Electrostatics

Study of stationary electric charges and the forces and fields they produce.

Definition

Electrostatics is the branch of physics that deals with phenomena due to electric charges at rest. It forms the foundation for understanding electric circuits, electromagnetic waves, and atomic structure.

Key Concepts

  • Electric Charge — fundamental property (positive/negative), quantized in units of $e$
  • Conservation of Charge — charge cannot be created or destroyed
  • Coulomb's Law — force between point charges: $$F = k\frac{q_1 q_2}{r^2}$$
  • Electric Field — force per unit charge: $$\vec{E} = \frac{\vec{F}}{q}$$
  • Electric Field Lines — visualization, density represents field strength
  • Electric Flux — $\Phi_E = \int \vec{E} \cdot d\vec{A}$
  • Gauss's Law — relates flux to enclosed charge: $$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$$
  • Electric Potential — potential energy per unit charge
  • Equipotential Surfaces — surfaces of constant potential
  • Motion in Uniform Electric Field — charged particle trajectories in parallel-plate fields
    • Perpendicular entry → parabolic path ($a_y = qE/m$)
    • Parallel entry → linear acceleration ($a = qE/m$)
    • Dynamic equilibrium → $qE = mg$

Concept Map

flowchart TD
    A[Electrostatics] --> B[Charges at Rest]
    B --> C[Coulomb's Law<br/>F = kq₁q₂/r²]
    C --> D[Electric Field<br/>E = F/q = kQ/r²]
    D --> E[Electric Flux<br/>Φ = ∫E·dA]
    E --> F[Gauss's Law<br/>∮E·dA = Q_enc/ε₀]
    D --> G[Electric Potential<br/>V = kq/r]
    G --> H[Potential Energy<br/>U = kq₁q₂/r]
    G --> I[Equipotential Surfaces]
    H --> J[Motion in Uniform E-field]
    D --> J

Charge Properties

Quantization

Charge is quantized. Any electric charge $Q$ occurs as integer multiples of the elementary charge $e$:

$$Q = ne$$

where $n = 1, 2, 3, \dots$ and $e = 1.6 \times 10^{-19} \text{ C}$.

Scalar Nature

Total charge $Q$ is a scalar quantity.

SI Unit Definition

1 Coulomb (C) is defined as the total charge transferred by a current of one ampere in one second.

Conductors and Insulators

flowchart LR
    subgraph Conductor["Conductor"]
        direction TB
        C1[Free electrons<br/>move easily] --> C2[Charge resides<br/>on surface]
        C2 --> C3[E = 0 inside<br/>in equilibrium]
        C3 --> C4[All points at<br/>same potential]
    end

    subgraph Insulator["Insulator"]
        direction TB
        I1[Electrons tightly<br/>bound] --> I2[Charge remains<br/>localized]
        I2 --> I3[E can be<br/>non-zero inside]
        I3 --> I4[Potential varies<br/>within material]
    end

Coulomb's Law Details

Coulomb's Constant

$$k = 8.9875 \times 10^{9} \text{ N m}^2 \text{ C}^{-2} \approx 9.0 \times 10^{9} \text{ N m}^2 \text{ C}^{-2}$$

Also expressed as: $$k = \frac{1}{4\pi\varepsilon_0}$$

where $\varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{ m}^{-2}$ is the permittivity of free space.

Vector Nature

The electrostatic force is a vector quantity with unit Newton (N).

Newton's Third Law

For two point charges, the force on $q_1$ due to $q_2$ equals in magnitude the force on $q_2$ due to $q_1$:

$$F_{12} = F_{21} = k\frac{q_1 q_2}{r^2}$$

Sign Conventions

  • The sign of charge can be ignored when substituting into Coulomb's law to calculate magnitude: $$F = k\frac{|q_1||q_2|}{r^2}$$
  • The sign is important for determining the direction of the force (attractive for opposite signs, repulsive for same signs).

