13.1 Coulomb's Law, Electric Fields, Gauss's Law & Capacitors

Coulomb's Law, Electric Fields, Gauss's Law & Capacitors

In the Intermediate Physics Part II syllabus and the University of Engineering and Technology (UET) ECAT examination, electrostatics constitutes one of the most conceptually demanding and heavily weighted topics. Engineering entrance questions test inverse-square vector superposition, dielectric polarization effects, potential gradients, Gaussian flux evaluations across geometric symmetries, and multi-capacitor transient networks without calculators. Developing a rigorous conceptual and mathematical mastery of these principles is critical for scoring top marks.

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1. Coulomb's Law in Vacuum and Material Media

In 1785, Charles-Augustin de Coulomb formulated the fundamental quantitative law governing the electrostatic force between two stationary point charges in an inertial reference frame.

Mathematical Statement of Coulomb's Law

The magnitude of the electrostatic force $F$ between two point charges $q_1$ and $q_2$ separated by a distance $r$ in vacuum is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them:

$F = k \frac{|q_1 q_2|}{r^2}$

where $k$ is the electrostatic constant of proportionality. In the International System of Units (SI):

$k = \frac{1}{4\pi\varepsilon_0} \approx 8.98755 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2 \approx 9.0 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2$

The constant $\varepsilon_0$ denotes the permittivity of free space (vacuum):

$\varepsilon_0 = 8.854 \times 10^{-12}\text{ C}^2/(\text{N}\cdot\text{m}^2) = 8.854 \times 10^{-12}\text{ F/m}$

Vector Form and Newton's Third Law

Electrostatic force is intrinsically a vector quantity directed along the line joining the centers of the two charges. Let $\hat{\mathbf{r}}_{12}$ be a unit vector pointing from charge $q_1$ toward charge $q_2$. The force $\mathbf{F}_{21}$ exerted on charge $q_2$ by charge $q_1$ is expressed as:

$\mathbf{F}_{21} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2} \hat{\mathbf{r}}_{12}$

Similarly, the force $\mathbf{F}_{12}$ exerted on $q_1$ by $q_2$ is:

$\mathbf{F}_{12} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2} \hat{\mathbf{r}}_{21} = -\mathbf{F}_{21}$

This explicit anti-symmetry demonstrates that Coulomb forces constitute an action-reaction pair complying strictly with Newton's Third Law, irrespective of whether the charges are identical or radically unequal in magnitude ($|q_1| \neq |q_2|$).

Effect of an Insulating Dielectric Medium

When the space between two charges is filled with an isotropic, homogeneous insulating medium (dielectric) of relative permittivity $\varepsilon_r$ (also known as the dielectric constant $\kappa$ or $K$), atomic dipoles within the insulator align against the external field. This dielectric polarization produces bound surface charges that establish an internal opposing electric field, attenuating the net force:

$F_{\text{med}} = \frac{1}{4\pi\varepsilon}\frac{q_1 q_2}{r^2} = \frac{1}{4\pi\varepsilon_0\varepsilon_r}\frac{q_1 q_2}{r^2} = \frac{F_{\text{vac}}}{\varepsilon_r}$

Because all material insulators possess relative permittivity greater than unity ($\varepsilon_r > 1$, e.g., $\varepsilon_r \approx 1.0006$ for dry air, $2.1$ for Teflon, $5$–$10$ for glass/mica, and $78.5$ for pure water at $25^\circ\text{C}$):

$\varepsilon_r = \frac{F_{\text{vac}}}{F_{\text{med}}} > 1 \implies F_{\text{med}} < F_{\text{vac}}$

ECAT Key Rule: The introduction of an insulating medium between two stationary charges always strictly reduces the mutual electrostatic force by a factor of $\varepsilon_r$. The relative permittivity $\varepsilon_r$ is a dimensionless scalar quantity with no units.

