surface charge density and ε0 is the permittivity of free space. (Inside the conductor E_n = 0, so the discontinuity equals σ/ε0.)
• Equipotential: The entire conductor (including its surface and interior) is at the same electric potential V. Moving a charge inside the conductor requires no work.
• Sharp points and curvature: Surface charge density and hence the local electric field are larger at regions of high curvature (sharp points). This is why corona discharge and strong fields occur at pointed conductors.
• Hollow conductor and induced charges: If a charge q is placed in a cavity inside a hollow isolated conductor (not touching the conductor), an induced charge −q appears on the inner surface and a charge +q appears on the outer surface, so the net conductor charge remains unchanged. If the conductor is grounded, the induced charge may flow to ground.
• Electrostatic shielding (Faraday cage): Because E = 0 inside a closed conductor, sensitive devices inside a conducting enclosure are shielded from external static electric fields.

Common derivations/arguments:

• Gauss's law: For a Gaussian surface just beneath the conductor surface, flux = 0 ⇒ field inside = 0 ⇒ any flux originates from surface charges.
• Work and potential: Since E = 0 inside, potential V is constant throughout the conductor. Therefore, V(surface) = V(internal point).

Practical consequences:

• Design of lightning rods and grounding systems uses the fact that charges concentrate at points and that grounding removes excess charge.
• Capacitors use conducting plates where charges live on surfaces, and the field between plates is used to store energy.
• Faraday cages protect electronics from external static fields and electrostatic discharge.

Fig 1.13 — Educational Diagram: Electric Dipole in Uniform Field (Torque τ = p×E & Potential Energy)

What is a continuous charge distribution?
When charge is spread over a line, surface or volume (not concentrated at points), we treat it as a continuous distribution described by a density: linear λ (C/m), surface σ (C/m²) or volume ρ (C/m³). To find the electric field E (or potential V) we split the distribution into infinitesimal elements dq, find the contribution dE from each element using Coulomb's law, and integrate over the object.

General method (step-by-step)

• Choose coordinates and an element of charge: dq = λ dx (line), dq = σ dA (surface), dq = ρ dV (volume).
• Write dE from Coulomb's law: dE = (1/4πε₀) (dq / r²) r̂. Express r (distance) and r̂ (direction cosines) in your variables.
• Resolve dE into components if symmetry causes cancellation; keep only the nonzero component(s).
• Integrate the component(s) over the entire distribution; apply limits and simplify. Use limits (→∞, R→∞) for special cases (infinite rod/plane).

Worked (canonical) examples — with key integrals and results

1. Field on the perpendicular bisector of a finite uniformly charged rod (rod of half-length L, linear density λ):

Geometry: rod along x from −L to +L, point P on y-axis at distance y from centre. Take dq = λ dx. Only the y-components add (x-components cancel).

Integral (vertical component): E_y = (1/4πε₀) ∫_−L^+L (λ y dx)/(x² + y²)^(3/2) = (1/4πε₀) · (2λL)/(y √(L² + y²)).

Limits: L → ∞ ⇒ infinite line: E = λ/(2πε₀ y) (field varies as 1/y).

2. Field at a point on the axis of a uniformly charged finite rod (rod on x from 0 to L, point P at x = x₀ > L):

dq = λ dx, distance r = x₀ − x. Integration gives E = (1/4πε₀) λ [1/(x₀ − L) − 1/(x₀)], directed away from the rod if λ>0.

3. Field on the axis of a uniformly charged ring (radius R, total charge Q):

Symmetry: horizontal components cancel; axial component adds. dq = (Q/2πR) R dθ. E(x) = (1/4πε₀) · Q x /(x² + R²)^(3/2), where x is distance from center along axis. Maximum of E occurs at x = R/√2. For x ≫ R, E ≈ (1/4πε₀) Q/x² (ring acts like point charge far away).

4. Field on the axis of a uniformly charged circular disk (radius R, surface density σ):

Consider the disk as concentric rings. Element ring of radius r, width dr has dq = σ (2πr dr). The axial field from the ring: dE = (1/4πε₀) (dq x)/(x² + r²)^(3/2). Integrate r from 0 to R to get

E(x) = (σ/(2ε₀)) [1 − x/√(x² + R²)], directed along axis. In the limit R → ∞ (infinite plane) this becomes E = σ/(2ε₀), independent of x.

5. Field of a uniformly charged thin spherical shell:

By symmetry and Gauss's law (or by integrating Coulomb contributions), for radius a and total charge Q: outside (r > a) E = (1/4πε₀) Q/r² (acts like point charge); inside (r < a) E = 0.

Important remarks

• Choose symmetry to simplify integrals: translational (infinite rod), cylindrical (ring/disk), spherical (sphere) symmetries reduce vector integrals to scalars.
• Always resolve vector dE into components when symmetry cancels some components.
• Keep track of limits when integrating (finite→infinite cases).

Fig 1.14 — Educational Diagram: Parabolic Motion of Charged Particle in Uniform Electric Field

What this topic covers
In electrostatics we use a small set of physical constants and SI units to express charge, force, field and potential. These constants appear in Coulomb's law, expressions for electric field and potential of point charges, and in relations that include the permittivity of the medium.

