Electric Charge and Coulomb's Law

• Electric charge is a fundamental property of matter that causes it to experience a force when near other electrically charged matter
• Electric charge exists in discrete units, as integer multiples of the elementary charge (e ≈ 1.602 × 10⁻¹⁹ Coulombs)
• Charges are of two types: positive and negative; like charges repel and unlike charges attract

Coulomb's Law

• Coulomb's Law quantifies the electrostatic force (F) between two point charges
• The electrostatic force (F) is directly proportional to the product of the magnitudes of the charges (|q1| and |q2|)
• The electrostatic force (F) is inversely proportional to the square of the distance (r) between them
• The electrostatic force (F) acts along the line joining the two charges
• Mathematically, Coulomb's Law is expressed as: F = k * |q1 * q2| / r², where k is Coulomb's constant, k ≈ 8.9875 × 10⁹ N⋅m²/C²
• The force is repulsive if the charges have the same sign and attractive if the charges have opposite signs
• In vector form, the electrostatic force on charge q1 due to charge q2 is F12 = k * q1 * q2 / r² * r̂12, where r̂12 is the unit vector pointing from q2 to q1
• The superposition principle applies to electrostatic forces: the total force on a charge due to multiple other charges is the vector sum of the individual forces

Electric Field Lines

• Electric field lines are a visual representation of the electric field in space
• They originate from positive charges and terminate on negative charges
• The density of field lines indicates the strength of the electric field: more lines per unit area indicate a stronger field
• Electric field lines never cross each other
• The direction of the electric field at any point is tangent to the field line at that point
• For a positive point charge, the electric field lines radiate outwards
• For a negative point charge, the electric field lines converge inwards
• The electric field lines are perpendicular to the surface of a charged conductor in electrostatic equilibrium
• The electric field is stronger where the field lines are closer together

Electric Field

• The electric field (E) is defined as the force per unit positive charge experienced by a test charge placed in the field
• Mathematically, E = F/q, where F is the electrostatic force on the test charge q
• The electric field points in the direction of the force on a positive test charge
• The electric field due to a point charge q at a distance r is E = k * q / r² * r̂, where r̂ is the unit vector pointing from the charge to the point where the field is being calculated
• The superposition principle applies to electric fields: the total electric field at a point due to multiple charges is the vector sum of the individual electric fields

Gauss's Law

• Gauss's Law relates the electric flux through a closed surface to the enclosed electric charge
• Electric flux (ΦE) is a measure of the number of electric field lines passing through a surface
• For a uniform electric field E passing through a flat surface of area A, the electric flux is ΦE = E ⋅ A = E A cos θ, 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 proportional to the enclosed electric charge: ΦE = ∮ E ⋅ dA = Qenc / ε₀, where Qenc is the net charge enclosed by the surface and ε₀ is the permittivity of free space (ε₀ ≈ 8.854 × 10⁻¹² C²/N⋅m²)
• Gauss's Law can be used to calculate the electric field for charge distributions with sufficient symmetry
• Common Gaussian surfaces include spheres, cylinders, and boxes

Applications of Gauss's Law

• Spherical Symmetry: For a spherically symmetric charge distribution, a spherical Gaussian surface is used to find the electric field at a distance r from the center
• Cylindrical Symmetry: For an infinitely long charged cylinder, a cylindrical Gaussian surface is used to find the electric field at a radial distance r from the axis
• Planar Symmetry: For an infinite charged plane, a cylindrical Gaussian surface is used to find the electric field near the plane
• Conductors: Inside a conductor in electrostatic equilibrium, the electric field is zero. All excess charge resides on the surface of the conductor
• The electric field at the surface of a charged conductor is perpendicular to the surface and has a magnitude of E = σ / ε₀, where σ is the surface charge density

Electric Dipole

• An electric dipole consists of two equal and opposite charges (+q and -q) separated by a distance d
• The dipole moment (p) is a vector pointing from the negative charge to the positive charge, with magnitude p = qd
• The electric field due to a dipole at a point far from the dipole (r >> d) depends on the dipole moment and the distance
• Torque on a dipole in an electric field: τ = p × E, where τ is the torque, p is the dipole moment, and E is the electric field
• Potential energy of a dipole in an electric field: U = -p ⋅ E = -pE cos θ, where U is the potential energy and θ is the angle between p and E

Conductors and Insulators

• Conductors are materials that allow electric charges to move freely through them (e.g., metals)
• In conductors, electrons are not bound to specific atoms and can move easily in response to an electric field
• Insulators (dielectrics) are materials that do not allow electric charges to move freely (e.g., glass, rubber)
• In insulators, electrons are tightly bound to atoms and cannot move easily
• When a conductor is placed in an electric field, charges redistribute themselves on the surface such that the electric field inside the conductor is zero in electrostatic equilibrium
• When an insulator is placed in an electric field, the molecules become polarized, creating an induced electric field that opposes the external field