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2. Electric Field Intensity, Field Lines & Dipoles

Electric Field Intensity ($\mathbf{E}$)

An electric charge modifies the physical state of the space surrounding it, establishing an electric field. The electric field intensity (or field strength) $\mathbf{E}$ at any spatial coordinate is defined as the electrostatic force experienced per unit positive test charge $q_0$ placed at that point, in the limit where the test charge vanishes to prevent disturbance of source charges:

$\mathbf{E} = \lim_{q_0 \to 0} \frac{\mathbf{F}}{q_0}$

For an isolated point charge $q$ located at the origin, the electric field intensity at distance $r$ is:

$\mathbf{E} = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2} \hat{\mathbf{r}}$

Electric field intensity is a vector quantity with SI units of newtons per coulomb (N/C), which is dimensionally and physically identical to volts per meter (V/m) ($1\text{ N/C} = 1\text{ V/m}$).

Electric Lines of Force (Field Lines)

Conceived by Michael Faraday, electric lines of force provide an intuitive geometric representation of vector fields:

1. Field lines originate at positive charges (or at infinity) and terminate on negative charges (or at infinity).
2. The tangent to an electric line of force at any point gives the direction of the vector $\mathbf{E}$ at that point.
3. The areal density of lines (number of lines passing perpendicularly through a unit cross-sectional area) is directly proportional to the magnitude of the electric field intensity $|\mathbf{E}|$.
4. No two electric field lines can ever intersect. If two lines crossed, a test charge at the intersection would experience two distinct directions of force simultaneously, which is physically impossible.
5. Electric lines of force are always perpendicular to the surface of a conductor in electrostatic equilibrium.

Electric Dipole Dynamics

An electric dipole consists of a pair of equal and opposite point charges $+q$ and $-q$ separated by a small vector distance $\mathbf{d}$ directed from the negative charge to the positive charge. The electric dipole moment $\mathbf{p}$ is defined as:

$\mathbf{p} = q\mathbf{d}$

When a dipole is immersed in a uniform external electric field $\mathbf{E}$:

• The net translational force is zero: $\mathbf{F}_{\text{net}} = (+q\mathbf{E}) + (-q\mathbf{E}) = \mathbf{0}$.
• Because the equal and opposite forces act along different lines of action, they constitute a couple that exerts a restoring mechanical torque:

$\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E} \implies |\boldsymbol{\tau}| = p E \sin\theta$

• The electrostatic potential energy $U$ stored in the dipole system as a function of orientation angle $\theta$ is:

$U = -\mathbf{p} \cdot \mathbf{E} = -p E \cos\theta$

• Stable Equilibrium ($\theta = 0^\circ$): Dipole moment aligns parallel to $\mathbf{E}$; torque $\tau = 0$ and potential energy is at its minimum ($U = -p E$).
• Unstable Equilibrium ($\theta = 180^\circ$): Dipole moment aligns antiparallel to $\mathbf{E}$; torque $\tau = 0$ and potential energy is at its maximum ($U = +p E$).
• Maximum Torque ($\theta = 90^\circ$): $\tau_{\max} = p E$, potential energy $U = 0$.

ECAT Trap: In a non-uniform electric field, the two charges experience unequal forces ($|\mathbf{F}_+| \neq |\mathbf{F}_-|$). Consequently, in a non-uniform field, an electric dipole experiences both a net torque and a net translational force.

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3. Electric Potential, Potential Gradient & the Electron-Volt

Electric Potential and Potential Difference

Because the electrostatic Coulomb force is strictly conservative, the work done moving a test charge between two points is path-independent. The electric potential difference $\Delta V = V_B - V_A$ between point $A$ and point $B$ is the external work $W_{A\to B}$ required to move a unit positive charge from $A$ to $B$ at constant kinetic energy:

$\Delta V = V_B - V_A = \frac{W_{A\to B}}{q_0} = -\int_A^B \mathbf{E} \cdot d\mathbf{r}$

Taking reference potential at infinity ($V_\infty = 0$), the absolute electric potential $V$ at distance $r$ from an isolated point charge $q$ is a scalar quantity:

$V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$

For a system of $n$ discrete point charges, the total potential is the algebraic scalar sum $V = \sum_{i=1}^n \frac{1}{4\pi\varepsilon_0}\frac{q_i}{r_i}$, eliminating the complex vector decompositions required for electric field intensity.

