Lec 1

Electric Charge and Coulomb’s Law

The Origin of Electricity

Electricity arises from the fundamental structure of atoms, which are composed of protons, neutrons, and electrons. The electrical nature of matter is due to the presence of charged particles within the atom. Protons carry a positive charge, electrons carry a negative charge, and neutrons are electrically neutral.

• Proton mass: $m_p = 1.673 \times 10^{-27}$ kg

• Neutron mass: $m_n = 1.675 \times 10^{-27}$ kg

• Electron mass: $m_e = 9.11 \times 10^{-31}$ kg

• Elementary charge: $e = 1.60 \times 10^{-19}$ C (Coulomb)

• Atoms are electrically neutral when they contain equal numbers of protons and electrons.

Quantization of Charge

Charge is quantized, meaning it exists in discrete amounts. The smallest non-zero charge is the elementary charge, $e$. The net charge $q$ of an object is given by:

• $q = Ne$ where $N$ is the number of electrons added or removed.

• Example: One coulomb of negative charge contains $N = \frac{1}{1.60 \times 10^{-19}} \approx 6.25 \times 10^{18}$ electrons.

Transferring Electric Charge

Electric charge can be transferred between objects, typically by moving electrons. The object losing electrons becomes positively charged, while the object gaining electrons becomes negatively charged.

• Law of Conservation of Electric Charge: The net electric charge of an isolated system remains constant during any process.

• Like charges repel, unlike charges attract.

Conductors, Insulators, and Semiconductors

Materials differ in their ability to conduct electric charge:

• Conductors: Substances that readily conduct electric charge (e.g., metals like copper, aluminum, gold).

• Insulators: Materials that conduct electric charge poorly (e.g., glass, plastics, rubber).

• Semiconductors: Materials with intermediate conductivity (e.g., silicon, germanium, gallium-arsenide).

Charging by Contact and Induction

Objects can be charged by direct contact or by induction:

• Charging by Contact: When a charged object touches another, electrons are transferred, resulting in both objects having similar charges.

• Charging by Induction: A charged object brought near a neutral object causes redistribution of charges within the neutral object, without direct contact. Grounding allows electrons to flow to or from the earth, stabilizing the charge.

Detection of Electric Charge: The Gold-Leaf Electroscope

The gold-leaf electroscope is an early device used to detect electric charge. When a net charge is present, the gold leaves repel each other due to like charges.

Polarization of Molecules

When a charged object is brought near a neutral insulator, it can polarize the molecules, causing a slight positive surface charge on the insulator.

Coulomb’s Law

Coulomb’s Law describes the electrostatic force between two point charges. The force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them:

• $F = k \frac{q_1 q_2}{r^2}$

• $k = 8.99 \times 10^9$ N·m$^2$/C$^2$ (Coulomb constant)

• $\varepsilon_0 = 8.85 \times 10^{-12}$ C$^2$/N·m$^2$ (permittivity of free space)

• Analogous to Newton’s law of gravitation: $F = G \frac{m_1 m_2}{r^2}$

Application: Bohr Model of the Hydrogen Atom

In the Bohr model, the electron orbits the proton at a fixed radius. The electrostatic force provides the centripetal force for circular motion:

• $F = k \frac{e^2}{r^2}$

• For $r = 5.29 \times 10^{-11}$ m, $m_e = 9.11 \times 10^{-31}$ kg, $e = 1.60 \times 10^{-19}$ C

• Speed of electron: $v = \sqrt{\frac{F r}{m_e}}$

Example: Three Charges on a Line

To find the net force on a charge due to other charges aligned on a line, calculate the individual forces using Coulomb’s Law and sum them vectorially.

• Forces: $F_{12}$ and $F_{13}$

• Net force: $F = F_{12} + F_{13}$

Example: Three Charges in Two Dimensions

When charges are arranged in two dimensions, the net force is found by vector addition of the individual forces. The direction is determined using trigonometry.

• $F = \sqrt{F_x^2 + F_y^2}$

• $\theta = \arctan\left(\frac{F_y}{F_x}\right)$

Summary Table: Properties of Materials

Type	Conductivity	Examples
Conductor	High	Copper, Aluminum, Gold
Insulator	Low	Glass, Plastic, Rubber
Semiconductor	Intermediate	Silicon, Germanium

Key Equations

• $q = Ne$

• $F = k \frac{q_1 q_2}{r^2}$

• $k = \frac{1}{4 \pi \varepsilon_0}$

• $F = \sqrt{F_x^2 + F_y^2}$

• $\theta = \arctan\left(\frac{F_y}{F_x}\right)$

Additional info:

• All examples and calculations are based on standard physics principles and typical introductory physics course content.