Electric Charge and Coulomb's Law

• Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field.
• There are two types of electric charges: positive and negative, conventionally named by Benjamin Franklin.
• Like charges repel each other, while unlike charges attract.
• The SI unit of electric charge is the coulomb (C).
• Charge is quantized, meaning it exists in discrete packets, which are integral multiples of the elementary charge (e), the charge of an electron or proton, approximately 1.602 × 10⁻¹⁹ C.
• Charge is conserved, meaning the total electric charge in an isolated system remains constant.
• Charging by friction involves the transfer of electrons between two initially neutral objects when they are rubbed together.
• Charging by induction involves the redistribution of charges in an object caused by the presence of a nearby charged object, without direct contact.

Coulomb's Law

• Coulomb's Law states that the electrostatic force between two point charges is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance between them.
• Mathematically, Coulomb's Law is expressed as F = k * |q1*q2| / r², where F is the electrostatic force, q1 and q2 are the magnitudes of the charges, r is the distance between the charges, and k is Coulomb's constant (approximately 8.9875 × 10⁹ N⋅m²/C²).
• The electrostatic force is a vector quantity, having both magnitude and direction; it acts along the line joining the two charges.
• In a medium other than vacuum, the electrostatic force is reduced by a factor of the relative permittivity (dielectric constant) of the medium.
• The principle of superposition states that the net force on a charge due to multiple charges is the vector sum of the individual forces exerted by each charge.

Electric Field

• An electric field is a region of space around a charged object in which another charged object would experience an electric force.
• The electric field intensity (E) at a point is defined as the force per unit positive charge that would be exerted on a test charge placed at that point: E = F/q, where F is the electric force and q is the test charge.
• The SI unit of electric field intensity is newtons per coulomb (N/C) or volts per meter (V/m).
• The electric field due to a point charge q at a distance r from the charge is given by E = k * |q| / r², directed radially away from the charge if it is positive, and radially towards the charge if it is negative.
• Electric field lines are a visual representation of the electric field, indicating the direction and strength of the field.
• Electric field lines originate from positive charges and terminate on negative charges; the density of lines indicates the strength of the field.
• The 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 distance between them: p = 2aq, directed from the negative charge to the positive charge.
• The electric field due to a dipole varies with position and is inversely proportional to the cube of the distance from the dipole at large distances.

Electric Flux

• Electric flux (Φ) is a measure of the number of electric field lines passing through a given area.
• For a uniform electric field E passing through a flat area A, the electric flux is given by Φ = E * A * cos(θ), where θ is the angle between the electric field and the normal to the area.
• Gauss's Law states that the total electric flux through a closed surface is proportional to the total charge enclosed within the surface.
• Mathematically, Gauss's Law is expressed as ∮ E ⋅ dA = Q_enclosed / ε₀, where ∮ E ⋅ dA is the surface integral of the electric field over the closed surface, Q_enclosed is the total charge enclosed by the surface, and ε₀ is the permittivity of free space (approximately 8.854 × 10⁻¹² C²/N⋅m²).
• Gauss's Law is particularly useful for calculating the electric field due to symmetric charge distributions, such as spherical, cylindrical, and planar charge distributions.
• Applications of Gauss's Law include determining the electric field due to a uniformly charged sphere, an infinitely long charged wire, and a large charged plane sheet.

Electric Potential

• Electric potential (V) at a point is the electric potential energy per unit charge at that point.
• The electric potential difference between two points is defined as the work done per unit charge to move a test charge from one point to the other.
• The SI unit of electric potential is the volt (V), defined as one joule per coulomb (1 V = 1 J/C).
• For a point charge q, the electric potential at a distance r from the charge is given by V = k * q / r.
• The electric field is related to the electric potential by E = -∇V, where ∇V is the gradient of the electric potential.
• Equipotential surfaces are surfaces on which the electric potential is constant.
• The electric field is always perpendicular to equipotential surfaces.
• Electric potential energy (U) of a system of charges is the work required to assemble the charges from infinity to their current positions.
• For two point charges q1 and q2 separated by a distance r, the electric potential energy is given by U = k * q1 * q2 / r.

Capacitance

• A capacitor is a device that stores electrical energy in an electric field.
• Capacitance (C) is defined as the ratio of the charge (Q) stored on a capacitor to the potential difference (V) between its plates: C = Q / V.
• The SI unit of capacitance is the farad (F), defined as one coulomb per volt (1 F = 1 C/V).
• For a parallel-plate capacitor with plate area A and separation d, the capacitance is given by C = ε₀ * A / d.
• If a dielectric material with dielectric constant κ is inserted between the plates, the capacitance increases to C = κε₀ * A / d.
• Capacitors can be connected in series or parallel.
• For capacitors in parallel, the equivalent capacitance is the sum of the individual capacitances: C_eq = C1 + C2 + C3 + ...
• For capacitors in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances: 1/C_eq = 1/C1 + 1/C2 + 1/C3 + ...
• The energy (U) stored in a capacitor is given by U = (1/2) * C * V² = (1/2) * Q * V = (1/2) * Q² / C.
• The energy density (u) in an electric field is given by u = (1/2) * ε₀ * E².

Dielectrics and Polarization

• A dielectric is a non-conducting material that can be polarized by an electric field.
• When a dielectric is placed in an electric field, its molecules align themselves with the field, reducing the overall electric field within the material.
• Polarization (P) is the dipole moment per unit volume in a dielectric material.
• The introduction of a dielectric between the plates of a capacitor increases the capacitance by a factor equal to the dielectric constant (κ) of the material.
• There are two main types of dielectric polarization: electronic polarization and ionic polarization.
• Electronic polarization occurs when the electron cloud around an atom is distorted by the electric field, creating an induced dipole moment.
• Ionic polarization occurs in ionic crystals when the positive and negative ions are displaced relative to each other by the electric field.