Electric Charge and Coulomb’s Law: Concepts, Properties, and Applications

Electric Charge; Coulomb’s Law

Introduction to Electromagnetism

Electromagnetism is a fundamental branch of physics that studies the forces arising from electric charge. These forces include the electric force and the magnetic force, both of which are mediated by electric and magnetic fields. The concepts introduced here expand upon classical mechanics, introducing new entities—fields—that are central to understanding phenomena such as electromagnetic radiation and visible light.

• Classical Physics: Describes matter and fields, sufficient for many engineering applications.

• Modern Physics: Includes relativity and quantum theory, which revise classical ideas for atomic-scale and high-speed phenomena.

Electric Charge

Electric charge is a physical quantity responsible for electric phenomena such as static electricity. It can be transferred between objects and comes in two types: positive and negative.

• Conductors: Materials where charges move freely.

• Insulators: Materials where charge movement is restricted.

• Charge Measurement: Charge is measured in coulombs (C), defined via electric current (1 ampere = 1 coulomb/second).

• Elementary Charge: $e = 1.602177 \times 10^{-19}\ \text{C}$

• Electron Charge: $-e$; Proton Charge: $+e$

• Quantization: All observed charges are integer multiples of $e$: $q = n e$, $n = 0, \pm1, \pm2, \ldots$

• Electron Mass: $m_e = 9.1094 \times 10^{-31}\ \text{kg}$

Coulomb’s Law

Coulomb’s Law describes the force between two point charges. The force can be attractive or repulsive depending on the sign of the charges.

• Formula: $F = k \left| q_1 \right| \left| q_2 \right| / r^2$

• Constant: $k = 8.9876 \times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2$

• Alternate Form: $k = 1/(4\pi\varepsilon_0)$, where $\varepsilon_0 = 8.85419 \times 10^{-12}\ \text{C}^2/\text{N}\cdot\text{m}^2$ (permittivity of free space)

• Direction: Like charges repel; unlike charges attract.

• Superposition Principle: The net force on a charge is the vector sum of all individual forces.

Worked Examples

Example 1: Total Charge of Electrons

• Given: 75.0 kg of electrons

• Number of electrons: $N = M / m_e = (75.0\ \text{kg}) / (9.11 \times 10^{-31}\ \text{kg}) = 8.23 \times 10^{31}$

• Total charge: $Q = N(-e) = (8.23 \times 10^{31})(-1.60 \times 10^{-19}\ \text{C}) = -1.32 \times 10^{13}\ \text{C}$

Example 2: Removing Electrons from a Penny

• Given: Penny mass = 3.11 g (copper), desired charge = $+1.0 \times 10^{-7}\ \text{C}$

• Electrons to remove: $N = q_{\text{Total}} / e = (1.0 \times 10^{-7}\ \text{C}) / (1.60 \times 10^{-19}\ \text{C}) = 6.2 \times 10^{11}$

• Total electrons in penny: $n_{\text{Cu}} = (3.11\ \text{g}) / (63.54\ \text{g/mol}) = 4.89 \times 10^{-2}\ \text{mol}$

• $N_{\text{Cu}} = n_{\text{Cu}} N_A = (4.89 \times 10^{-2}\ \text{mol})(6.022 \times 10^{23}\ \text{mol}^{-1}) = 2.95 \times 10^{22}$

• $N_e = 29 N_{\text{Cu}} = 8.55 \times 10^{23}$

• Fraction removed: $f = (6.2 \times 10^{11}) / (8.55 \times 10^{23}) = 7.3 \times 10^{-13}$

Example 3: Force Between Two Point Charges

• Charges: $q_1 = +3.00 \times 10^{-6}\ \text{C}$, $q_2 = -1.50 \times 10^{-6}\ \text{C}$, $r = 12.0\ \text{cm} = 0.12\ \text{m}$

• Force: $F = k |q_1 q_2| / r^2 = (8.99 \times 10^9)(3.00 \times 10^{-6})(1.50 \times 10^{-6}) / (0.12)^2 = 2.81\ \text{N}$

• Direction: Attractive (opposite signs)

