Electrostatics: Charge, Coulomb's Law, and the Electric Field

Introduction to Electric Charge and Fundamental Particles

Properties of Subatomic Particles

Electric charge is a fundamental property of matter, carried by subatomic particles such as electrons, protons, and neutrons. Understanding their properties is essential for studying electrostatics.

• Electron (e-): Charge = -e, Mass = $9.11 \times 10^{-31}$ kg

• Proton (p): Charge = +e, Mass = $1.67 \times 10^{-27}$ kg

• Neutron (n): Charge = 0, Mass = $1.67 \times 10^{-27}$ kg

Elementary charge: $e = 1.602 \times 10^{-19}$ C (coulombs)

• Electric charge is a scalar quantity.

Rules of Electrical Interaction

• Like charges (+ + or - -) repel each other.

• Unlike charges (+ -) attract each other.

Quantitative Description of Electric Interaction

Point Charges and Charge Quantization

A point charge is an idealized entity with no size or structure, carrying a discrete amount of electric charge. All observable charges are integer multiples of the elementary charge $e$.

• For every charge, there is an equal and opposite charge (conservation of charge).

• Charge is quantized: $q = n e$, where $n$ is an integer.

Example: Electron-Proton Interaction

• Electron and proton have equal magnitude but opposite sign charges.

• Forces between them are equal in magnitude and opposite in direction (Newton's Third Law).

Coulomb's Law

Statement and Formula

Coulomb's Law describes the force between two point charges:

• The force is along the line joining the charges.

• It is attractive for opposite charges and repulsive for like charges.

Formula:

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

• $F$ = magnitude of force (N)

• $k_e$ = Coulomb's constant $= 8.99 \times 10^9$ N·m2/C2

• $q_1, q_2$ = charges (C)

• $r$ = distance between charges (m)

Comparison: Electric vs. Gravitational Force

Both forces are inverse-square laws, but the electric force is much stronger for elementary particles.

Force	Formula	Constant
Electric	$F_e = k_e \frac{|q_1 q_2|}{r^2}$	$k_e = 8.99 \times 10^9$ N·m2/C2
Gravitational	$F_g = G \frac{m_1 m_2}{r^2}$	$G = 6.67 \times 10^{-11}$ N·m2/kg2

Example: The electric force between two electrons is about $10^{42}$ times stronger than their gravitational attraction.

Electric Field

Definition and Properties

The electric field is a vector field that describes the force per unit charge at each point in space due to a source charge distribution.

• Symbol: $\vec{E}$

• Unit: N/C (newtons per coulomb)

• Direction: The direction of the force on a positive test charge.

Formula for a point charge:

$\vec{E}(\vec{r}) = k_e \frac{q}{r^2} \hat{r}$

• $q$ = source charge

• $r$ = distance from the charge

• $\hat{r}$ = unit vector pointing from the source to the field point

Force on a Test Charge

The force experienced by a test charge $q_0$ in an electric field $\vec{E}$ is:

$\vec{F} = q_0 \vec{E}$

Superposition Principle

The net electric field at a point due to multiple charges is the vector sum of the fields produced by each charge individually.

$\vec{E}_{\text{total}} = \sum_i \vec{E}_i$

Worked Examples and Applications

Example 1: Calculating Electric Field from a Point Charge

• Given $q = -1.6 \times 10^{-19}$ C at a distance $r = 0.001$ m:

$\vec{E} = k_e \frac{q}{r^2} = 8.99 \times 10^9 \frac{-1.6 \times 10^{-19}}{(0.001)^2}$

$\vec{E} = -1.44 \times 10^{-6}$ N/C

Example 2: Components of Electric Field

• Decompose $\vec{E}$ into $x$ and $y$ components using trigonometry:

$E_x = E \cos \theta$

$E_y = E \sin \theta$

Example 3: Force on a Charge in an Electric Field

• Given $q_0$ and $\vec{E}$, calculate $\vec{F}$:

$\vec{F} = q_0 \vec{E}$

Limitations and Extensions

• Coulomb's Law applies strictly to point charges or spherically symmetric charge distributions, and when the separation is much greater than the size of the charges.

• For atomic-scale systems (e.g., hydrogen atom), more sophisticated quantum models are required.

Summary Table: Key Constants and Units

Quantity	Symbol	Value	Unit
Elementary charge	e	$1.602 \times 10^{-19}$	C
Coulomb's constant	$k_e$	$8.99 \times 10^9$	N·m2/C2
Gravitational constant	G	$6.67 \times 10^{-11}$	N·m2/kg2
Electron mass	$m_e$	$9.11 \times 10^{-31}$	kg
Proton mass	$m_p$	$1.67 \times 10^{-27}$	kg

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and calculations have been generalized for study purposes.