Physics II

Electric charge is a fundamental property of matter, like mass. There are two types of charge, conventionally called positive and negative. Ordinary matter is electrically neutral because it contains equal numbers of protons (charge $+e$) and electrons (charge $-e$).

Charge is quantized: the elementary charge $e = 1.602\times10^{-19}\,\mathrm{C}$ is the fundamental quantum. Every observable charge in nature is an integer multiple of $e$. Quarks have fractional charges ($\pm e/3$, $\pm 2e/3$) but are never observed in isolation due to colour confinement.

Charge is conserved: in every physical process the total charge of an isolated system remains constant. This principle, together with conservation of energy and momentum, underlies all electromagnetic reactions.

Coulomb's law: $$F = \frac{1}{4\pi\varepsilon_0}\frac{|q_1 q_2|}{r^2}$$ where $k = 1/(4\pi\varepsilon_0) = 8.99\times10^9\,\mathrm{N\cdot m^2/C^2}$ is Coulomb's constant and $\varepsilon_0 = 8.85\times10^{-12}\,\mathrm{F/m}$ is the vacuum permittivity. The force is directly proportional to the product of the charges and inversely proportional to the square of the distance. Its direction lies along the line joining the charges: attractive for opposite signs, repulsive for like signs. In vector form: $$\vec{F}_{12} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}\hat{r}_{12}$$ where $\hat{r}_{12}$ is the unit vector pointing from $q_1$ to $q_2$.

Superposition principle: when more than two charges are present, the total force on a given charge is the vector sum of the forces exerted by every other charge: $$\vec{F}_i = \sum_{j\neq i} \frac{1}{4\pi\varepsilon_0}\frac{q_i q_j}{r_{ij}^2}\hat{r}_{ij}$$

Real-world examples:

• Static electricity: rubbing a plastic comb through hair transfers electrons, creating a charge imbalance — the comb becomes negatively charged and attracts the hair.
• Lightning: friction between ice crystals and hydrometeors inside storm clouds separates charges; the enormous potential difference generates a discharge (lightning bolt).
• Particle accelerators: use electric fields to accelerate beams of charged particles by exploiting Coulomb's force.

The electric field $\vec{E}$ is a vector quantity that describes the influence a charge (or charge distribution) exerts on the surrounding space. It is defined as the force per unit positive test charge: $$\vec{E} = \frac{\vec{F}}{q_0} = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat{r}$$ The electric field is therefore independent of the test charge and depends only on the source charge and the location.

Field lines are a powerful visual tool: they originate from positive charges and terminate on negative ones. Their density indicates the field strength. Field lines never cross (the field at a point is unique). The vector $\vec{E}$ is always tangent to the field lines at every point.

Key fields (obtained by integrating Coulomb's law or via Gauss):

• Infinite wire with linear density $\lambda$: $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$, directed radially. The field decays as $1/r$, slower than a point charge because the wire is extended.
• Infinite plane with surface density $\sigma$: $E = \dfrac{\sigma}{2\varepsilon_0}$, uniform and independent of distance. The field is the same everywhere — a consequence of planar symmetry.
• Parallel-plate capacitor: $E = \dfrac{\sigma}{\varepsilon_0}$. The superposition of the fields of the two plates (with opposite charge) doubles the field in the interior region and cancels it outside.
• Uniformly charged sphere (radius $R$, total charge $Q$): outside ($r>R$) $E = \dfrac{Q}{4\pi\varepsilon_0 r^2}$ (equivalent to a point charge); inside a conductor $E=0$; inside a uniformly charged dielectric sphere $E = \dfrac{\rho r}{3\varepsilon_0}$ (grows linearly with $r$).

Superposition principle for fields: the total field at a point is the vector sum of the fields generated by each charge in the distribution.

Applications:

• Capacitors — store energy in the electric field between the plates.
• Faraday cage — a hollow conductor shields the external electric field ($E=0$ inside).
• Electrostatic precipitators — remove suspended particles from exhaust gases by ionising them and attracting them with electric fields.

The electric potential $V$ is a scalar field that greatly simplifies the study of electrical phenomena compared with the vector field $\vec{E}$. The potential difference between two points is the work per unit charge required to move a test charge from one point to the other: $$V(P) = -\int_\infty^P \vec{E}\cdot d\vec{l}$$ with the convention $V(\infty)=0$. For a point charge $q$: $$V(r) = \frac{q}{4\pi\varepsilon_0 r}$$ The potential is positive for $q>0$, negative for $q<0$. For a charge distribution, the total potential is the (scalar) sum of the individual potentials — much simpler than the vector field.

Field-potential relation: $$\vec{E} = -\nabla V$$ In Cartesian coordinates: $E_x = -\partial V/\partial x$, $E_y = -\partial V/\partial y$, $E_z = -\partial V/\partial z$. The electric field points in the direction of steepest decrease of the potential — positive charges "fall" toward decreasing potentials.

Equipotential surfaces are loci of constant potential. They are always perpendicular to field lines. Moving along an equipotential surface requires no work.

Electric dipole: two charges $\pm q$ separated by a distance $d$ form a dipole with dipole moment $\vec{p} = q\vec{d}$ (directed from $-q$ to $+q$). The potential at a distance $r$ (with $r\gg d$) is: $$V_{dip} = \frac{p\cos\theta}{4\pi\varepsilon_0 r^2}$$ while the electric field decays as $1/r^3$, faster than for a point charge.