Potential Gradient and Field Relation

Differentiating the work integral in one dimension yields the fundamental differential relationship between electric field and potential:

$E = -\frac{dV}{dr}$

In three-dimensional Cartesian space, the electric field vector equals the negative spatial gradient of scalar potential:

$\mathbf{E} = -\boldsymbol{\nabla}V = -\left(\frac{\partial V}{\partial x}\hat{\mathbf{i}} + \frac{\partial V}{\partial y}\hat{\mathbf{j}} + \frac{\partial V}{\partial z}\hat{\mathbf{k}}\right)$

Physical Sign Significance: The negative sign dictates that the electric field vector $\mathbf{E}$ always points in the direction of maximum rate of decrease of electric potential. Positive charges naturally accelerate toward regions of lower potential, whereas negative charges accelerate toward regions of higher potential.

The Electron-Volt (eV)

In atomic and nuclear physics, the standard SI unit of work (joule) is inconveniently large. The electron-volt (eV) is defined as the amount of kinetic energy gained or lost by an electron when it accelerates through an electrostatic potential difference of exactly one volt:

$1\text{ eV} = q_e \times \Delta V = (1.602 \times 10^{-19}\text{ C}) \times (1\text{ V}) = 1.602 \times 10^{-19}\text{ J}$

Common multiples: $1\text{ keV} = 10^3\text{ eV}$, $1\text{ MeV} = 10^6\text{ eV}$, $1\text{ GeV} = 10^9\text{ eV}$.

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4. Gauss's Law and Electrostatic Applications

Electric Flux ($\Phi_e$)

Electric flux quantifies the total number of electric field lines penetrating a given surface area. For a planar surface of vector area $\mathbf{A} = A\hat{\mathbf{n}}$ (where $\hat{\mathbf{n}}$ is the outward unit normal vector) immersed in a uniform field $\mathbf{E}$:

$\Phi_e = \mathbf{E} \cdot \mathbf{A} = E A \cos\theta$

where $\theta$ is the angle between the electric field vector $\mathbf{E}$ and the outward normal vector $\hat{\mathbf{n}}$:

• When $\mathbf{E}$ is parallel to the outward surface normal (perpendicular to surface, $\theta = 0^\circ$): $\Phi_e = E A$ (maximum flux).
• When $\mathbf{E}$ is perpendicular to outward normal (parallel/glancing along surface, $\theta = 90^\circ$): $\Phi_e = 0$.

For arbitrary curved surfaces in non-uniform fields, total flux is the surface integral:

$\Phi_e = \iint_S \mathbf{E} \cdot d\mathbf{A}$

SI units of electric flux: $\text{N}\cdot\text{m}^2/\text{C}$ or $\text{V}\cdot\text{m}$.

Statement of Gauss's Law

Karl Friedrich Gauss established that the total outward electric flux through any arbitrary closed hypothetical surface (termed a Gaussian surface) is equal to the total net enclosed electric charge $Q_{\text{enc}}$ divided by the permittivity of free space $\varepsilon_0$:

$\Phi_{\text{total}} = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$

Gauss's law is valid for any closed geometry and depends exclusively on the enclosed charge, remaining completely independent of the size, shape, or geometric location of charges within the surface.

Three Standard Applications of Gauss's Law

1. Electric Field Inside a Hollow Charged Conducting Sphere

Consider a hollow metallic spherical shell of radius $R$ bearing a uniform surface charge $Q$. To determine the electric field at an interior point at radial distance $r < R$, construct a concentric spherical Gaussian surface of radius $r$.

Because all static charge resides exclusively on the outer metallic surface of the conductor, the Gaussian surface contains zero net charge:

$Q_{\text{enc}} = 0 \implies \oint \mathbf{E} \cdot d\mathbf{A} = E(4\pi r^2) = \frac{0}{\varepsilon_0} = 0 \implies E = 0$

• The electric field inside any hollow charged conductor is identically zero ($E_{\text{inside}} = 0$).
• Because $E = -\frac{dV}{dr} = 0$, the electric potential throughout the interior is strictly constant and equal to its value on the surface:

$V_{\text{inside}} = V_{\text{surface}} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{R}$

This principle provides electrostatic shielding (Faraday Cage), ensuring sensitive electronic apparatus or vehicle occupants remain immune to external high-voltage electrical fields and lightning strikes.

