Electric Field and Gauss's Law

Electrostatics investigates electric charges at rest and the static electric fields they establish in vacuum and material media.

1. The Fundamental Nature of Electric Charge

Electric charge is an intrinsic fundamental property of subatomic matter. Matter exhibits two complementary types of charge:

• Positive Charge ($+q$): Borne by protons ($+e$).
• Negative Charge ($-q$): Borne by electrons ($-e$).

Like charges repel one another with mutual electrostatic forces; unlike charges attract.

2. Quantization of Electric Charge

Robert A. Millikan's oil-drop experiment (1909) experimentally confirmed that electric charge is not continuous, but exists exclusively in discrete, integer multiples of the elementary charge $e$: $$q = \pm n e, \quad n = 0, 1, 2, 3, \dots$$ where the elementary quantum of charge is defined by the 2019 SI standard: $$e = 1.602176634 \times 10^{-19} \text{ Coulombs (exact)}$$ (Note: Although quarks carry fractional charges $\pm \frac{1}{3}e$ and $\pm \frac{2}{3}e$, quark confinement strictly forbids isolated fractional charges under ordinary conditions).

3. The Law of Conservation of Electric Charge

In any closed, isolated physical system, the algebraic sum of electric charges remains strictly constant over time: $$\sum q_i = \text{constant}, \quad \frac{d Q_{\text{total}}}{dt} = 0$$ In relativistic and high-energy particle physics, while particles can be created or annihilated (e.g., pair production $\gamma \to e^- + e^+$ or electron-positron annihilation $e^- + e^+ \to 2\gamma$), net electric charge is conserved in every known physical interaction.

4. Continuous Charge Distributions

On macroscopic scales where individual charges cannot be resolved, charge is modeled via continuous spatial density distributions:

• Linear Charge Density ($\lambda$): $\lambda = \frac{dq}{dl}$ (Units: C/m)
• Surface Charge Density ($\sigma$): $\sigma = \frac{dq}{dA}$ (Units: C/m²)
• Volume Charge Density ($\rho$): $\rho = \frac{dq}{dV}$ (Units: C/m³)

Charles-Augustin de Coulomb (1785) established the fundamental quantitative law of electrostatic force using a precision torsion balance.

1. Coulomb's Law in Vector Form

The electrostatic force $\vec{F}_{12}$ exerted by a point charge $q_1$ located at position $\vec{r}_1$ on a second point charge $q_2$ located at $\vec{r}_2$ in vacuum is: $$\vec{F}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{|\vec{r}_2 - \vec{r}_1|^2} \hat{r}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{|\vec{r}_2 - \vec{r}_1|^3} (\vec{r}_2 - \vec{r}_1)$$ where:

• $\epsilon_0$: Permittivity of Free Space (Vacuum Permittivity): $$\epsilon_0 = 8.8541878128 \times 10^{-12} \text{ F/m (or C}^2/(\text{N}\cdot\text{m}^2))$$
• Coulomb's constant: $$k_e = \frac{1}{4\pi\epsilon_0} \approx 8.98755 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$$
• By Newton's third law of action and reaction: $\vec{F}_{21} = -\vec{F}_{12}$.

2. Coulomb's Force in a Dielectric Medium

When point charges are embedded in a linear, isotropic, homogeneous dielectric medium of relative permittivity $\epsilon_r$ (dielectric constant $\kappa$): $$\vec{F} = \frac{1}{4\pi\epsilon} \frac{q_1 q_2}{r^2} \hat{r} = \frac{1}{4\pi\epsilon_0 \epsilon_r} \frac{q_1 q_2}{r^2} \hat{r} = \frac{\vec{F}_{\text{vacuum}}}{\epsilon_r}$$ Because $\epsilon_r > 1$ for all physical media (e.g., $\epsilon_r \approx 80$ for pure water at 20°C), polarization of the surrounding dielectric shields the charges, reducing the electrostatic force significantly.

3. The Superposition Principle

The net electrostatic force exerted on a test charge $q_0$ by an assembly of $N$ discrete point charges $q_1, q_2, \dots, q_N$ is the vector sum of the individual Coulomb forces: $$\vec{F}_{\text{net}} = \sum_{i=1}^N \vec{F}_i = \frac{q_0}{4\pi\epsilon_0} \sum_{i=1}^N \frac{q_i}{|\vec{r}_0 - \vec{r}_i|^3} (\vec{r}_0 - \vec{r}_i)$$ For a continuous volume charge distribution $\rho(\vec{r}')$: $$\vec{F}_{\text{net}} = \frac{q_0}{4\pi\epsilon_0} \int_V \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|^3} (\vec{r} - \vec{r}') dV'$$

The concept of the electric field, introduced by Michael Faraday, replaces direct action-at-a-distance with a local physical intermediary.