Applications:

• Polar molecules (water, HCl) — possess a permanent dipole moment and align with external electric fields.
• Electrocardiography — the heart generates an electrical signal that can be modelled as a time-varying dipole.
• Dielectrophoresis — manipulation of neutral particles using non-uniform electric fields.

Gauss's law establishes a fundamental relation between the flux of the electric field through a closed surface and the charge contained within it: $$\oint_S \vec{E}\cdot d\vec{A} = \frac{Q_{int}}{\varepsilon_0}$$ The electric flux $\Phi_E = \oint_S \vec{E}\cdot d\vec{A}$ measures how many field lines "pierce" the surface.

Physical intuition: if a closed surface contains net positive charge, the total flux is outward (field lines "exit" the surface). If it contains net negative charge, the flux is inward. If $Q_{int}=0$, the flux is zero — as many lines enter as leave.

Gauss's law is especially useful for computing the electric field in the presence of symmetries (spherical, cylindrical, planar): one chooses a Gaussian surface that exploits the symmetry, computes the flux, and solves for $E$.

Differential form: applying the divergence theorem yields: $$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}$$ which expresses locally the relation between field and charge density. Combining it with $\vec{E} = -\nabla V$ gives Poisson's equation: $$\nabla^2 V = -\frac{\rho}{\varepsilon_0}$$ In regions with no charge ($\rho=0$) this reduces to Laplace's equation $\nabla^2 V = 0$.

Applications:

• Faraday cage — inside a hollow conductor the field is zero, shielding external charges.
• Coaxial cables — the field between inner conductor and outer braid is easily computed using Gauss.
• Atomic force microscopy — measures electrostatic forces between probe and sample.

A conductor in electrostatic equilibrium exhibits the following fundamental properties:

• $\vec{E} = 0$ everywhere inside — if a field existed, free charges would move until they cancelled it.
• Any excess charge resides on the surface.
• The conductor is equipotential: $V = \text{constant}$ throughout the volume and on the surface.
• The field just outside the surface is perpendicular and equals $E = \sigma/\varepsilon_0$, where $\sigma$ is the local surface charge density.

Capacitance: the capacitance $C = Q/V$ measures how much charge a conductor can store at a given potential. It depends only on geometry.

For a parallel-plate capacitor (area $A$, separation $d$): $$C = \frac{\varepsilon_0 A}{d}$$ Capacitance increases with plate area and decreases with separation.

Electrostatic energy stored in a capacitor: $$U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$$ This energy is the work done to charge the capacitor by transferring charge from one plate to the other.

Electric field energy density: $$u = \frac{1}{2}\varepsilon_0 E^2$$ The energy is not "concentrated on the plates" but is distributed throughout all space where the electric field exists. This local view is fundamental in electrodynamics.

Applications:

• Camera flash — a capacitor charges slowly and discharges rapidly through the xenon lamp.
• DRAM — each memory cell is a tiny capacitor storing one bit.
• Defibrillators — discharge a large amount of energy in a short pulse to restart the heart.
• Filters and timers in electronic circuits.

A dielectric is an insulating material that, when placed in an external electric field, becomes polarised: the bound charges (electrons and nuclei) shift slightly, creating a polarisation field that opposes the external field. The net result is a reduction of the field inside the material.

Electric displacement field: to describe the field in the presence of dielectrics, the vector $\vec{D}$ (also called dielectric induction) is introduced: $$\vec{D} = \varepsilon_0\varepsilon_r\vec{E}$$ The $\vec{D}$ field accounts only for free charges (not polarisation charges): $$\nabla\cdot\vec{D} = \rho_{free}$$ where $\rho_{free}$ is the free charge density.

Relative permittivity $\varepsilon_r$: measures how much the material reduces the electric field compared with vacuum. $\varepsilon_r > 1$ for all materials:

• Air: $\varepsilon_r \approx 1.0006$
• Silica (SiO₂): $\varepsilon_r \approx 3.9$
• Water: $\varepsilon_r \approx 80$ (water is an excellent solvent because it strongly screens electric charges)
• Strontium titanate: $\varepsilon_r \approx 300$

Capacitor with a dielectric: inserting a dielectric between the plates increases the capacitance by a factor $\varepsilon_r$: $$C_{diel} = \varepsilon_r C_0$$ Since $C = Q/V$, at the same charge the voltage decreases (or at the same voltage more charge is stored).

Dielectric strength: the maximum electric field a material can withstand before becoming conductive (dielectric breakdown). Examples: air $\sim 3\,\mathrm{MV/m}$, mica $\sim 100\,\mathrm{MV/m}$, transformer oil $\sim 15\,\mathrm{MV/m}$.

Applications:

• Electrolytic capacitors — use a thin oxide layer as a dielectric to achieve high capacitance.
• Modern transistors — use high-k materials (hafnium, zirconium) as the gate dielectric to reduce leakage currents.
• Dielectric oil — in transformers as an insulator and coolant.
• Microwaves — the microwave oven heats water (a polar dipole) through molecular rotation induced by the oscillating electric field.