2. Electric Field Near an Infinite Sheet of Charge

Consider an infinite non-conducting planar sheet having a uniform positive surface charge density $\sigma = \frac{Q}{A}\text{ (C/m}^2\text{)}$. By planar symmetry, $\mathbf{E}$ points perpendicularly away from both sides of the sheet.

Construct a cylindrical Gaussian pillbox of cross-sectional end-cap area $A$ piercing perpendicularly through the sheet. The electric lines of force run parallel to the curved cylindrical walls, yielding zero curved-wall flux. Flux emerges exclusively through the two flat end caps:

$\Phi_{\text{total}} = \Phi_{\text{left}} + \Phi_{\text{right}} + \Phi_{\text{curved}} = E A + E A + 0 = 2EA$

The enclosed charge is $Q_{\text{enc}} = \sigma A$. Applying Gauss's Law:

$2EA = \frac{\sigma A}{\varepsilon_0} \implies E = \frac{\sigma}{2\varepsilon_0}$

Remarkably, the electric field near an infinite sheet of charge is strictly uniform and independent of distance from the sheet.

3. Electric Field Between Oppositely Charged Parallel Conducting Plates

Consider two large, flat parallel conducting plates bearing equal and opposite uniform surface charge densities $+\sigma$ and $-\sigma$. By vector superposition of the fields from each plate:

• In the region outside the plates, the individual fields point in opposite directions and cancel identically: $E_{\text{outside}} = \frac{\sigma}{2\varepsilon_0} - \frac{\sigma}{2\varepsilon_0} = 0$.
• In the region between the plates, both fields point from the positive plate toward the negative plate, reinforcing each other:

$E_{\text{between}} = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{\sigma}{\varepsilon_0}$

Alternatively, constructing a rectangular Gaussian box with one face inside the metallic conductor ($E = 0$) and one face in the gap yields $\Phi = E A = \frac{\sigma A}{\varepsilon_0} \implies E = \frac{\sigma}{\varepsilon_0}$.

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5. Capacitance, Dielectrics & Energy Density

Definition of Capacitance

A capacitor is an electrical component engineered to store electric charge and electrostatic potential energy. The charge $Q$ accumulated on either plate is directly proportional to the applied potential difference $V$ across the plates:

$Q = C V \implies C = \frac{Q}{V}$

The constant of proportionality $C$ is the capacitance, measured in SI units of farads (F) ($1\text{ F} = 1\text{ coulomb/volt}$). Practical laboratory values are measured in microfarads ($\mu\text{F} = 10^{-6}\text{ F}$), nanofarads ($\text{nF} = 10^{-9}\text{ F}$), or picofarads ($\text{pF} = 10^{-12}\text{ F}$). Capacitance depends exclusively on conductor geometry and dielectric properties, never on $Q$ or $V$.

Parallel-Plate Capacitor Formula

For two parallel conducting plates each of area $A$ separated by a small gap distance $d$ in vacuum ($d \ll \sqrt{A}$ to minimize edge fringing effects):

$E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}, \quad V = E d = \frac{Q d}{\varepsilon_0 A}$

Substituting into $C = \frac{Q}{V}$ gives the vacuum capacitance $C_{\text{vac}}$:

$C_{\text{vac}} = \frac{\varepsilon_0 A}{d}$

When the entire gap between plates is filled with a dielectric medium of relative permittivity $\varepsilon_r$:

$C_{\text{med}} = \frac{\varepsilon_0\varepsilon_r A}{d} = \varepsilon_r C_{\text{vac}}$

Dielectric Polarization Mechanics

When a dielectric slab is inserted into the electric field $\mathbf{E}_0$ of a charged capacitor, external field stresses displace atomic nucleus charges relative to electron clouds, creating billions of microscopic dipole moments $\mathbf{p}$. The vector sum per unit volume is the polarization vector $\mathbf{P}$. This creates induced bound surface charges ($+\sigma_p$ and $-\sigma_p$) on dielectric faces adjoining the plates, establishing an opposing internal electric field $\mathbf{E}'$. The net resultant field is reduced:

$E_{\text{net}} = E_0 - E' = \frac{E_0}{\varepsilon_r}$

Battery Connected vs. Battery Disconnected (Critical ECAT Comparison)

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6. Capacitor Combinations, Stored Energy & RC Transient Circuits

Combinations of Capacitors

1. Series Combination

Capacitors connected end-to-end along a single path. By electrostatic induction, each capacitor carries identical charge $Q$:

$Q_1 = Q_2 = \dots = Q_n = Q$

The total applied terminal voltage is divided across the series capacitors: $V = V_1 + V_2 + \dots + V_n$.

