Electric Charges and Coulomb's Law

• Electric charges are fundamental properties of matter that cause it to experience a force when placed in an electromagnetic field.
• Charges are of two types: positive and negative, carried by protons and electrons, respectively.
• Like charges repel each other, while unlike charges attract.
• Charge is quantized, meaning it exists in discrete packets.
• The smallest unit of free charge is the elementary charge, denoted by 'e', approximately equal to 1.602 × 10⁻¹⁹ coulombs.
• Charge is conserved, meaning the total charge in an isolated system remains constant.
• Charge can be transferred from one object to another through various methods like friction, conduction, and induction.

Coulomb's Law

• Coulomb's Law quantifies the electrostatic force between two point charges.
• The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
• Mathematically, the electrostatic force F between two charges q₁ and q₂ separated by a distance r is given by F = k * (|q₁q₂| / r²), where k is Coulomb's constant.
• Coulomb's constant k is approximately 8.9875 × 10⁹ N⋅m²/C².
• In vacuum, k is often expressed as 1 / (4πε₀), where ε₀ is the vacuum permittivity, approximately 8.854 × 10⁻¹² F/m.
• Coulomb's Law is a vector law; the force acts along the line joining the two charges.
• The force is repulsive if the charges have the same sign and attractive if the charges have opposite signs.

Superposition Principle

• The superposition principle states that the total force on a charge due to multiple other charges is the vector sum of the individual forces exerted by each charge.
• If there are n charges q₁, q₂, ..., qₙ exerting forces on a charge q, then the total force F on q is given by F = F₁ + F₂ + ... + Fₙ, where Fᵢ is the force exerted by qᵢ on q.

Electric Field

• An electric field is a region of space around an electric charge or a distribution of charges where another charge would experience a force.
• The electric field E at a point is defined as the force F experienced by a small positive test charge q₀ placed at that point, divided by the magnitude of the test charge: E = F / q₀.
• The electric field is a vector quantity; its direction is the direction of the force on a positive test charge.
• The SI unit of electric field is N/C (newtons per coulomb) or V/m (volts per meter).
• The electric field due to a point charge q at a distance r is given by E = k * (q / r²) in magnitude, and its direction is radially outward from the charge if q is positive and radially inward if q is negative.

Electric Field Lines

• Electric field lines are a visual representation of the electric field.
• They are imaginary lines that indicate the direction of the electric field at various points in space.
• Electric field lines originate from positive charges and terminate at negative charges.
• The density of electric field lines indicates the strength of the electric field; closer lines indicate a stronger field.
• Electric field lines never intersect each other.
• For a positive point charge, the electric field lines are radially outward, and for a negative point charge, they are radially inward.

Electric Dipole

• An electric dipole consists of two equal and opposite charges, +q and -q, separated by a small distance 2a.
• The dipole moment p is a vector quantity defined as the product of the magnitude of either charge and the separation between them: p = 2aq.
• The direction of the dipole moment is from the negative charge to the positive charge.
• The electric field due to a dipole varies with position.
• On the axial line (the line passing through the charges), the electric field at a distance r from the center of the dipole is approximately E = (1 / 4πε₀) * (2p / r³) for r >> a.
• On the equatorial line (the line perpendicular to the axial line and passing through the center), the electric field at a distance r from the center of the dipole is approximately E = (1 / 4πε₀) * (p / r³) for r >> a and is opposite in direction to the dipole moment.

Dipole in a Uniform Electric Field

• When an electric dipole is placed in a uniform electric field E, it experiences a torque τ but no net force.
• The torque is given by τ = p × E, where p is the dipole moment.
• The magnitude of the torque is τ = pEsinθ, where θ is the angle between p and E.
• The torque tends to align the dipole with the electric field.
• The potential energy U of a dipole in an electric field is given by U = -p ⋅ E = -pEcosθ.
• The potential energy is minimum (stable equilibrium) when θ = 0° (dipole aligned with the field) and maximum (unstable equilibrium) when θ = 180° (dipole anti-aligned with the field).

Continuous Charge Distribution

• Continuous charge distributions involve charge spread continuously over a region, rather than discrete point charges.
• Linear charge density (λ) is the charge per unit length: λ = dQ / dl.
• Surface charge density (σ) is the charge per unit area: σ = dQ / dA.
• Volume charge density (ρ) is the charge per unit volume: ρ = dQ / dV.
• To find the electric field due to a continuous charge distribution, one must integrate the contributions from infinitesimal charge elements over the entire distribution.
• The electric field dE due to an infinitesimal charge element dQ is given by dE = k * (dQ / r²), where r is the distance from the charge element to the point where the field is being calculated.
• The total electric field is then found by integrating dE over the entire charge distribution, taking into account the vector nature of the field.

Gauss's Law

• Gauss's Law relates the electric flux through a closed surface to the enclosed electric charge.
• Electric flux (Φ) through a surface is a measure of the number of electric field lines passing through that surface.
• For a uniform electric field E passing through a flat surface of area A, the electric flux is given by Φ = E ⋅ A = EAcosθ, where θ is the angle between the electric field and the normal to the surface.
• Gauss's Law states that the total electric flux through a closed surface (Gaussian surface) is equal to the total charge enclosed by the surface divided by the vacuum permittivity ε₀: ∮ E ⋅ dA = Q_enclosed / ε₀.
• Gauss's Law is a powerful tool for calculating electric fields in situations with high symmetry, such as spherical, cylindrical, or planar symmetry.
• By choosing an appropriate Gaussian surface, the electric field can often be determined without directly integrating over the charge distribution.

Applications of Gauss's Law

• Electric Field due to a uniformly charged spherical shell:
  • Outside the shell (r > R, where R is the radius of the shell), the electric field is the same as if all the charge were concentrated at the center: E = (1 / 4πε₀) * (Q / r²).
  • Inside the shell (r < R), the electric field is zero.
• Electric Field due to a uniformly charged solid sphere:
  • Outside the sphere (r > R), the electric field is the same as if all the charge were concentrated at the center: E = (1 / 4πε₀) * (Q / r²).
  • Inside the sphere (r < R), the electric field is proportional to the distance from the center: E = (1 / 4πε₀) * (Qr / R³).
• Electric Field due to an infinitely long charged wire:
  • At a distance r from the wire, the electric field is E = λ / (2πε₀r), where λ is the linear charge density.
  • The electric field is radial and perpendicular to the wire.
• Electric Field due to a uniformly charged infinite plane sheet:
  • The electric field is uniform and perpendicular to the sheet, with magnitude E = σ / (2ε₀), where σ is the surface charge density.
  • The field points away from the sheet if the charge is positive and towards the sheet if the charge is negative.
• Electric Field between two parallel charged sheets:
  • If the sheets have equal and opposite charge densities ±σ, the electric field between the sheets is E = σ / ε₀ and is zero outside the sheets.
  • If the sheets have the same charge density σ, the electric field is zero between the sheets and E = σ / ε₀ outside the sheets.