Discovering Maxwell's Equations: As an Infant

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com) 

Introduction

From the earliest moments of my life, I was immersed in a world filled with mysterious forces and unseen interactions. The gentle static that made my hair stand on end, the invisible pull of magnets, and the sparks of lightning during a storm all captivated my curiosity. Driven by a desire to understand these phenomena, I embarked on a journey of exploration and logical reasoning. Through observations, experiments, and the evolution of concepts grounded in my experiences, I began to unravel the fundamental laws governing electricity and magnetism.

In this narrative, I will detail how, starting from basic sensory experiences as an infant, I independently observed, experimented, and logically deduced what would later be known as Maxwell's Equations. This journey illustrates how profound scientific principles can emerge from simple observations, without relying on subjective definitions or prior formal education.

Chapter 1: Observing Static Electricity1.1 Early Encounters with Electric ChargesExperiencing Static Cling

• Observation: When I rubbed a balloon against my hair, it caused my hair to stand up and cling to the balloon.

• Reflection: There is an invisible force causing attraction between the balloon and my hair.

Experimenting with Materials

• Experiment: Rubbing different materials together, such as wool and plastic.

• Observation: Some combinations led to attraction, others did not.

• Semantics: The materials acquire a property that causes them to attract or repel each other.

1.2 Conceptualizing Electric ChargeDefining Charge

• Concept: An intrinsic property of matter that causes it to experience a force when near other charged matter.

• Semantics: Electric charge comes in two types, which I will label positive and negative, based on their interactions.

Understanding Attraction and Repulsion

• Observation: Like charges repel; unlike charges attract.

• Logical Proposition:$$\text{If charges are the same type}\to \text{Repulsion}$$$$\text{If charges are opposite types}\to \text{Attraction}$$

Chapter 2: Investigating Electric Forces2.1 Quantifying the Electric ForceExperimenting with Charged Objects

• Setup: Suspend a lightweight, charged object (e.g., a pith ball) and bring another charged object close to it.

• Observation: The force between the objects changes with distance.

• Reflection: The force seems to decrease as the distance increases.

2.2 Formulating Coulomb's LawHypothesizing the Relationship

• Hypothesis: The electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Mathematical Expression:

F=kq1q2r2F = k \frac{q_1 q_2}{r^2}F=kr2q1​q2​​

• $F$: Electric force

• q1,q2q_1, q_2q1​,q2​: Magnitudes of the charges

• $r$: Distance between the charges

• $k$: Proportionality constant (Coulomb's constant)

Testing the Hypothesis

• Experiment: Varying the charges and distances, observing the changes in force.

• Conclusion: The proposed relationship holds true across different scenarios.

Chapter 3: Exploring Electric Fields3.1 Concept of the Electric FieldDefining the Field

• Observation: A charged object creates a region around it where other charges experience a force.

• Concept: The electric field is a vector field representing the force per unit charge at each point in space.

Mathematical Representation:

E⃗=F⃗q\vec{E} = \frac{\vec{F}}{q}E=qF​

• E⃗\vec{E}E: Electric field vector

• F⃗\vec{F}F: Force experienced by a test charge

• $q$: Magnitude of the test charge

3.2 Visualizing Electric Field LinesUsing Test Charges

• Experiment: Mapping the direction of the electric force on small positive test charges placed around a source charge.

• Observation: Field lines point away from positive charges and toward negative charges.

Semantics:

• Field lines represent the direction of the electric field.

• The density of lines indicates the field's strength.

Chapter 4: Investigating Magnetism4.1 Early Encounters with MagnetsPlaying with Magnets

• Observation: Magnets attract certain metals and can attract or repel other magnets.

• Reflection: Magnets have two poles, labeled north and south.

Understanding Magnetic Poles

• Observation: Like poles repel; unlike poles attract.

• Semantics: Similar to electric charges, but with magnetic poles.

4.2 Concept of the Magnetic FieldDefining the Field

• Observation: A magnet creates a region around it where magnetic materials experience a force.

• Concept: The magnetic field is a vector field representing the magnetic influence at each point in space.

Visualizing Magnetic Field Lines

• Experiment: Sprinkling iron filings around a magnet to reveal the field pattern.

• Observation: Field lines emerge from the north pole and enter the south pole.

