Properties of Electric Charge

• Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field

  • Charges are of two kinds: positive and negative.
  • Like charges repel and unlike charges attract.
  • The SI unit of electric charge is the coulomb (C).

• Quantization of electric charge:

  • Electric charge is quantized, meaning it exists in discrete packets.
  • The smallest unit of charge is the elementary charge, e, which is the magnitude of the charge of an electron or proton, approximately equal to 1.602 × 10⁻¹⁹ C
  • Any charge, q, can be expressed as an integer multiple of e: q = ne, where n is an integer.

• Conservation of electric charge:

  • The total electric charge in an isolated system remains constant.
  • Charge can neither be created nor destroyed, but it can be transferred from one body to another.

• Additivity of electric charges:

  • The total charge of a system is the algebraic sum of all individual charges located at different points in the system.
  • If a system contains charges q₁, q₂, q₃, ..., qn, the total charge Q of the system is Q = q₁ + q₂ + q₃ + ... + qn.

Coulomb's Law

• Coulomb's law describes 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 force F between two charges q₁ and q₂ separated by a distance r is given by: F = k * (|q₁q₂| / r²) where k is the electrostatic constant.
  • The electrostatic constant k is given by k = 1 / (4πε₀), where ε₀ is the permittivity of free space, approximately equal to 8.854 × 10⁻¹² C²/Nm².
  • The value of k is approximately 8.9875 × 10⁹ Nm²/C².
  • In a medium other than vacuum, the force is reduced by a factor of the relative permittivity (dielectric constant) εr of the medium: F_medium = F / εr.

• Coulomb's law in vector form:

  • The force F₁₂ exerted on charge q₁ by charge q₂ is given by F₁₂ = k * (q₁q₂ / r²₁₂) * r̂₁₂, where r̂₁₂ is the unit vector pointing from q₂ to q₁.

• Superposition principle:

  • The net 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 q₀, the net force F₀ on q₀ is F₀ = F₀₁ + F₀₂ + ... + F₀n, where F₀i is the force exerted on q₀ by qi.

Electric Field

• An electric field is a region around an electric charge in which another charge experiences 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 test charge: E = F / q₀.

  • The SI unit of electric field is newtons per coulomb (N/C) or volts per meter (V/m).

• Electric field due to a point charge:

  • The electric field E at a distance r from a point charge q is given by E = k * (|q| / r²) * r̂, where r̂ is the unit vector pointing from the charge to the point.
  • In vector form: E = k * (q / r²) * r̂.

• Superposition principle for electric fields:

  • The net electric field at a point due to multiple charges is the vector sum of the electric fields due to each individual charge.
  • If there are n charges, the net electric field E at a point is E = E₁ + E₂ + ... + En, where Ei is the electric field due to charge qi.

• Electric field lines:

  • Electric field lines are a visual representation of the electric field.
  • They start on positive charges and end on negative charges.
  • The direction of the electric field at any point is tangent to the field line at that point.
  • The density of field lines indicates the strength of the electric field; closer lines indicate a stronger field.
  • Electric field lines do not form closed loops.
  • Electric field lines never intersect.

• 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 p = 2aq, directed from the negative charge to the positive charge.

• Electric field due to a dipole:

  • At a point on the axial line (end-on position) at a distance r from the center of the dipole: E = (2kp) / r³ (for r >> a).
  • At a point on the equatorial line (broadside-on position) at a distance r from the center of the dipole: E = (kp) / r³ (for r >> a).

• Dipole in a uniform electric field:

  • When a dipole is placed in a uniform electric field E, it experiences a torque τ given by τ = p × E.
  • The magnitude of the torque is τ = pEsinθ, where θ is the angle between the dipole moment p and the electric field E.
  • The potential energy U of a dipole in an electric field is given by U = -p ⋅ E = -pEcosθ.

Continuous Charge Distribution

• Linear charge density:

  • For a charge distributed along a line, the linear charge density λ is defined as the charge per unit length: λ = dq / dl, where dq is the charge on a small length element dl.

• Surface charge density:

  • For a charge distributed on a surface, the surface charge density σ is defined as the charge per unit area: σ = dq / dA, where dq is the charge on a small area element dA.

• Volume charge density:

  • For a charge distributed throughout a volume, the volume charge density ρ is defined as the charge per unit volume: ρ = dq / dV, where dq is the charge in a small volume element dV.

• Electric field due to continuous charge distributions:

  • The electric field due to a continuous charge distribution is found by integrating the contributions from all the infinitesimal charge elements.
  • For a linear charge distribution: E = ∫ k (λ dl / r²) r̂.
  • For a surface charge distribution: E = ∫ k (σ dA / r²) r̂.
  • For a volume charge distribution: E = ∫ k (ρ dV / r²) r̂.

Gauss's Law

• Gauss's law relates the electric flux through a closed surface to the charge enclosed by that surface.

• Electric flux ΦE is defined as the surface integral of the electric field over a closed surface: ΦE = ∮ E ⋅ dA, where dA is a vector normal to the surface element.

• Gauss's law states that the total electric flux through a closed surface is equal to the total charge Q enclosed by the surface divided by the permittivity of free space ε₀: ΦE = ∮ E ⋅ dA = Q_enclosed / ε₀.

• Applications of Gauss's law:

  • Electric field due to an infinitely long straight wire with uniform linear charge density λ: E = (λ / (2πε₀r)) r̂, where r is the distance from the wire.
  • Electric field due to an infinite plane sheet of charge with uniform surface charge density σ: E = (σ / (2ε₀)) n̂, where n̂ is a unit vector normal to the sheet.
  • Electric field inside a uniformly charged spherical shell is zero.
  • Electric field outside a uniformly charged spherical shell is E = (Q / (4πε₀r²)) r̂, where Q is the total charge on the shell and r is the distance from the center of the shell.
  • Electric field inside a uniformly charged solid sphere with total charge Q and radius R at a distance r from the center (for r ≤ R) is E = (Qr / (4πε₀R³)) r̂.
  • Electric field outside a uniformly charged solid sphere (for r ≥ R) is E = (Q / (4πε₀r²)) r̂.

Electric charge is a fundamental property of matter. Charges are of two kinds: positive and negative. Electric charge is quantized, meaning it exists in discrete packets.