Example 4: Distance for Given Force

• Charges: $q_1 = 26.0\ \mu\text{C}$, $q_2 = -47.0\ \mu\text{C}$, $F = 5.70\ \text{N}$

• Distance: $r^2 = k |q_1 q_2| / F$

• Plug values: $r^2 = (8.99 \times 10^9)(26.0 \times 10^{-6})(47.0 \times 10^{-6}) / 5.70 = 1.93\ \text{m}^2$

• $r = \sqrt{1.93} = 1.39\ \text{m}$

Example 5: Nuclear Fission Fragments

• Each fragment: $Z = 46$, $q = 46e = 7.369 \times 10^{-18}\ \text{C}$

• Separation: $r = 2R = 2 \times 5.9 \times 10^{-15}\ \text{m} = 1.18 \times 10^{-14}\ \text{m}$

• Force: $F = k q^2 / r^2 = (8.99 \times 10^9)(7.369 \times 10^{-18})^2 / (1.18 \times 10^{-14})^2 = 3.5 \times 10^3\ \text{N}$

Example 6: Charges on Two Spheres

• Combined charge: $q_1 + q_2 = 5.0 \times 10^{-5}\ \text{C}$

• Force: $F = k q_1 q_2 / r^2 = 1.0\ \text{N}$, $r = 2.0\ \text{m}$

• Product: $q_1 q_2 = (1.0)(2.0)^2 / (8.99 \times 10^9) = 4.449 \times 10^{-10}\ \text{C}^2$

• Solving quadratic: $q_1 = 3.84 \times 10^{-5}\ \text{C}$, $q_2 = 1.16 \times 10^{-5}\ \text{C}$ (or vice versa)

Example 7: Identical Conducting Spheres

• Initial force (attraction): $F = 0.108\ \text{N}$, $r = 0.50\ \text{m}$

• Product: $q_1 q_2 = -3.00 \times 10^{-12}\ \text{C}^2$

• After wire: $Q = \pm1.00 \times 10^{-6}\ \text{C}$, $q_1 + q_2 = \pm2.00 \times 10^{-6}\ \text{C}$

• Solving: Possible pairs: $(-1.00\ \mu\text{C}, +3.00\ \mu\text{C})$ or $(+1.00\ \mu\text{C}, -3.00\ \mu\text{C})$

Example 8: Maximizing Repulsive Force

• Charge division: $Q$ split into $q$ and $Q-q$

• Force: $F = k (Q-q)q / r^2$

• Maximum: $q = Q/2$

Example 9: Quarks in a Neutron

• Down quark charge: $q = -e/3 = -5.33 \times 10^{-20}\ \text{C}$

• Separation: $r = 2.6 \times 10^{-15}\ \text{m}$

• Force: $F = k q^2 / r^2 = (8.99 \times 10^9)(5.33 \times 10^{-20})^2 / (2.6 \times 10^{-15})^2 = 3.8\ \text{N}$

Example 10: Force in the XY Plane

• Charges: $q_1 = +3.0\ \mu\text{C}$ at $(3.5\ \text{cm}, 0.50\ \text{cm})$, $q_2 = -4.0\ \mu\text{C}$ at $(-2.0\ \text{cm}, 1.5\ \text{cm})$

• Distance: $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = 5.59\ \text{cm}$

• Force: $F = k |q_1||q_2| / r^2 = 35\ \text{N}$

• Direction: Vector from $q_2$ to $q_1$: $\theta = \tan^{-1}(-1.0/5.5) = -10.3^\circ$

• Placement of third charge: $q_3 = +4.0\ \mu\text{C}$, placed at $(-8.35\ \text{cm}, 2.65\ \text{cm})$ to cancel net force on $q_2$

Summary Table: Key Constants and Properties

Quantity	Symbol	Value	Units
Elementary charge	e	1.602177 × 10-19	C
Electron mass	me	9.1094 × 10-31	kg
Coulomb’s constant	k	8.9876 × 109	N·m2/C2
Permittivity of free space	ε0	8.85419 × 10-12	C2/N·m2

Additional info:

• All worked examples are based on standard applications of Coulomb’s Law and quantization of charge.

• Vector directions and placement calculations use basic coordinate geometry and trigonometry.

• Superposition principle is foundational for later topics in electric fields and potential.