Chapter 5: Connecting Electricity and Magnetism5.1 Discovering ElectromagnetismOersted's Experiment

• Observation: A compass needle deflects when placed near a current-carrying wire.

• Reflection: Electric currents produce magnetic fields.

Conceptualizing the Relationship

• Hypothesis: Moving electric charges (currents) create magnetic fields.

5.2 Ampère's LawFormulating the Law

• Statement: The integrated magnetic field around a closed loop is proportional to the electric current passing through the loop.

Mathematical Expression:

∮B⃗⋅dl⃗=μ0I\oint \vec{B} \cdot d\vec{l} = \mu_0 I∮B⋅dl=μ0​I

• B⃗\vec{B}B: Magnetic field

• dl⃗d\vec{l}dl: Differential length element along the closed loop

• μ0\mu_0μ0​: Permeability of free space

• $I$: Electric current enclosed by the loop

Testing the Law

• Experiment: Measuring magnetic fields around wires with different currents.

• Conclusion: The relationship holds, confirming the law.

Chapter 6: Faraday's Law of Induction6.1 Observing Electromagnetic InductionInducing Current

• Experiment: Moving a magnet through a coil of wire induces a current in the wire.

• Observation: A changing magnetic field produces an electric current.

6.2 Formulating Faraday's LawConceptualizing the Relationship

• Hypothesis: The induced electromotive force (emf) in a closed loop is proportional to the rate of change of the magnetic flux through the loop.

Mathematical Expression:

E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}E=−dtdΦB​​

• $\mathcal{E}$: Induced emf

• ΦB\Phi_BΦB​: Magnetic flux through the loop

• The negative sign indicates the direction of the induced emf (Lenz's Law).

Understanding Magnetic Flux

• Definition:ΦB=∫B⃗⋅dA⃗\Phi_B = \int \vec{B} \cdot d\vec{A}ΦB​=∫B⋅dA

• dA⃗d\vec{A}dA: Differential area element perpendicular to the surface.

Chapter 7: Gauss's Laws7.1 Gauss's Law for ElectricityObserving Electric Flux

• Concept: The total electric flux out of a closed surface is proportional to the enclosed electric charge.

Mathematical Expression:

∮E⃗⋅dA⃗=Qencε0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}∮E⋅dA=ε0​Qenc​​

• E⃗\vec{E}E: Electric field

• dA⃗d\vec{A}dA: Differential area element on the closed surface

• QencQ_{\text{enc}}Qenc​: Total enclosed charge

• ε0\varepsilon_0ε0​: Permittivity of free space

7.2 Gauss's Law for MagnetismObserving Magnetic Monopoles

• Observation: Magnetic field lines are continuous loops; there are no isolated magnetic charges (monopoles).

Mathematical Expression:

∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0∮B⋅dA=0

• Interpretation: The net magnetic flux through any closed surface is zero.

Chapter 8: Maxwell's Addition to Ampère's Law8.1 Identifying the InconsistencyProblem with Ampère's Law

• Observation: In situations involving changing electric fields, Ampère's Law alone does not satisfy the conservation of charge.

Concept of Displacement Current

• Hypothesis: A changing electric field contributes to the magnetic field, similar to a current.

8.2 Modifying Ampère's LawMathematical Expression:

∮B⃗⋅dl⃗=μ0(I+ε0dΦEdt)\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right)∮B⋅dl=μ0​(I+ε0​dtdΦE​​)

• dΦEdt\frac{d\Phi_E}{dt}dtdΦE​​: Rate of change of electric flux

• Displacement Current Term: ε0dΦEdt\varepsilon_0 \frac{d\Phi_E}{dt}ε0​dtdΦE​​

Conclusion:

• Maxwell's Correction: Including the displacement current term resolves the inconsistency and completes the symmetry between electricity and magnetism.

Chapter 9: Formulating Maxwell's Equations9.1 Summarizing the EquationsGauss's Law for Electricity:

∮E⃗⋅dA⃗=Qencε0\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}∮E⋅dA=ε0​Qenc​​

• Interpretation: Electric charges produce electric fields.

Gauss's Law for Magnetism:

∮B⃗⋅dA⃗=0\oint \vec{B} \cdot d\vec{A} = 0∮B⋅dA=0

• Interpretation: There are no magnetic monopoles; magnetic field lines are continuous.