1. Definition of the Electric Field Vector

The electric field $\vec{E}(\vec{r})$ at a point in space is defined as the electrostatic force experienced per unit positive test charge placed at that location, in the limit where the test charge $q_0$ approaches zero to prevent perturbing the source charge distribution: $$\vec{E}(\vec{r}) = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0}$$ SI Unit: $\text{N/C}$ (equivalent to $\text{Volts/meter}$, $\text{V/m}$).

2. Field of a Point Charge and Continuous Distribution

For an isolated point charge $q$ at the origin: $$\vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}$$ For an arbitrary continuous charge distribution occupying volume $V'$: $$\vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{V'} \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|^3} (\vec{r} - \vec{r}') dV'$$

3. Motion of a Point Charge in an Electric Field

A particle of mass $m$ and charge $q$ placed in an electric field $\vec{E}$ experiences acceleration: $$\vec{a} = \frac{\vec{F}}{m} = \frac{q \vec{E}}{m}$$ In a uniform electric field $\vec{E} = E_0 \hat{j}$:

• A charged particle launched perpendicular to $\vec{E}$ executes a parabolic trajectory, completely analogous to projectile motion under uniform gravity (the operational principle of cathode ray oscilloscopes and ink-jet printers).

4. Electric Field Lines (Lines of Force)

Electric field lines are imaginary curves whose tangent at any point indicates the direction of the local electric field vector $\vec{E}$:

1. Field lines originate on positive charges and terminate on negative charges (or extend to infinity).
2. The local spatial density of lines (number of lines per unit area normal to $\vec{E}$) is directly proportional to field magnitude $|\vec{E}|$.
3. Field lines never intersect in free space, because the electric field vector is uniquely defined at every point.

An electric dipole consists of two equal and opposite point charges $+q$ and $-q$ separated by a fixed distance $2a$.

1. The Electric Dipole Moment Vector

The electric dipole moment $\vec{p}$ is defined as: $$\vec{p} = q \vec{d}$$ where $\vec{d}$ is the displacement vector directed from the negative charge $-q$ toward the positive charge $+q$. SI Unit: $\text{Coulomb}\cdot\text{meter}$ (C·m). (In molecular physics, the Debye unit is commonly used: $1 \text{ D} = 3.33564 \times 10^{-30} \text{ C}\cdot\text{m}$).

2. Torque on a Dipole in a Uniform Electric Field

When placed in a uniform external field $\vec{E}$, the net translational force on the dipole vanishes: $$\vec{F}_{\text{net}} = (+q)\vec{E} + (-q)\vec{E} = 0$$ However, the forces act along different lines of action, producing a net mechanical restoring torque: $$\vec{\tau} = \vec{r}_+ \times (q\vec{E}) + \vec{r}_- \times (-q\vec{E}) = (\vec{r}_+ - \vec{r}_-) \times q\vec{E} = \vec{d} \times q\vec{E}$$ $$\vec{\tau} = \vec{p} \times \vec{E}$$ Magnitude: $\tau = p E \sin\theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$. The torque acts to align the dipole moment parallel to the external field ($\theta = 0$).

3. Potential Energy of an Electric Dipole

The external work required to rotate the dipole from reference angle $\theta_0 = 90^{\circ}$ to angle $\theta$ is: $$U(\theta) = \int_{90^{\circ}}^\theta \tau_{\text{ext}} d\theta' = \int_{90^{\circ}}^\theta (p E \sin\theta') d\theta' = -p E \cos\theta$$ $$U = -\vec{p} \cdot \vec{E}$$

• Stable Equilibrium ($\theta = 0^{\circ}$): Dipole aligned with $\vec{E}$; minimum potential energy $U_{\min} = -p E$.
• Unstable Equilibrium ($\theta = 180^{\circ}$): Dipole antiparallel to $\vec{E}$; maximum potential energy $U_{\max} = +p E$.

4. Dipole in a Non-Uniform Electric Field

In an inhomogeneous field where $\vec{E}$ varies spatially, the forces on $+q$ and $-q$ do not cancel. The net translational force is: $$\vec{F} = (\vec{p} \cdot \nabla) \vec{E} = \nabla (\vec{p} \cdot \vec{E})$$ A neutral dipole is always pulled toward regions of stronger electric field strength.

Carl Friedrich Gauss (1835) formulated Gauss's law, which relates the total electric flux passing through a closed geometric surface to the enclosed net electric charge.

1. Definition of Electric Flux

The electric flux $d\Phi_E$ through an infinitesimal oriented surface element $d\vec{A} = \hat{n} dA$ is: $$d\Phi_E = \vec{E} \cdot d\vec{A} = E \cos\theta \, dA$$ where $\hat{n}$ is the outward unit normal vector. For an arbitrary closed Gaussian surface $S$: $$\Phi_E = \oint_S \vec{E} \cdot d\vec{A}$$ SI Unit: $\text{N}\cdot\text{m}^2/\text{C} = \text{V}\cdot\text{m}$.

