Chapter 107: The Unveiling of the Grand Palace

Klein’s low voice resonated through the silent study. Turning his head, his deep blue eyes clearly mirrored Lia’s form, along with the four world-altering equations that lay before her.

Lia’s heart quickened its pace. The weight of his gaze, imbued with an unfamiliar intensity she had never before encountered, settled upon her.

Her cheeks flushed with heat, and her breathing grew subtly unsteady.

Almost simultaneously, their gazes averted.

Klein’s gaze returned to the black slate, but his knuckles unconsciously tightened, betraying the turmoil within him.

Lia, meanwhile, lowered her head to stare at her toes, her vision filled only with the simple tops of her shoes.

Yet, the heat radiating from her cheeks was impossible to ignore; she could even feel her earlobes tingling with warmth.

An unsettling silence descended upon the study, broken only by chalk dust drifting silently in the shafts of light, which seemed to elongate the awkward atmosphere indefinitely.

“We…” Lia began, her voice an attempt to dispel the suffocating tension that rendered her helpless.

Her gaze darted frantically about the study, finally settling on the four equations etched into the black slate—the sole clear anchor in her chaotic thoughts.

“We could… try to verify them.”

She raised a hand, pointing to the first equation on the black slate.

“If this system is correct, then it must be capable of encompassing all known laws.”

Lia’s voice gradually regained its composure.

“For instance, the law discovered by Charles Coulomb concerning the interaction between stationary charges.”

Klein’s body imperceptibly eased.

He refocused his attention on the black slate, his diffuse aura coalescing once more, becoming intensely focused and sharp.

“You are right.”

He picked up a new piece of chalk and moved towards another blank slate.

“Then, let us try.”

The value of a theory lies not merely in its ability to explain new phenomena, but more profoundly in its perfect compatibility with established truths.

Without the slightest hesitation, he immediately began to derive.

“Let us assume an isolated point charge, with charge Q, is placed in space.” Klein dotted a point at the center of the slate, labeling it with the letter Q.

“According to this formula, the divergence of the electric field… we want to calculate the force experienced at a distance r from this point charge.”

The chalk in his hand glided across the slate, leaving smooth lines.

Around the point charge, he drew a sphere with radius r.

“We will integrate this formula over the entire spherical surface.”

His movements flowed effortlessly, clearly indicating he was fully absorbed in the realm of theory.

Lia stood quietly to the side, observing his profile.

His focused demeanor held a unique allure, seeming to eclipse everything around him.

This thought, utterly unbidden, instantly left Lia flustered and disoriented.

She immediately forced herself to avert her gaze from Klein’s profile, refocusing instead on the equations unfolding on the slate.

Klein, employing his newly grasped divergence theorem, deftly transformed the volume integral into a surface integral.

“According to the formula, the result of the integral on the left side equals the total charge within the volume divided by the dielectric constant.”

He murmured to himself as he wrote.

“Which is… And the right side…”

He paused, glancing at Lia as if seeking confirmation, yet also as if sharing his thought process.

“Due to symmetry, the magnitude of the electric field intensity at every point on the spherical surface should be the same, and its direction should be radially outward.”

Lia nodded gently.

Reassured by her affirmation, Klein’s hand moved even faster.

“The electric field strength can be directly factored out of the integral sign, leaving the surface area of the entire sphere as the result.”

A simple relation appeared on the slate. From it, Klein solved for the expression of the electric field intensity E.

“This is the electric field intensity generated by a point charge at distance r.” He murmured to himself, then scribbled down the definition of force, substituting the expression for E.

The instant the formula was complete, the chalk in Klein’s hand stilled.

He stared, transfixed, at the final expression.

In his memory, the formula Charles Coulomb had published in *On Arcane Dynamics* surfaced: The forms were perfectly identical.

That empirical formula, painstakingly derived from countless torsion balance experiments, from tedious data and precise measurements, had now been deduced through pure logic, from a more fundamental, more abstract divergence equation.

Furthermore, it also revealed the physical meaning of the empirical constant *k*.

“Oh, heavens…”

Klein’s breath hitched; he reached out, his fingertips lightly traced the equations on the slate, as if caressing a sacred work of art.

In his eyes, these were no longer mere cold symbols.

This was the fundamental logic of the universe revealing its true visage to him.

“Lia!”

He spun around abruptly, his eyes ablaze with a brilliant light. He even took a step forward, seizing Lia’s arm with a grip so strong it caused her a pang of pain.

“We did it! It’s… it’s not just correct, it’s perfect!”

Lia watched his unbridled excitement, a profound sense of satisfaction surging within her.

She was familiar with these theories. Yet, to witness their recreation in this world through her and Klein’s hands, and to see a brilliant mind so profoundly moved by them, was a sensation truly incomparable.

Klein quickly released Lia’s arm and stood before the blackboard once more.

Armed with the results of this derivation, Klein’s gaze immediately swept across the other three equations.

His eyes fell upon the second one:

“The divergence of the magnetic field is always zero…”

He murmured.

“Doesn’t this imply… that magnetic field lines are always closed curves, possessing neither a beginning nor an end? Therefore… therefore, isolated magnetic poles can never exist!”

His gaze then shifted to the third equation:

“The curl of the electric field equals the negative rate of change of the magnetic field… This—this is the very law I discovered before! A changing magnetic field generates a vortex electric field!”

The core tenet of his treatise, On the Fundamental Laws of Magnetic Induction, had been expressed by this equation in an even more concise and profound manner.

Finally, his gaze settled on the fourth equation, which, after correction, appeared the most intricate.

“The curl of the magnetic field is proportional to the sum of the conduction current and… the displacement current.”

He saw Lia’s initial concept of ‘electricity begetting magnetism’.

He saw his own research on ‘magnetism generating electricity’.

And, most importantly, he saw the crucial piece of the puzzle introduced by Lia—the changing electric field—which seamlessly mended the fissures in the entire theoretical framework.

Electrostatic fields, steady magnetic fields, electromagnetic induction, the magnetic effects of current…

From the annals of history to the present moment, all human endeavors, all fragmented discoveries, all debates and conjectures, were, in this instant, unified by these four simple lines of equations.

They were no longer isolated phenomena, but rather different facets of a singular, magnificent existence.

A wave of dizziness washed over Klein.

He felt as though he no longer stood upon the solid floor of the study, but was instead adrift in the profound expanse of the starry cosmos, witnessing firsthand the grand symphony of universal laws.

“The law of electrostatic interaction, the negation of magnetic monopoles, the law of magnetic induction, the magnetic effects of current…”

His voice hoarse, he enumerated each theory that had once been hailed as a monumental breakthrough.

“All of these… all the achievements that the various schools of electromagnetism have prided themselves on, are merely corollaries of these four equations under specific conditions.”

He took a step back, bracing himself against the edge of the desk to steady his trembling body.

He looked at Lia, standing so close to him, and in her eyes, a confluence of shock, ecstasy, and awe intertwined, ultimately dissolving into an ineffable complexity of emotion.

“Lia,” he began, his voice trembling slightly with emotion.

“Have we… have we truly glimpsed that magnificent palace, hidden beyond the mists?”