Important constants

• Elementary charge (e): magnitude of charge on a proton (and magnitude of electron charge). Exact SI value (2019 redefinition): e = 1.602176634 × 10^-19 C. Electron charge = −e.
• Permittivity of free space (ε₀): characterizes how electric field lines behave in vacuum. ε₀ ≈ 8.8541878128 × 10^-12 F m^-1 (farad per metre). It is also often written as units C² N^-1 m^-2.
• Coulomb constant (k): k = 1/(4π ε₀) ≈ 8.9875517923 × 10⁹ N m² C^-2 (commonly approximated as 9.0 × 10⁹). It appears in Coulomb's law as the proportionality constant.
• Relative permittivity (ε_r): dimensionless dielectric constant of a medium. Absolute permittivity ε = ε_r ε₀.

SI units used in electrostatics (quick list)

• Charge: coulomb (C). 1 C = 1 A · s (ampere-second).
• Force: newton (N) = kg · m · s^-2.
• Electric field: newton per coulomb (N/C) or volt per metre (V/m).
• Electric potential (voltage): volt (V) = J/C.
• Capacitance: farad (F) = C/V.
• Electric flux: N · m² / C.

How the constants appear in formulas (conceptual)
Coulomb's law: the force between two point charges q₁ and q₂ separated by r is F = k q₁ q₂ / r². The constant k (or equivalently ε₀) sets the strength of the electrostatic interaction in vacuum. For a medium with permittivity ε, replace ε₀ by ε (or equivalently divide k by ε_r).

Practical notes for students
When solving problems remember: (a) use sign for charge when calculating direction of field/force/potential; (b) use k ≈ 9 × 10⁹ N m² C^-2 for quick calculations or compute from ε₀ when accuracy is needed; (c) express charge in coulombs (convert μC, nC etc. to C: 1 μC = 10^-6 C, 1 nC = 10^-9 C).

Fig 1.15 — Educational Diagram: Dielectric Polarization & Net Reduced Field (E = E0 / K)

Overview: In electrostatics many problems become tractable by combining basic problem-solving techniques with symmetry arguments. Symmetry tells you the direction of the electric field and which coordinates it can depend on, so you can choose methods (Coulomb's law or Gauss's law) and coordinate systems that simplify calculations.

Key ideas and steps to solve problems:

1. Identify the charge distribution and any symmetry: spherical, cylindrical (axial), planar (translational), or discrete (dipole, ring).
2. Decide if direct superposition (Coulomb's law) or Gauss's law is appropriate. Use Gauss's law when symmetry makes the field constant on a chosen surface or normal to it.
3. Choose a coordinate system and a Gaussian surface that matches the symmetry (sphere for spherical symmetry, cylinder for cylindrical symmetry, plane/box for planar symmetry).
4. Use symmetry to determine field direction and dependence: e.g., spherical symmetry → E radial and depends only on r; infinite plane → E perpendicular and constant; infinite line → E radial in plane perpendicular to line and depends on r only.
5. Write down Coulomb's law or the integral form of Gauss's law. If using Gauss's law, compute flux = ∮ E·dA = Q_enclosed/ε0. Solve for E and check units and limiting behavior.
6. For configurations without sufficient symmetry use superposition: break distribution into simple parts, compute fields (analytically or by symmetry) and add vectorially. Use components and unit vectors as needed.
7. Check special cases and limits: far-field behavior, r→0 or r→∞, field continuity at boundaries, and physical constraints (e.g., field inside conductor = 0).

Symmetry types and consequences:

• Spherical symmetry: Field is radial, depends only on radial distance r. Gauss's law with spherical surface gives E(r) easily. Example: uniformly charged sphere (insulating) and conducting sphere.
• Cylindrical symmetry: Field is radial outward from axis, depends only on radial distance ρ (in cylindrical coordinates). Use a coaxial Gaussian cylinder to find E(ρ).
• Planar/translational symmetry: For an infinite plane of charge field is perpendicular to plane and independent of distance. Use a pillbox Gaussian surface.
• Reflection/rotational symmetry: Helps place zero components: e.g., for an infinite charged sheet with a hole or for rings/dipoles along symmetry axes.

Other useful techniques: superposition principle, decomposition into components, use of image charges for conductors with simple planar or spherical boundaries (method of images), dimensional analysis and limiting cases, and sketching field lines to visualize direction and relative strength.

When to use Gauss's law vs Coulomb's law:

• Use Gauss's law when symmetry forces E constant on portions of the Gaussian surface or E is everywhere normal (spheres, infinite cylinders, infinite planes).
• Use Coulomb's law (integration) or superposition when the distribution lacks the requisite symmetry (finite rods, rings, dipoles) or for on-axis fields of symmetric finite objects.

Common pitfalls: assuming symmetry that does not exist (e.g., finite sheet ≠ infinite sheet), wrong Gaussian surface, ignoring vector nature of fields, failing to include sign of enclosed charge, and not checking boundary/limit behavior.