Units, Physical Quantities, and Vectors

Scalars and Vectors

In physics, quantities are classified as either scalars or vectors. Scalars have only magnitude, while vectors possess both magnitude and direction. Understanding the distinction is essential for analyzing physical phenomena, especially in electricity and magnetism.

• Scalar: Examples include mass, temperature, and charge.

• Vector: Examples include displacement, velocity, and force.

Vector Representation and Operations

Vectors can be represented in Cartesian coordinates as \( \vec{d} = d_x \mathbf{i} + d_y \mathbf{j} \), where \( d_x \) and \( d_y \) are the components along the x and y axes, respectively. The magnitude and direction of a vector are given by:

• Magnitude:

• Direction (angle):

Vector operations include addition, subtraction, dot product, and cross product, which are foundational for analyzing forces in electrostatics.

Electric Charge

Nature and Properties of Electric Charge

Electric charge is a fundamental property of matter that enables it to experience electromagnetic interactions. Charges are either positive or negative, and their interactions are governed by the following rules:

• Like charges repel; unlike charges attract.

• Charge is quantized and conserved (Law of Charge Conservation).

• The SI unit of charge is the coulomb (C).

Conservation and Transfer of Charge

Charge can be transferred between objects but cannot be created or destroyed. The net charge in an isolated system remains constant. The conventional symbols for charge are ±Q or ±q.

Historical Context

Charles-Augustin de Coulomb made significant contributions to the understanding of electric charge and formulated Coulomb's Law.

Charging by Friction: Experimental Examples

Rubbing Materials

When two different materials are rubbed together, electrons may be transferred from one to the other, resulting in one object becoming negatively charged and the other positively charged. For example:

• Rubbing fur and plastic: Plastic becomes negatively charged, fur becomes positively charged.

• Rubbing silk and glass: Glass becomes positively charged, silk becomes negatively charged.

Repeating the experiment with two rods of the same material demonstrates that like charges repel.

Microscopic Origin of Charge

Atomic Structure

All matter is composed of atoms, which consist of a nucleus (protons and neutrons) surrounded by electrons. Atoms are electrically neutral when the number of protons equals the number of electrons.

• Proton: Positive charge, mass kg, charge C

• Electron: Negative charge, mass kg, charge C

• Neutron: No charge, similar mass to proton

When an atom gains electrons, it becomes a negative ion; when it loses electrons, it becomes a positive ion.

Conductors and Insulators

Definitions and Examples

Materials are classified based on their ability to allow charge movement:

• Conductors: Allow easy movement of charge (e.g., metals).

• Insulators: Do not allow easy movement of charge (e.g., rubber, glass).

Charging by Conduction and Induction

Conduction

Charging by conduction involves direct contact between a charged object and a neutral conductor, resulting in the transfer of charge.

Induction

Charging by induction involves bringing a charged object near a conductor, causing a redistribution of charges within the conductor without direct contact. This process can induce a net charge on the conductor if it is grounded.

Coulomb’s Law

Quantifying Electric Force

Coulomb’s Law describes the force between two point charges. The magnitude of the force is given by:

• Where N·m2/C2 and

• is the electric constant (permittivity of free space)

• The direction is along the line joining the charges, repulsive if charges are alike, attractive if opposite.

The vector form is:

• is the unit vector from to

Steps for Calculating Coulomb Force

1. Identify and

2. Calculate the magnitude:

3. Determine the direction: if (repulsive), if (attractive)

Example Problem: Superposition Principle

For multiple charges, the net force on a charge is the vector sum of the forces from all other charges (superposition principle).

• Given: C at (0,0), C at (0, 0.5), C at (-1.2, 0)

• Calculate and using the steps above, then sum to find .

• Result: N,

Additional info: The superposition principle is fundamental in electrostatics and applies to all vector forces.