Electrostatics: Charge, Coulomb's Law, and Gauss's Law

Irreversible Processes and Entropy (S)

An irreversible process is one that cannot be reversed by means of small changes in the environment. The direction in which an irreversible process proceeds is set by the change in entropy ($S$) of the system undergoing the process. Entropy $S$ is a state property of the system.

Fundamentals of Electric Charge

The strength of a particle's electrical interaction with objects around it depends on its electric charge, which can be either positive or negative. Charges with the same sign repel each other, and charges with opposite signs attract each other.

An object with equal amounts of the two kinds of charge is electrically neutral, whereas one with an imbalance is electrically charged.

Conductors and Insulators

• Conductors are materials in which a significant number of charged particles (such as electrons in metals) are free to move.
• The charged particles in nonconductors, or insulators, are not free to move.

SI Units: The Coulomb (C) and Ampere (A)

The SI unit of charge is the coulomb (C). It is defined in terms of the unit of current, the ampere (A), as the charge passing a particular point in 1 second when there is a current of 1 ampere at that point:

$$1\text{ C} = 1\text{ A} \cdot 1\text{ s}$$

This definition is based on the relation between current $i$ and the rate of charge flow $dq/dt$.

Coulomb's Law and Electrostatic Force

Coulomb's Law describes the electrostatic force ($F$) between small (point) electric charges $q_1$ and $q_2$ separated by a distance $r$:

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

where $k$ is the electrostatic constant (or $\frac{1}{4\pi\epsilon_0}$, where $\epsilon_0$ is the permittivity constant). The force of attraction or repulsion between point charges at rest acts along the line joining the two charges.

The Superposition Principle

If more than two charges are present, Coulomb's Law holds for each pair of charges. The net force on each charge is then found, using the superposition principle, as the vector sum of the forces exerted on that charge by all the others.

Gauss's Law

Gauss's Law and Coulomb's Law are different ways of describing the relation between charge and the electric field in static situations. Gauss's Law is expressed as:

$$\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$$

in which $\oint \vec{E} \cdot d\vec{A}$ is the net flux ($\Phi_E$) of the electric field through the closed Gaussian surface, and $q_{enc}$ is the net charge enclosed.

Key Applications of Gauss's Law

Using Gauss's Law and, in some cases, symmetry arguments, we can derive several important results in electrostatic situations. Among these are:

1. Charge Distribution on Conductors: An excess charge on an isolated conductor is located entirely on the outer surface of the conductor.
2. Field Near a Charged Conductor: The external electric field ($\vec{E}$) near the surface of a charged conductor is perpendicular to the surface and has magnitude: $$\left| \vec{E} \right| = \frac{\sigma}{\epsilon_0}$$ where $\sigma$ is the surface charge density.
3. Infinite Line of Charge: The electric field at any point due to an infinite line of charge with uniform linear charge density $\lambda$ is perpendicular to the line of charge and has magnitude: $$\left| \vec{E} \right| = \frac{\lambda}{2\pi\epsilon_0 r}$$ where $r$ is the perpendicular distance from the line of charge to the point.
4. Infinite Nonconducting Sheet: The electric field due to an infinite nonconducting sheet with uniform surface charge density $\sigma$ is perpendicular to the plane of the sheet and has magnitude: $$\left| \vec{E} \right| = \frac{\sigma}{2\epsilon_0}$$