Premed · Premed · Physics 2
Lecture 1: Electric Charge and Coulomb's Law
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the fundamental properties of electric charge, including quantization and conservation
- Distinguish between conductors and insulators and explain charging mechanisms
- Apply Coulomb's law to calculate the force between point charges
- Use the principle of superposition to find the net force on a charge due to multiple charges
- Solve problems involving charge distributions in one and two dimensions
Lecture Content
I. Electric Charge: Fundamental Properties
All matter contains electric charge, which is carried by subatomic particles. Protons carry positive charge (+e), electrons carry negative charge (-e), and neutrons are electrically neutral. The elementary charge has a magnitude of e = 1.602 x 10^-19 C.
Charge is quantized, meaning that any observable charge is always an integer multiple of the elementary charge, expressed as Q = ne, where n is an integer. Charge is also conserved: the total charge in an isolated system remains constant. Charge can be transferred between objects, but it can never be created or destroyed. Like charges repel one another, while opposite charges attract.
The SI unit of charge is the coulomb (C). One coulomb represents an enormous amount of charge, corresponding to roughly 6.24 x 10^18 elementary charges.
<image>Panel A: Diagram showing the structure of an atom with protons (labeled +) and neutrons in the nucleus, and electrons (labeled -) in orbital shells around the nucleus. Panel B: Two pairs of charges — a pair of like charges (both positive) with arrows pointing away from each other showing repulsion, and a pair of unlike charges (one positive, one negative) with arrows pointing toward each other showing attraction.</image>
II. Conductors, Insulators, and Semiconductors
Conductors are materials in which charge moves freely. Metals such as copper, aluminum, and silver are excellent conductors because their outer electrons are delocalized, forming an "electron sea" that can flow in response to electric fields.
Insulators, by contrast, are materials in which charge does not move freely. Common examples include glass, rubber, plastic, and wood. In these materials, electrons are tightly bound to individual atoms and cannot migrate through the bulk of the material.
Semiconductors occupy a middle ground, exhibiting intermediate conductivity that can be precisely adjusted through a process called doping. Silicon and germanium are the most common semiconductors and form the basis of modern electronics. Finally, superconductors are special materials that exhibit zero electrical resistance below a critical temperature, enabling persistent currents without energy loss.
III. Charging Mechanisms
There are several ways to charge an object. Charging by friction (the triboelectric effect) involves rubbing two different materials together, which transfers electrons from one surface to the other. For example, rubbing a glass rod with silk leaves the rod positively charged because electrons transfer to the silk. The triboelectric series ranks materials by their tendency to gain or lose electrons.
Charging by contact (conduction) occurs when a charged object touches a neutral conductor. Charge flows from one to the other until both reach the same electric potential, leaving both objects with the same sign of charge.
Charging by induction is a subtler process. A charged object is brought near, but does not touch, a neutral conductor. The presence of the nearby charge causes the charges in the conductor to redistribute, a phenomenon called polarization. If the conductor is then grounded, charge flows to or from the ground, and when the ground connection is removed, the conductor is left with a net charge that is opposite in sign to the inducing charge.
Even insulators can be polarized. An external charge causes slight displacement of electron clouds within the molecules, inducing temporary dipole moments. This explains why a charged object can attract neutral insulators, such as when a charged comb picks up bits of paper.
<image>A four-step sequential diagram showing charging by induction. Step 1: A neutral metal sphere on an insulating stand with evenly distributed charges. Step 2: A negatively charged rod is brought near the sphere, causing positive charges to migrate toward the rod and negative charges to move to the far side. Step 3: A ground wire is connected to the far side of the sphere, allowing negative charges to escape to ground. Step 4: The ground wire is removed, then the rod is removed, leaving the sphere with a net positive charge uniformly distributed.</image>
IV. Coulomb's Law
The electrostatic force between two point charges q1 and q2 separated by a distance r is given by Coulomb's law: F = k|q1||q2| / r^2. Here, k = 8.99 x 10^9 N m^2/C^2 is Coulomb's constant, which can also be written as k = 1/(4 pi epsilon_0), where epsilon_0 = 8.85 x 10^-12 C^2/(N m^2) is the permittivity of free space.
The force is directed along the line connecting the two charges. It is attractive for unlike charges and repulsive for like charges. In vector form, F_12 = k q1 q2 / r^2 * r_hat_12, where r_hat_12 is the unit vector pointing from charge 1 to charge 2. When the charges carry the same sign, the force is positive (repulsive); when they carry opposite signs, the force is negative (attractive). Coulomb's law obeys Newton's third law, so F_12 = -F_21.
It is instructive to compare Coulomb's law with Newton's law of gravitation. Both are inverse-square laws, but there are important differences. The electrostatic force can be either attractive or repulsive, whereas gravity is always attractive. Furthermore, the electrostatic force is enormously stronger than gravity: for an electron-proton pair, the electric force exceeds the gravitational force by a factor of roughly 10^36.
<image>A diagram showing two positive point charges q1 and q2 separated by distance r along a horizontal line. Force vectors are drawn on each charge: F_12 points to the right (away from q1) on q2, and F_21 points to the left (away from q2) on q1. The distance r is marked with a double-headed arrow between the charges. Below, the equation F = k|q1||q2|/r^2 is displayed with each variable labeled.</image>
V. Principle of Superposition
The net force on a charge due to multiple other charges is the vector sum of the individual forces: F_net = F_1 + F_2 + F_3 + ... Each pairwise force is calculated independently using Coulomb's law, and then the results are combined using vector addition.
To solve superposition problems, begin by identifying all charges and their positions. Calculate the magnitude of each individual force using Coulomb's law, and determine whether each force is attractive or repulsive. Then resolve every force into its x and y components, sum the components separately (F_net,x = sum of F_ix, F_net,y = sum of F_iy), and finally compute the magnitude and direction of the resultant force from those summed components.
VI. Applications and Problem-Solving Strategies
Symmetry arguments can simplify calculations significantly. If charges are arranged symmetrically, certain force components cancel by symmetry, reducing the amount of computation required.
Common problem types include finding the force on a charge at the corner of a geometric shape such as a triangle or square, determining the force on a charge along a line of other charges, and solving equilibrium problems where you must find the position at which a third charge experiences zero net force.
As a preview for later lectures, when charges are spread continuously over a line, surface, or volume rather than concentrated at discrete points, the summation in the superposition principle is replaced by an integration: F = integral of k dq / r^2. This technique will be developed more fully in the context of electric fields.
<image>A diagram showing three charges arranged at the vertices of a right triangle. Charge q1 (positive) is at the lower left, q2 (negative) is at the lower right, and q3 (positive) is at the top. Force vectors on q3 are shown: F_13 directed upward and to the right (repulsion from q1), and F_23 directed downward and to the right (attraction toward q2). The vector sum F_net is shown as a dashed arrow representing the resultant force on q3. Component decomposition (x and y axes) is indicated with dotted lines.</image>
