Lecture Notes on Coulomb's Law and Electrostatics

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Physical Contact for Net Charge: Changing the net charge of an object typically requires physical contact.
Electrostatic Induction: Bringing a charged rod near a conducting sphere redistributes charge without changing the net charge. The sphere must be insulated to prevent charge flow.
Charging by Contact:
- Friction can add or remove electrons.
- Connecting a conductor to an asymmetrically charged sphere allows electron removal, resulting in a net positive charge after the wire is removed.
Exception: In a sufficiently large electric field, objects can lose electrons (discharge) without physical contact, as seen with the Van de Graaff generator.

Positive Charge Movement: Positive charges (protons) are generally stationary within a conductor's lattice structure.
Electron Movement: Electrons are the primary mobile charge carriers in conductors.
Positive Charge Representation: Depicting positive charge often indicates an absence of electrons that would neutralize the atom.
Ionized Hydrogen Example: Ionized hydrogen (free protons) can move.
Non-Conductors: In materials like fur and rubber, electrons do not flow freely; thus, charge transfer is less reversible.

Triboelectric Effect: Charging objects through friction is not fully understood, despite its discovery millennia ago.
Ongoing Research: Current research explores various theories (e.g., contact welding, friction, quantum mechanical tunneling).
Unpredictability: No existing theory can predict whether a material will gain or lose electrons through friction; it is primarily determined experimentally.

Bohr-Rutherford Model: Used to visualize electron movement in conductors, but it's a simplified model.
Quantum Mechanical Explanation: Quantum mechanics explains why electrons do not simply fall into the nucleus.
Electron Orbitals: Electrons exist in probability clouds, not fixed orbits.
- Darker regions indicate higher probability of finding the electron.
Ground State of Hydrogen: In the ground state, the probability of an electron being at the nucleus is zero.
Schrödinger Equation: Not covered in this course.

Force Vector Notation: F_{q2 \rightarrow q1} denotes the force of charge q2 on charge q1.
Unit Vector Definition: \hat{r}_{q2 \rightarrow q1} is a unit vector pointing from q2 to q1.
Axis Conventions: Positive x-axis goes to the right, positive y-axis goes up.
Coulomb's Law Equation: F = k \frac{q1 q2}{r^2} \hat{r}
- k is Coulomb's constant (8.99 \times 10^9).
- r is the distance between charges.
- q1 and q2 are the charges, including their signs.
Unit Vector Importance: Ensures force direction is correct based on charge signs.
Units: Charge in Coulombs, force in Newtons, distance in meters; k is in Newton-meters squared per Coulomb squared.
Like Charges: Repel; opposite charges attract.
Unit Vector Direction: Points from the charge exerting the force to the charge receiving the force.

Vector Components: In 2D, unit vectors can be written as \hat{i} (x-direction) and \hat{j} (y-direction).
Example: A vector pointing along the negative x-axis is represented as -1 \hat{i} + 0 \hat{j}.

Linear Force: Coulomb force is linear, so net force is the vector sum of individual forces.
Net Force Calculation: F{net} = F{1 \rightarrow 3} + F_{2 \rightarrow 3} + …
Component-wise Addition: Add x-components and y-components separately to find the net force vector.
Distance Labeling: The distance between charges must be correctly labeled in calculations (e.g., r_{13} for the distance between q1 and q3).

Problem Setup: Three charges q1, q2, and q3 are positioned on a grid; find the net force on q3.
Diagram: Draw a diagram showing the charges and force vectors.
Vector Sum: The net force on q3 is F{1 \rightarrow 3} + F{2 \rightarrow 3}.
Coulomb's Law Application: F = k \frac{q1 q3}{r{13}^2} \hat{r}{13} + k \frac{q2 q3}{r{23}^2} \hat{r}{23}
Unit Vector Calculation: Calculate unit vectors from charge positions using \hat{r} = \frac{\vec{r}}{|\vec{r}|}.
Component Addition: Add force components in x and y directions to find the net force.
Final Answer Format: F_{net} = (x \hat{i} + y \hat{j}), where x and y are the net force components in Newtons.

Superposition Principle: The net force on a charge is the sum of all individual forces.
Thin Shell of Charge: For a uniformly charged thin shell, the net force on an external electron is the same as if all the shell's charge were concentrated at its center.
Charge Inside the Shell:
- At the center of the shell, forces cancel out due to symmetry.
- Anywhere inside the shell, the net force on a charge is zero.
Inverse Square Law: The shell theorem relies on the inverse square relationship of the Coulomb force; it is also applicable to gravitational forces.

Problem: Given charges +2q and -q, find a point along the axis where a positive charge +q experiences zero net force.
Solution: The point is to the right of the positive charge, where the forces from both charges can balance due to differing distances.

Size Consideration: The charged objects must be small relative to the distance separating them.
Stationary Charges: Charges must be stationary; moving charges emit radiation, complicating force calculations.
Dipole Effects: For neutral objects like hydrogen atoms, the Coulomb force is zero at large distances, but close up, dipole forces come into play, which do not follow the inverse square law.