Faraday's Law of Induction:

∮E⃗⋅dl⃗=−dΦBdt\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}∮E⋅dl=−dtdΦB​​

• Interpretation: Changing magnetic fields induce electric fields.

Ampère-Maxwell Law:

∮B⃗⋅dl⃗=μ0(I+ε0dΦEdt)\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right)∮B⋅dl=μ0​(I+ε0​dtdΦE​​)

• Interpretation: Electric currents and changing electric fields produce magnetic fields.

9.2 Writing the Equations in Differential FormUsing Vector Calculus

• Gauss's Law for Electricity:

  ∇⋅E⃗=ρε0\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}∇⋅E=ε0​ρ​

• $\rho$: Charge density

• Gauss's Law for Magnetism:

  ∇⋅B⃗=0\nabla \cdot \vec{B} = 0∇⋅B=0

• Faraday's Law:

  ∇×E⃗=−∂B⃗∂t\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}∇×E=−∂t∂B​

• Ampère-Maxwell Law:

  ∇×B⃗=μ0J⃗+μ0ε0∂E⃗∂t\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}∇×B=μ0​J+μ0​ε0​∂t∂E​

• J⃗\vec{J}J: Current density

Chapter 10: Deducing Electromagnetic Waves10.1 Wave Equations from Maxwell's EquationsCombining Equations

• Process: Take the curl of Faraday's Law and substitute using the other equations.

Deriving the Wave Equation for Electric Field

∇2E⃗−μ0ε0∂2E⃗∂t2=0\nabla^2 \vec{E} - \mu_0 \varepsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = 0∇2E−μ0​ε0​∂t2∂2E​=0

• Interpretation: Electric fields propagate as waves at speed c=1μ0ε0c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}c=μ0​ε0​​1​.

10.2 Understanding Light as an Electromagnetic WaveCalculating the Speed of Light

• Using Known Constants:c=1μ0ε0c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}c=μ0​ε0​​1​

• Result: The calculated speed matches the known speed of light.

Conclusion:

• Insight: Light is an electromagnetic wave, unifying optics with electromagnetism.

Chapter 11: Reflecting on the Discovery11.1 The Elegance of Maxwell's EquationsSymmetry and Unification

• Observation: Maxwell's Equations elegantly unify electricity and magnetism.

• Reflection: The equations are symmetrical and interrelated.

11.2 The Power of Logical ReasoningFrom Observations to Laws

• Process: Starting from simple experiments, I developed concepts, formulated hypotheses, and derived mathematical expressions.

• Semantics: Each concept evolved explicitly from my experiences, avoiding subjective definitions.

Conclusion

Through a journey of curiosity, experimentation, and logical reasoning, I was able to discover and formulate Maxwell's Equations. Starting from basic observations of electric and magnetic phenomena, I developed the concepts of electric charge, electric and magnetic fields, and their interactions. By systematically exploring these ideas and unifying them through mathematical expressions, I arrived at a comprehensive framework that describes the fundamental laws of electromagnetism.

This exploration demonstrates that profound scientific principles can emerge naturally from simple experiences. By grounding each concept in reality and evolving the semantics explicitly, complex ideas become accessible without relying on subjective definitions or advanced prior knowledge.

Maxwell's Equations not only provided a deeper understanding of the natural world but also paved the way for technological advancements and the unification of physics. This journey highlights the power of curiosity and logical thinking in uncovering the fundamental truths of the universe.

Epilogue: Implications for Learning and AI

This narrative illustrates how foundational scientific principles can be understood through direct interaction with the environment and logical reasoning. In the context of artificial intelligence and cognitive development, it emphasizes the importance of experiential learning and the evolution of semantics from core experiences.

By enabling AI systems to observe patterns, formulate hypotheses, and derive laws from observations, we can foster the development of intuitive understanding similar to human learning. This approach avoids reliance on predefined definitions and promotes the natural discovery of scientific relationships.

Note: This detailed narrative presents the conceptualization of Maxwell's Equations as if I, an infant, independently observed and reasoned them out. Each concept is derived from basic experiences, emphasizing the natural progression from simple observations of electric and magnetic phenomena to the understanding of fundamental electromagnetic laws. This approach demonstrates that with curiosity and logical thinking, foundational knowledge about physics can be accessed and understood without relying on subjective definitions.

References

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".