Electric Charges, Fields, and Gauss’s Law

Charge comes in positive and negative amounts, and the elementary charge e (the size of an electron’s or proton’s charge) is the smallest piece you’ll work with. Coulomb’s law, $F = \frac{1}{4\pi\varepsilon_0}\frac{\lvert q_1 q_2\rvert}{r^2}$, gives the force between two point charges, where $\varepsilon_0$ is the permittivity of free space: the force acts along the line between them, repels like charges, attracts opposite ones, and is usually far stronger than the gravity between the same particles. In a conductor charges move easily; in an insulator they mostly stay put.

Charge is never created or destroyed: an object’s net charge changes only when charge, usually electrons, moves onto or off it. You can charge things by friction, by contact, or by induction, where a nearby charge shoves charges around inside an object (polarization) and a ground connection lets charge flow in or out.

The electric field is the force per unit charge a small test charge would feel, $\vec{E} = \frac{\vec{F}}{q}$, pointing away from positive charges and toward negative ones. The net field at a point is the vector sum of the fields from every charge, and field-line diagrams show its direction and, by how crowded the lines are, its strength. Inside a conductor in electrostatic equilibrium the field is zero and the extra charge sits on the surface, while an insulator can hold charge all through it.

For charge spread along a line, ring or arc, you split it into tiny pieces $dq$, write each piece’s field $dE = \frac{1}{4\pi\varepsilon_0}\frac{dq}{r^2}$, use symmetry to cancel components, and integrate. On the exam this covers an infinite line or cylinder, a ring (on its axis), a semicircular arc (at its center), and a finite line charge (on its line or on its perpendicular bisector).

Electric flux measures how much electric field passes through a surface. For a uniform field through a flat area, $\Phi_E = \vec{E}\cdot\vec{A} = EA\cos\theta$, and in general $\Phi_E = \int \vec{E}\cdot d\vec{A}$. The area vector points straight out of the surface (outward for a closed one), so field leaving a closed surface gives positive flux and field entering gives negative flux.

Gauss’s law says the total flux through any closed surface equals the charge inside divided by $\varepsilon_0$: $\oint \vec{E}\cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}$. For spherical, cylindrical or planar symmetry you pick a Gaussian surface where the field is either constant and straight through the surface or runs along it, so the integral becomes simple algebra; if the charge density varies, you integrate it to find the enclosed charge. Gauss’s law is the first of Maxwell’s four equations of electromagnetism.