Point Charge Approximation

Charges can be treated as point-like when their physical size is negligible compared to the separation distance $r$ between them.

Graphical Relationships

Graph Shape Physical Meaning
$F$ vs $r$ Inverse-square curve Force falls off as $1/r^2$
$F$ vs $1/r^2$ Straight line through origin Gradient $= kq_1q_2$

Electric Field Strength

The electric field strength $E$ at a point is defined as the electric force per unit positive test charge:

$$E = \frac{F}{q_0}$$

  • It is a vector quantity.
  • Units: $\text{N C}^{-1}$ or $\text{V m}^{-1}$
  • Derived from Coulomb's law for a point charge $Q$ at distance $r$: $$E = \frac{kQ}{r^2}$$
  • The direction of $\vec{E}$ depends on the sign of the source charge: radially outward for positive, radially inward for negative.

Direction of Force vs. Field

Test Charge Direction of $\vec{F}$ relative to $\vec{E}$
Positive ($+q$) Same direction as $\vec{E}$
Negative ($-q$) Opposite direction to $\vec{E}$

Examples:

  • Electron: $F = Ee$, force opposite to field direction.
  • Proton: $F = Ee$, force in same direction as field.
  • Alpha particle ($^4_2\text{He}$): $F = 2Ee$, force in same direction as field.

Electric Field Lines

Michael Faraday introduced electric field lines as a visualization tool with these properties:

  1. The field vector $\vec{E}$ is tangent to the field line at every point.
  2. The magnitude of $E$ is proportional to the density of lines (number per unit area perpendicular to the lines). Closer lines = stronger field.
  3. Field lines start on positive charges and end on negative charges. The number of lines is proportional to the magnitude of the charge.
  4. Field lines never cross because the electric field has a unique value at each point.

Field Patterns

Configuration Field Pattern
Single positive charge Radially outward
Single negative charge Radially inward
Two equal opposite charges ($+q$, $-q$) Curved lines from $+q$ to $-q$
Two equal positive charges ($+q$, $+q$) Bulging outward; neutral point exists where $\vec{E} = 0$
Two opposite unequal charges ($+2q$, $-q$) Lines proportional to charge magnitude
Opposite charged parallel plates Uniform, parallel, perpendicular to plates (except near edges)

Neutral Point

A neutral point is a point (or region) in space where the resultant electric field is zero. For example, it lies along the perpendicular bisector between two equal like charges.

Key Formulas

Formula Description
$F = k\frac{q_1 q_2}{r^2}$ Coulomb's Law
$E = \frac{F}{q_0}$ Electric field strength (definition)
$E = k\frac{q}{r^2}$ Electric field of point charge
$E = \frac{\sigma}{\varepsilon_0}$ Infinite plane of charge
$E = \frac{\lambda}{2\pi\varepsilon_0 r}$ Infinite line of charge
$V = k\frac{q}{r}$ Electric potential
$U = k\frac{q_1 q_2}{r}$ Potential energy
$a = \frac{qE}{m}$ Acceleration in uniform field
$v_y = \frac{qEx}{mv_0}$ Vertical velocity (perpendicular entry)
$t = \frac{x}{v_0}$ Transit time between plates
$\theta = \tan^{-1}!\left(\frac{v_y}{v_x}\right)$ Deflection angle
$qE = mg$ Electrostatic-weight equilibrium

Problem-Solving Flowchart

flowchart TD
    A[Read Problem] --> B["Identify given quantities<br/>q, r, E, V, d, m, v₀"]
    B --> C{What is asked?}
    C -->|Force / Field| D[Use Coulomb's Law<br/>or E = kQ/r²]
    C -->|Potential / Energy| E[Use V = kq/r<br/>or U = kq₁q₂/r]
    C -->|Motion / Trajectory| F[Use F = qE = ma<br/>Kinematics]
    D --> G[Check units & signs]
    E --> G
    F --> G
    G --> H[Verify magnitude<br/>& direction]
    H --> I[Final Answer]

Related Concepts

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