Chapter 105: The Field and the Flaw

Lia Farrien’s heart rippled with emotion at the unexpected compliment.

She buried her face in the pages of On Arcane Arts and Martial Prowess, the cool paper pressing against her flushed skin, yet unable to dissipate the heat that surged within her.

Klein’s hurried departure, and the words he’d stumbled over, replayed endlessly in her mind. It was more perplexing than any complex formula.

She had no idea how to respond, nor could she pinpoint the source of this unfamiliar emotion, leaving her no choice but to flee from it.

Fortunately, this strange feeling was soon drowned out by the relentless tide of academic pursuits.

***

Klein plunged headfirst into his laboratory. Lia’s conjecture—‘a changing electric field produces a magnetic field’—felt to him like the key that had unlocked a door to a new world.

The words ‘symmetry,’ ‘harmony,’ and ‘unity’ were deeply etched into his mind. He was convinced that truth lay hidden behind this very door.

Yet, the path to truth was fraught with thorns. For the first few days, Klein attempted to verify this conjecture using known experimental methods.

He constructed various devices capable of generating rapidly changing electric fields, such as quickly charging and discharging alchemical capacitors, and arranged the most sensitive magnetic needles and micro-ammeters around them.

The results, however, yielded nothing.

No matter how the electric field varied, the surrounding magnetic needles remained motionless, showing no indication of a new magnetic field being generated.

‘Is the magnetic field too weak for our equipment to detect? Or is the conjecture itself flawed?’

A low pressure permeated Klein’s study due to the continuous failures. Although he didn’t pull all-nighters and maintained his regular routine, the hours he spent in his study during the day grew longer and longer.

His desk was piled high with calculation manuscripts, each sheet covered with complex force field models and integral paths, all ultimately pointing to the same disheartening conclusion.

His theoretical derivations, too, had reached an impasse.

He had tried to incorporate the variable of a “changing electric field” into the existing magnetic field circuit law, but no matter how he attempted it, he couldn’t make both sides of the equation logically self-consistent.

The old theoretical framework was like a sturdy cage, trapping his thoughts firmly within its bars.

That afternoon, when Lia walked into Klein’s study carrying a plate of sliced fruit, this was the scene that greeted her:

Klein stood before a massive black slate, covered densely with formulas scribbled in white chalk. He held a piece of chalk suspended in mid-air, his brows deeply furrowed, his entire being frozen like a statue.

Lia placed the fruit platter on a nearby table, careful not to disturb him. She walked to the other side of the black slate, her gaze sweeping over the intricate calculations.

She could discern Klein’s line of thought: he was attempting to explain the new phenomena using the concept of ‘force,’ seeking to calculate the precise force exerted by a changing electric field on the surrounding space, thereby exciting a magnetic field.

“Klein,” Lia said softly.

Klein’s thoughts were broken. He turned, his blue eyes still clouded with unresolved confusion. “Hmm?”

“Have you ever considered that the object of our study might have been wrong from the very beginning?”

Lia walked to the black slate, extending a finger to tap the symbol representing force.

“We have constantly discussed the force between charges, and the force between magnetic poles and currents. But have you ever considered that the field itself might be the fundamental building block of this world?”

“Field?” Klein echoed the word. Although the concepts of electric and magnetic fields had long been introduced in their papers, more often than not, they existed merely as auxiliary tools for calculating forces.

“Yes, the field,” Lia affirmed with conviction.

“A charge does not directly act upon another charge. Instead, it excites an electric field in the surrounding space, and the other charge experiences a force because it enters this electric field.

The electric field, I believe, is a true form of existence, independent of matter.”

She picked up a piece of chalk and, in a blank space on the black slate, drew a dot to represent a charge, then sketched lines of force radiating outwards.

“We’ve always focused our attention on these lines,” she said, gesturing to the lines of force.

“But the true actor is the space itself that carries these lines. It is the nature of the ‘field’ that determines everything.”

Klein’s gaze followed her finger, a thoughtful expression dawning in his eyes.

“If we take the ‘field’ as our fundamental object of study, then our previous integral formulas, which describe overall effects, seem rather clumsy,” Lia continued to guide him.

