Electric fields
Electric charges create electric fields that exert forces on other charges. Field concepts explain capacitors, insulation, and the foundations of every electrical device.
Coulomb's law
Charles-Augustin de Coulomb measured in 1785 that the force between two point charges is proportional to the product of the charges and inversely proportional to the square of their distance: \(F = k\frac{q_1 q_2}{r^2}\), with \(k \approx 8.99 \times 10^{9}\) N m\(^2\)/C\(^2\). Like charges repel and unlike charges attract. Electric forces are enormous: they hold atoms and molecules together.
The electric field
The electric field E at a point is the force per unit charge a small test charge would feel there: \(\mathbf{E} = \frac{\mathbf{F}}{q}\), measured in newtons per coulomb or, equivalently, volts per metre. A point charge's field points radially outwards (for positive charge) and weakens with the square of distance. Field lines visualise E: they start on positive charges, end on negative ones, and never cross.
Potential and field
Electric potential, voltage, is potential energy per unit charge. The field points from high to low potential, and its strength equals how fast potential changes with distance. Between parallel plates separated by d with voltage V, the field is uniform with E = V/d: 100 V across 1 mm gives 100,000 V/m. Equipotential surfaces are perpendicular to field lines.
Gauss's law
Gauss's law, one of Maxwell's equations, states that the electric flux out of any closed surface equals the enclosed charge divided by the permittivity of free space: \(\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0}\), with \(\varepsilon_0 \approx 8.85 \times 10^{-12}\) F/m. For symmetric charge distributions, such as spheres, long wires, and planes, it gives fields quickly, and it explains why charge on a conductor resides on its surface.
Conductors and shielding
In electrostatic equilibrium, the field inside a conductor is zero, and any excess charge sits on its surface. A closed conducting shell, a Faraday cage, shields its interior from external electric fields, which is why cars protect occupants from lightning and why sensitive electronics and microwave ovens use metal enclosures and meshes.
Dielectrics and breakdown
Insulating materials, dielectrics, become polarised in a field, reducing the field inside them and increasing capacitance by their relative permittivity: about 1 for air, 3 to 4 for many plastics, and over 1,000 for some ceramics. Every insulator has a dielectric strength beyond which it breaks down and conducts: about 3 million V/m for dry air, which is why sparks jump and lightning strikes.
Electrostatic phenomena and uses
Static electricity builds when materials rub and exchange electrons; discharges can damage electronic chips, so electronics workers use anti-static wrist straps and bags. Electrostatic forces are used in laser printers and photocopiers, electrostatic precipitators that clean smokestack gases, spray painting, and touchscreens that sense a finger's effect on electric fields.
Worked example
Two charges of 1 \(\mu\)C each are 10 cm apart. \(F = 8.99 \times 10^{9} \times \frac{(10^{-6})^2}{0.1^2} = 8.99 \times 10^{9} \times 10^{-10} \approx 0.9\) N, about the weight of a 90-gram object, from tiny charges. A spark across a 1 cm gap in dry air requires roughly \(3 \times 10^{6} \times 0.01 = 30{,}000\) V, the order of voltage a static shock reaches.
Key takeaways
Coulomb's law (1785): F = k q1 q2 / r^2, with k about 9 x 10^9; like charges repel. The electric field is force per unit charge, in N/C or V/m, shown by field lines from positive to negative charges. Field points from high to low potential; between plates E = V/d. Gauss's law relates flux through a closed surface to enclosed charge over epsilon0, about 8.85 x 10^-12 F/m. Conductors have zero internal field, and Faraday cages shield. Dielectrics raise capacitance by their relative permittivity and break down above their dielectric strength, about 3 MV/m for air. Static discharges damage chips; electrostatics powers printers, precipitators, and touchscreens.
Flashcard
F = k q1 q2 / r^2
Check your understanding
Why are people inside a car generally safe if lightning strikes it?