2. Gauss's Law in Integral Form

For an isolated point charge $q$ enclosed by a concentric sphere of radius $r$: $$\oint_S \vec{E} \cdot d\vec{A} = \oint_S \left( \frac{q}{4\pi\epsilon_0 r^2} \hat{r} \\right) \cdot (\hat{r} dA) = \frac{q}{4\pi\epsilon_0 r^2} (4\pi r^2) = \frac{q}{\epsilon_0}$$ By superposition, for any arbitrary closed surface enclosing net charge $Q_{\text{encl}}$: $$\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$$ This is **Gauss's Law** (the first of Maxwell's four foundational equations of electromagnetism).

3. Key Properties of Gauss's Law

• Gauss's law holds for *any* closed surface of arbitrary shape (called a Gaussian surface).
• Charges located *outside* the closed surface contribute zero net flux through the surface (any flux entering must leave).
• Gauss's law is a direct mathematical consequence of the inverse-square nature of Coulomb's force ($F \propto 1/r^2$). If the force followed $1/r^{2+\delta}$, Gauss's law would fail.

4. Differential Form of Gauss's Law

Applying Gauss's Divergence Theorem: $$\oint_S \vec{E} \cdot d\vec{A} = \int_V (\nabla \cdot \vec{E}) dV = \frac{1}{\epsilon_0} \int_V \rho \, dV$$ Since this holds for any arbitrary volume $V$: $$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$ This is the differential form of Gauss's law, stating that positive charge density acts as a local source (divergence) of electric field lines, and negative charge acts as a sink.

Gauss's law enables direct calculation of electric field configurations when the charge distribution possesses high geometric symmetry.

1. Spherically Symmetric Charge Distribution

Consider a solid insulating sphere of radius $R$ with uniform volume charge density $\rho$ (total charge $Q = \frac{4}{3}\pi R^3 \rho$). By spherical symmetry, $\vec{E} = E(r) \hat{r}$. Construct a concentric spherical Gaussian surface of radius $r$: $$\oint_S \vec{E} \cdot d\vec{A} = E(r) \oint_S dA = E(r) (4\pi r^2)$$

• Outside the Sphere ($r \ge R$): $$Q_{\text{encl}} = Q \implies E(r) (4\pi r^2) = \frac{Q}{\epsilon_0} \implies E(r) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$$ The external field is identical to that of a point charge $Q$ at the center.
• Inside the Sphere ($r < R$): $$Q_{\text{encl}} = \rho \left( \frac{4}{3}\pi r^3 \\right) = Q \left( \frac{r^3}{R^3} \\right)$$ $$E(r) (4\pi r^2) = \frac{Q r^3}{\epsilon_0 R^3} \implies E(r) = \frac{Q}{4\pi\epsilon_0 R^3} r = \frac{\rho}{3\epsilon_0} r$$ Inside the sphere, the electric field increases linearly from zero at the center to maximum at the surface!

2. Infinitely Long Line of Charge (Cylindrical Symmetry)

Consider an infinite line carrying uniform linear charge density $\lambda$ (C/m). Construct a coaxial cylindrical Gaussian surface of radius $r$ and length $L$. Flux through the two flat end caps is zero because $\vec{E} \perp \hat{n}_{\text{caps}}$. Flux through the curved cylindrical jacket of area $2\pi r L$: $$\oint_S \vec{E} \cdot d\vec{A} = E(r) (2\pi r L) = \frac{Q_{\text{encl}}}{\epsilon_0} = \frac{\lambda L}{\epsilon_0}$$ $$E(r) = \frac{\lambda}{2\pi\epsilon_0 r}$$ The field decays inversely with radial distance ($E \propto 1/r$).

3. Infinite Plane Sheet of Charge (Planar Symmetry)

Consider an infinite non-conducting flat plane with uniform surface charge density $\sigma$ (C/m²). Construct a cylindrical Gaussian pillbox of cross-sectional area $A$ piercing the sheet perpendicularly. Flux through the cylindrical mantle is zero ($\vec{E} \parallel \text{mantle}$). Flux passes equally out of both end faces: $$\oint_S \vec{E} \cdot d\vec{A} = E A + E A = 2 E A = \frac{Q_{\text{encl}}}{\epsilon_0} = \frac{\sigma A}{\epsilon_0}$$ $$E = \frac{\sigma}{2\epsilon_0}$$ Remarkably, the field is completely **uniform and independent of distance** from the sheet! For a conducting surface where all charge resides on the outer boundary and $\vec{E}_{\text{inside}} = 0$: $$E_{\text{conductor}} = \frac{\sigma}{\epsilon_0}$$