Coulomb's law and Electric Fields

PHS 122: Electricity and Magnetism - Study Summary

Coulomb's Law and Electric Fields (Chapter 24)

Coulomb's Law

Describes the force between two point charges, q₁ and q₂, separated by a distance r.

• Nature of Force:
  • Same signs (positive-positive or negative-negative) → repulsion.
  • Opposite signs (positive-negative) → attraction.
• Formulas:
  • In vacuum: F_e = k₀ |q₁q₂| / r² (Traditional and common, sometimes without absolute values, which can lead to negative force for attraction).
  • Generalized for sign: F = k q₁q₂ / r² (Force is positive for repulsion, negative for attraction).
  • Alternative with absolute values to avoid sign confusion: F = k |q₁||q₂| / r² (Direction determined by inspection).
• Constants and Units:
  • Force (F): Newtons (N).
  • Distance (r): meters (m).
  • Charge (q): Coulombs (C), the SI unit.
  • Coulomb constant (k₀):
    • In vacuum: k₀ = 8.988 × 10⁹ N·m²/C² (often approximated as 9.0 × 10⁹ N·m²/C²).
    • Alternatively, k₀ = 1 / (4πε₀).
  • Permittivity of free space (ε₀): ε₀ = 8.85 × 10⁻¹² C²/N·m².
  • In a material medium:
    • The force is reduced due to induced charges.
    • Dielectric constant (K): A dimensionless factor specific to the material. For vacuum, K=1; for air, K≈1.0006.
    • Permittivity of the material (ε): ε = Kε₀.
    • Coulomb's Law in a material medium: F_e = 1 / (4πε) * q₁q₂ / r² = 1 / (4πKε₀) * q₁q₂ / r².
• Applicability: Applies to point charges, charged conducting spheres, and spherical shells (provided separation is much larger than radii).

Charge Properties

• Charge Is Quantized:
  • Smallest measurable charge: e = 1.60218 × 10⁻¹⁹ C (the quantum of charge).
  • All free charges are integer multiples of e.
  • Electron charge: -e; Proton charge: +e.
• Conservation of Charge: The algebraic sum of charges in the universe is constant. Charges are created/destroyed in pairs (+e and -e).

Electric Field (E)

• Test-Charge Concept: A very small, tiny charge (negligible effect on environment) used to measure electric systems.
• Definition: An electric field exists at a point if a test charge placed there experiences an electrical force.
• Direction: Same as the force experienced by a positive test charge.
• Electric Field Lines:
  • Sketch electric fields; lines indicate field direction.
  • Density of lines indicates field strength (closer lines = stronger field).
  • Originate from positive charges (repel positive test charge).
  • Terminate on negative charges (attract positive test charge).
• Strength of the Electric Field (E): Force per unit positive test charge.
  • E = F_E / q' (where q' is the test charge).
  • Units: N/C or V/m.
  • Vector Quantity: Electric field (E) is a vector.
• Force in an Electric Field: If a charge q is in an electric field E, it experiences a force F_E = qE.
  • If q is negative, F_E is opposite to E.
• Electric Field Due to a Point Charge (q):
  • In a material medium: E = 1 / (4πε) * q / r².
  • For positive q, E is directed radially outward.
  • For negative q, E is directed radially inward.
• Superposition Principle:
  • The net force on a charge is the vector sum of individual Coulomb forces.
  • The net electric field at a point is the vector sum of individual electric fields due to other charges.

Solved Problems (Chapter 24 examples)

Illustrate applications of Coulomb's Law for calculating forces between charges, electric fields due to point charges, and forces in atomic models (e.g., Bohr model of hydrogen).

Electric Potential; Capacitance (Chapter 25)

Electric Potential and Potential Energy

• Potential Difference (V or ΔV):
  • Definition: Work done against electrical forces in carrying a unit positive test charge from point A to point B (V_B - V_A).
  • Units: Volts (V), where 1 V = 1 J/C.
  • Work Done (W): In transporting a charge q from A to B: W = q(V_B - V_A) = qV.
• Absolute Potential (V):
  • Definition: Work done against electric forces in carrying a unit positive test charge from infinity (where potential is zero) to a given point.
  • Due to a point charge q₀ (in vacuum): V = k₀ q₀ / r.
  • Due to multiple point charges: Scalar sum V = k₀ Σ (qᵢ / rᵢ) (where rᵢ is distance from charge qᵢ).
  • Due to a uniformly charged sphere: Same as a point charge at its center for points outside or on the surface.
• Electrical Potential Energy (PE_E):
  • Definition: Work done to carry a charge q from infinity to a point with absolute potential V: PE_E = qV.
  • Change in PEE when moved through potential difference V: ΔPE_E = qV.
• Relation between V and E (for uniform electric field):
  • If E is uniform in x-direction with magnitude E_x, then the potential difference V across a distance x is V = E_xx.
  • For parallel plates separated by distance d, V = Ed.
• Electron Volt (eV):
  • Definition: Work done to carry a charge +e through a potential rise of 1 Volt.
  • Conversion: 1 eV = (1.602 × 10⁻¹⁹ C)(1 V) = 1.602 × 10⁻¹⁹ J.

Capacitance

• Capacitor: A device that stores charge, typically two conductors separated by an insulator (dielectric).
• Capacitance (C):
  • Definition: Ratio of magnitude of charge (Q) on either conductor to the magnitude of potential difference (V) between conductors: C = Q / V.
  • Units: Farads (F). Common practical units are microfarads (µF = 10⁻⁶ F) and nanofarads (nF = 10⁻⁹ F).
• Parallel-Plate Capacitor:
  • Formula: C = Kε₀A / d (where A is plate area, d is separation, K is dielectric constant, ε₀ is permittivity of free space).
  • For vacuum, K=1. A dielectric increases capacitance by a factor of K.
• Equivalent Capacitance (C_eq): Simplification of multiple capacitors in a circuit.
  • Capacitors in Parallel:
    • Connected to the same two nodes (same potential difference across each).
    • Formula: C_eq = C₁ + C₂ + C₃ + ... (capacitances add).
  • Capacitors in Series:
    • Connected end-to-end, sharing the same charge.
    • Formula: 1 / C_eq = 1 / C₁ + 1 / C₂ + 1 / C₃ + ... (reciprocal capacitances add).
    • For two capacitors in series: C_eq = (C₁C₂) / (C₁ + C₂).
  • Nodes: Points where three or more terminals attach. Important for identifying series/parallel connections.
  • Shorted Capacitors: If a wire is placed across a capacitor, there is no voltage across it, and it can be removed from analysis (though the wire remains).
• Energy Stored in a Capacitor (PE_E):
  • Formulas: PE_E = ½qV = ½CV² = ½Q²/C.

Solved Problems (Chapter 25 examples)

Cover calculations involving potential difference, work done, electric field in parallel plates, absolute potential due to point charges, and energy transformations (e.g., electron speed due to potential difference).

Supplementary Problems (Chapters 24 & 26)

A range of problems covering:

• Coulomb's Law applications (force calculation, finding separation, charge changes).
• Electric field and force calculations involving multiple charges.
• Charge quantization and related calculations (number of electrons, mass of electrons).
• Concepts of current, resistance, internal resistance, and terminal voltage (from Chapter 26, not fully detailed in the provided text, but problems are listed).