Chapter 106: The Language of Gods

The fissure manifested on the blackboard, and it etched itself upon Klein’s very perception.

The grand edifice he had meticulously constructed, just as it neared completion, was struck a devastating blow by a seemingly innocuous question from Lia, carving a fissure through its very foundation.

The divergence of current density is not zero.

This stark reality shattered the elegant simplicity of Ampère’s circuital law, exposing the profound contradiction lurking within.

***

The air in the study thickened, heavy and still. Klein stood before the blackboard, the chalk poised motionless in mid-air, absolutely still.

His face held none of the expected despair or anger Lia had anticipated; only an overwhelming bewilderment.

His blue eyes, usually alight with an inner storm, now held a disquieting blankness, as if he had suddenly lost his bearings in a landscape he thought he knew intimately.

The intricate logic he had painstakingly built, the deductions he had so proudly championed, crumbled into dust before the more fundamental axiom of “charge conservation.”

A considerable silence stretched, then Klein finally stirred.

He gently returned the chalk stub to its groove, then, with a silent, deliberate motion, took up the eraser and swept away every formula from the blackboard.

Those symbols, which had so recently shimmered with the light of intellect, under his steady hand, dissolved into a chaotic swirl of white dust, fluttering softly downward until they settled into a final, quiet oblivion.

“I need to rest for a while.”

He turned to Lia, his voice devoid of inflection.

Yet Lia sensed the profound weariness coiled beneath that placid exterior. It was not a physical exhaustion, but the temporary shock of a mind reeling from a violent upheaval.

With that, he walked directly out of the study, without a backward glance at the now-empty blackboard.

For the next few days, Klein ceased all research into electromagnetism.

He resumed his regular routine, appearing punctually for his lessons each day, after which he would retreat to the terrace atop the Mage Tower, pull up a chair, and quietly immerse himself in a book.

His chosen texts were not arcane magical theories, but rather weighty historical chronicles of the kingdom, or heroic biographies detailing ancient conflicts.

On several occasions, Lia observed him deeply absorbed in these narratives utterly divorced from formulas and laws. Sunlight bathed him, gilding him in a warm, golden hue.

He appeared less like a mage grappling with theoretical conundrums and more like a nobleman enjoying a leisurely holiday.

It was not until that afternoon, as Lia organized the latest issues of Magical Theory and their new papers in the library, that Klein entered.

He seemed to have fully recovered, his gaze clear and sharp, and the quiet, incisive aura that typically surrounded him had coalesced once more.

“I’ve realized the problem isn’t with the mathematical tools.”

He spoke without preamble, walking directly to another blackboard beside Lia.

“The root of the contradiction lies in our own incomplete definition of ‘current’.”

Lia set down the paper she was holding and stood.

Klein picked up the chalk and drew the familiar model of a capacitor on the blackboard. Two parallel metal plates, separated by a vacuum.

“Current flows from the wire and stops at the plates.”

He tapped the left plate with his chalk.

“Charge accumulates here. According to our previous laws, since there’s no steady current loop here, there shouldn’t be a circulating magnetic field in the space between the two plates.”

He paused, looking at Lia, “But his experimental results tell us there *is* a magnetic field there.”

“So, it’s not that there’s no current.”

Lia picked up his train of thought, stepping closer.

“Rather, there exists a type of ‘current’ that we have yet to define.”

Her elbow swept across the empty space between the two plates.

“As charge accumulates on the plates, the electric field strength here changes with time.

Since a changing magnetic field can induce a circulating electric field, then for the sake of the world’s symmetry and harmony, a changing electric field should also play a role.”

“The role it plays is that of a current.”

Klein’s thoughts seamlessly intertwined with hers, his eyes alight with the brilliance of sudden comprehension.

“A changing electric field is equivalent to a type of current. It can produce precisely the same magnetic effects as a real current.”

“Yes.”

Lia picked up a second piece of chalk.

“We can call it displacement current, because it isn’t generated by the actual movement of charge, but rather originates from the change in the electric field itself.”

‘Displacement current…’

‘If it’s equivalent to a current, then it must have its own expression.’ Klein’s mind raced, already formulating the next steps. “We must find it.”

This time, however, it was not Klein alone who would embark on the derivation.

On the other side of the blackboard, Lia inscribed the differential form of charge conservation: The divergence of current density plus the time rate of change of charge density, their sum is zero.

Lia elaborated.

“Charge conservation tells us that charge cannot be created or destroyed. If more current flows into a region than flows out, then the amount of charge in that region must decrease.”

“And Gauss’s Law for electrostatic fields is—”

Klein immediately picked up the thread, writing down another formula. The two formulas stood side-by-side on the blackboard.

Lia looked at them. Then, she performed a maneuver that left Klein utterly astonished. She took the partial derivative with respect to time on both sides of Gauss’s Law.

As spatial and temporal derivatives can be interchanged, the equation transformed into:

“We now have another expression for the rate of change of charge density.”

Lia’s chalk tip tapped the formula, producing a crisp, decisive sound. Klein’s breath hitched.

He gazed at the new relationship Lia had derived, then at the adjacent law of charge conservation; a single thought, like a bolt of lightning, cleaved through all the fog in his mind.

Swiftly, he substituted this new relation into the law of charge conservation.

Extracting the divergence operator, he revealed: A completely new vector field materialized within the parentheses. The divergence of this field was, incredibly, always zero.

Whether within the conducting wire or in the vacuum between the capacitor plates, it perfectly adhered to the law of charge conservation.

“This is it.”

Klein’s voice quivered with barely suppressed excitement. He pointed to the expression within the parentheses, “This is the complete, the true ‘total current’!”

The knot of frustration that had plagued his chest for days dissolved entirely, replaced by a sense of profound exhilaration he had never before experienced.

The fissure that had tormented him for days was, in this singular moment, perfectly mended.

The chalk in his hand moved across the slate, producing a clear, rhythmic scratching sound.

The curl of the magnetic field is proportional to the sum of the conduction current density and the displacement current density.

The instant the equation was complete, the entire study seemed to gain a subtle luminosity.

Klein took a step back, standing shoulder-to-shoulder with Lia, as they gazed at the four concise yet profound equations etched upon the blackboard.

There they stood, silent and immutable, like four lines of divine prophecy.

From electrostatics to magnetostatics, from electromagnetic induction to the generation of magnetic fields by changing electric fields, all known electromagnetic phenomena in the world were encompassed within these four elegant differential equations.

Interdependent, echoing one another, they formed a magnificent system, logically self-contained and perfectly unified.

“It… it looks like the language of gods.”

Klein murmured.

He turned his head to look at Lia. In his blue eyes, the light reflecting off the formulas on the blackboard mingled with the reflection of her own figure.

Lia met his gaze; the lake within her heart, which had only just settled into a fragile calm, once again stirred with ripples.