Electrostatics

1. Electric Charge

Electric charge is a fundamental property of matter responsible for electric forces and interactions. It is quantized, conserved, and invariant under changes in reference frame.

• Definition: Electric charge (q or Q) is measured in coulombs (C).

• Quantization: All observable charges are integer multiples of the elementary charge e:

• Carriers: Protons carry +e, electrons carry -e.

• Conservation: The total charge in an isolated system remains constant:

• Invariance: Charge does not depend on the observer's frame of reference.

Properties of Charge

• Like charges repel, unlike charges attract.

• Field lines: Originate from positive charges and terminate on negative charges.

• SI unit: Coulomb (C).

2. Methods of Charging

Materials can be charged by transferring electrons through various mechanisms.

2.1 Charging Conductors

• Friction (Triboelectric Effect): Rubbing two materials transfers electrons.

• Conduction (Contact): Touching a charged object to a conductor redistributes charge.

• Induction: Bringing a charged object near a conductor causes charge separation; grounding removes one type of charge.

2.2 Charging Insulators (Dielectrics)

• Insulators have bound electrons; charge remains localized.

• Polarization: External fields displace electron clouds, creating electric dipoles aligned with the field (dielectric polarization).

3. Electrostatic Force — Coulomb’s Law

Coulomb’s Law describes the force between two point charges.

• Vector Form:

• (permittivity of free space)

• points from to

Key Concepts

• Inverse-square law:

• Superposition Principle: The net force on a charge is the vector sum of forces from all other charges:

• Central force: Acts along the line joining the charges.

• Conservative force: Work done is path-independent; potential energy can be defined:

• Long-range force: Acts over infinite distances.

4. The Electric Field

The electric field describes the influence a charge exerts on the space around it, defined as force per unit positive test charge.

• Definition:

• Units:

• For a point charge at the origin:

Electric Field Lines

• Originate on positive charges, terminate on negative charges or at infinity.

• The tangent at any point gives the direction of .

• Density of lines ∝ .

• Field lines never cross.

• Perpendicular to conducting surfaces in electrostatic equilibrium.

Superposition of Electric Fields

• For N point charges:

5. Effects of the Electric Field on Point Charges

5.1 Force and Acceleration

• A charge in field experiences:

5.2 Motion in a Uniform Electric Field

• Between parallel plates separated by :

• Equations of motion for a particle entering with velocity perpendicular to :

Eliminating :

• Trajectory is a parabola (analogous to projectile motion).

5.3 Motion in a Non-Uniform Electric Field

• Equation of motion:

• Requires numerical integration in general.

• Dipole in a Uniform Field:

• Dipole in a Non-Uniform Field:

6. Electric Field of Continuous Charge Distributions

For extended objects, replace the sum over point charges with an integral over the charge distribution.

where and is the infinitesimal charge element.

6.1 Example: Ring of Charge

• Symmetry: Only the axial component survives at a point on the axis.

• Result:

• For : (point charge behavior).

• For : (center of ring).

• Maximum at .

6.2 Infinite Line Charge

Using Gauss’s Law:

6.3 Infinite Plane of Charge

Using a Gaussian pillbox:

6.4 Uniformly Charged Non-conducting Sphere

For a sphere of radius and total charge :

7. Summary Table

Example Application: Calculating the electric field at a point on the axis of a charged ring uses symmetry to simplify the integral, resulting in a field that points along the axis and depends on the distance from the center.

Additional info: These notes are designed for calculus-based physics students and assume familiarity with vector calculus, integrals, and basic mechanics. Worked examples and derivations should be added for practice and deeper understanding.