Chapter 2: Coulomb’s Law Study Notes

Chapter 2: Coulomb’s Law

2.1 Electric Charge

• Types of Charge: Positive and negative.
  • Derived from Benjamin Franklin's experiments.
  • Rubbing glass rod with silk → positive charge.
  • Rubbing sealing wax with fur → negative charge.
• Law of Charges:
  • Like charges repel.
  • Opposite charges attract.
• Unit of Charge: Coulomb (C).
• Elementary Charge:
  • Charge of an electron or proton:
    $e = 1.602 imes 10^{-19} C$
  • Charge quantization: Charge of matter is in integral multiples of $e$.
  • An electron: $-e$
  • A proton: $+e$
  • Charge conservation: In a closed system, total charge is conserved (cannot be created or destroyed, only transferred).

2.2 Coulomb's Law

• Definition: The force exerted by a point charge on another point charge separated by distance r in vacuum is given by: extbf{F}_{12} = k rac{q_1 q_2}{r^2} extbf{{e}}
  • Where:
  • $k$ is the Coulomb constant.
  • extbf{{e}} is the unit vector from charge 1 to charge 2.
• Vector Nature of Force: Electric force has both magnitude and direction.
• Coulomb Constant in SI Units: k = rac{1}{4 ext{π} ext{ε}_0} = 8.9875 imes 10^9 N ext{m}^2/ ext{C}^2
  • Where $ext{ε}_0 = 8.85 imes 10^{-12} ext{C}^2/ ext{N m}^2$ is the permittivity of free space.
• Example: In a hydrogen atom, the electrostatic force between the electron and proton at a distance $r ext{ (approximately } 5.3 imes 10^{-11} ext{ m)}$ is given by F_e = k rac{e^2}{r^2} ext{, while gravitational force } F_g = rac{G m_p m_e}{r^2}
  • Where
  • $G = 6.674 imes 10^{-11} ext{ N m}^2/ ext{kg}^2$
  • Gravitational force can be neglected in comparison to electrostatic force.

2.3 Principle of Superposition

• Definition: For multiple point charges, the net force on charge 3 due to charges 1 and 2 is the vector sum of the individual forces.
  • $extbf{F}<em>3 = extbf{F}</em>{31} + extbf{F}_{32}$
• Example 2.1: Arrangement of three charges.
  • Charge values:
  • $q_1 = -6.0 imes 10^{-6} C$
  • $q_2 = -6.0 imes 10^{-6} C$
  • $q_3 = 3.0 imes 10^{-6} C$
  • Distance between charges:
  • $a = 2.0 imes 10^{-2} ext{ m}$
• Use superposition to calculate forces on the charge.

2.4 Electric Field

• Definition: Electric field at a point due to a charge is defined as the force experienced by a unit positive charge placed at that point.
  • Given by:
    extbf{E} = rac{ extbf{F}}{q_0}
  • Where $extbf{F}$ is the force and $q_0$ is the test charge.
• Electric Field from a Point Charge: At a distance r from a charge q, the electric field is given by: extbf{E} = k rac{q}{r^2} extbf{{e}}
  • Superposition principle applies for multiple point charges.

2.5 Electric Field Lines

• Properties:
  • Field lines are drawn to represent the strength and direction of electric fields.
  • Direction of field lines:
  • Outward for positive charges.
  • Inward for negative charges.

2.6 Force on a Charged Particle in an Electric Field

• Force on Charge q in Electric Field E:
  • Given by
    $extbf{F} = q extbf{E}$
• Acceleration:
  a = rac{ extbf{F}}{m} = rac{q extbf{E}}{m}

2.7 Electric Dipole

• Definition: Consists of two equal and opposite charges separated by distance d.
  • Dipole moment $extbf{p} = q extbf{d}$, where d is the vector pointing from negative to positive charge.
• Electric Field of a Dipole:
  • At a point in space, the electric field due to a dipole can be expressed in terms of dipole moment and distance.
• Torque on Dipole:
  • When placed in an electric field, dipoles experience a torque given by:
    $au = extbf{p} imes extbf{E}$ .

2.8 Dipole in Electric Field

• Torque on a dipole:
  • A dipole in a uniform electric field experiences torque but no net force.
  • Torque increases potential energy when aligned parallel to the field.

2.9 Charge Density

• Definitions:
  • *Volume charge density:
    ho = rac{dq}{dV}
  • *Surface charge density: ext{σ} = rac{dq}{dA}
  • *Line charge density: ext{λ} = rac{dq}{dl}

2.10 Electric Fields from Continuous Charge Distributions

• Electric Field from Continuous Charge:
• Use integration to find electric field contributions from infinitesimal charge elements.

2.11 Summary

• Coulomb's law states that the force between two point charges is inversely proportional to the square of the distance between them.
• The electric field is defined based on the force acting on a test charge placed in the field.

2.12 Problem-Solving Strategies

• Discussed how to calculate electric fields for discrete and continuous distributions via the superposition principle.

2.13 Solved Problems

• In-depth solutions provided for several problems including hydrogen atom, Millikan experiment, and interactions of charges.

2.14 Conceptual Questions

• Explore fundamental concepts of electric forces and fields through comparative questions and electric field line behavior.

2.15 Additional Problems

• Suggested additional problems for deeper understanding of electric charges, dipoles, and their interactions with electric fields.