$\frac{Q}{C_{\text{eq}}} = \frac{Q}{C_1} + \frac{Q}{C_2} + \dots + \frac{Q}{C_n} \implies \frac{1}{C_{\text{eq}}} = \sum_{i=1}^n \frac{1}{C_i}$

For two series capacitors: $C_{\text{eq}} = \frac{C_1 C_2}{C_1 + C_2}$. The equivalent capacitance is always smaller than the smallest individual capacitance in the series chain.

2. Parallel Combination

Capacitors connected across common voltage nodes. All capacitors share identical terminal potential difference $V$:

$V_1 = V_2 = \dots = V_n = V$

The total stored charge is the sum of individual charges: $Q = Q_1 + Q_2 + \dots + Q_n$.

$C_{\text{eq}} V = C_1 V + C_2 V + \dots + C_n V \implies C_{\text{eq}} = \sum_{i=1}^n C_i$

The equivalent capacitance is always larger than the largest individual capacitance in the parallel group.

Electrostatic Potential Energy Stored in a Capacitor

Charging a capacitor involves transferring incremental charge $dq$ against an opposing instantaneous potential difference $v = \frac{q}{C}$. The total mechanical work done by the charging source is:

$W = \int_0^Q v \, dq = \int_0^Q \frac{q}{C} \, dq = \frac{1}{C} \left[\frac{q^2}{2}\right]_0^Q = \frac{1}{2}\frac{Q^2}{C}$

Using $Q = CV$, the stored electrostatic potential energy $U$ can be written in three equivalent forms:

$U = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2} C V^2 = \frac{1}{2} Q V$

Volumetric Electrostatic Energy Density ($u$)

Electrostatic energy is not stored mechanically on conductor surfaces; it resides intrinsically within the electric field occupying the dielectric volume between plates ($V_{\text{vol}} = A d$):

$u = \frac{U}{\text{Volume}} = \frac{\frac{1}{2} C V^2}{A d} = \frac{\frac{1}{2} \left(\frac{\varepsilon_0\varepsilon_r A}{d}\right) (E d)^2}{A d}$

$u = \frac{1}{2}\varepsilon_0\varepsilon_r E^2$

This expression demonstrates that electrostatic energy density is directly proportional to the square of the electric field intensity ($u \propto E^2$), a fundamental result holding true for all electrostatic and electromagnetic wave fields.

Transient RC Charging and Discharging Kinetics

In a series circuit consisting of a DC voltage source $\mathcal{E}$, a resistor $R$, and an uncharged capacitor $C$:

1. Charging Phase

When the switch closes at $t = 0$, current rushes in. Applying Kirchhoff's loop rule: $\mathcal{E} - iR - \frac{q}{C} = 0$. Solving the first-order differential equation yields instantaneous charge:

$q(t) = Q_0\left(1 - e^{-t / RC}\right)$

where $Q_0 = C\mathcal{E}$ is maximum equilibrium charge. The product $\tau = RC$ is defined as the capacitive time constant (measured in seconds: $[\Omega][\text{F}] = [\text{s}]$):

• At $t = \tau = RC$:

$q(\tau) = Q_0(1 - e^{-1}) = Q_0\left(1 - \frac{1}{2.718}\right) \approx 0.632 Q_0 = 63.2\% Q_0$

• The charging current decays exponentially: $i(t) = I_0 e^{-t/RC}$, where $I_0 = \frac{\mathcal{E}}{R}$. At $t = \tau$, current drops to $36.8\% I_0$.

2. Discharging Phase

Disconnecting the battery and shorting through resistor $R$ produces exponential charge drainage:

$q(t) = Q_0 e^{-t / RC}$

• At $t = \tau = RC$, the remaining charge drops to $q(\tau) = Q_0 e^{-1} \approx 0.368 Q_0 = 36.8\% Q_0$.

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7. Electrostatics and Capacitance Summary Table