“A grand theory should be able to describe the properties of any point in space, not merely the sum of a closed circuit.”

She looked at Klein and asked, “Can we not create a new mathematical tool to describe the characteristics of a ‘field’ at a specific point? For instance, whether it is diverging or rotating at that point?”

Diverging? Rotating?

These two words instantly brought clarity to Klein’s muddled thoughts. He thought of water flow—its direction, shape, and power.

He envisioned water, with some areas serving as sources where water surged outwards; this was ‘divergence.’ Other areas formed whirlpools, where water spun in place; this was ‘rotation.’

If the force field were likened to water flow, then indeed, every point in space should possess similar properties!

‘I understand…’

Klein murmured to himself, the light in his eyes growing brighter. “We need an operator capable of describing the changes in a field within an infinitesimal space!”

He began to write furiously on the black slate. By introducing partial derivatives in three directions and combining them into a brand-new mathematical symbol, he created a directional differential operator.

Lia watched quietly. She knew that Klein had touched the threshold of vector analysis.

“We could try to define two operations for it,” Lia added opportunely during his pause.

“One is the dot product with a vector field, used to describe the intensity of the field source, or the degree of ‘divergence.’ We can call this the divergence. Divergence.”

“The other is the cross product with a vector field, used to describe the rotational intensity of the field at a certain point. We can call this the ‘curl.’ Divergence, curl.”

Through continuous contemplation and experimentation, Klein wrote these two terms and their corresponding mathematical expressions on the black slate.

He no longer focused on the tedious integrals but began to re-examine the known laws of electromagnetism from a fresh perspective.

He first rewrote Gauss’s Law for electrostatics using the new ‘divergence’ tool.

The previously complex surface integral instantly transformed into an incredibly concise formula: the divergence of the electric field is proportional to the charge density at that point.

Next was Gauss’s Law for magnetism.

Magnetic field lines are always closed, implying the absence of isolated magnetic poles. Expressed with the new tool: the divergence of the magnetic field is always zero.

Then came Faraday’s Law of Electromagnetic Induction.

A changing magnetic field can excite a swirling electric field, which was the perfect application of ‘curl’: the curl of the electric field equals the negative of the rate of change of the magnetic field.

Three concise, elegant, and symmetrical formulas appeared on the black slate. They described the intrinsic connection between electricity and magnetism at every point in space, radiating the brilliance of truth.

Klein’s breathing grew somewhat ragged. He could feel himself drawing infinitely closer to that grand theory.

Finally, he turned his gaze to the magnetic field circuit law, which he had only recently perfected. He picked up the chalk and, using the language of ‘curl,’ translated this law as well.

The curl of the magnetic field is proportional to the current density at that point.

Everything seemed so perfect. Four concise formulas appeared to encompass all known electromagnetic phenomena.

Klein’s face broke into a slight smile. He turned to Lia, eager to share his joy.

However, Lia’s expression remained calm.

“The last formula poses a question.”

“If we first take the curl of a vector field, and then take its divergence, what do we get?”

Klein paused, momentarily taken aback; it was a purely mathematical question. He quickly performed the calculation on the blackboard and arrived at a definitive answer.

“It equals zero. The divergence of the curl of any field is always zero.”

“Is that so?” Lia nodded, then pointed to the newly written magnetic field circuit law on the black slate.

“Then, according to this equation, what do we get if we take the divergence of both sides simultaneously?”

Klein’s heart sank abruptly. “If we take the divergence of both sides, it must lead to. That is to say, the divergence of the current density must always be zero.”

“The divergence of the current density…” Klein repeated the phrase, his face gradually changing.

He recalled the scenario of a capacitor charging. Current flowed into one plate, but no current flowed out, causing charge to accumulate on the plate.

This meant that in the region of the plate, the current was interrupted, and the current density was not zero!

“It only equals zero in the case of a steady current…” Klein’s voice grew dry. He stared intently at the formula on the black slate, as if looking at a most familiar stranger.

Klein’s breathing hitched.

His proud magnetic field circuit law, the theoretical cornerstone he had personally laid, now, under his gaze, revealed an undeniable crack.

This law, in the case of non-steady currents, contradicted the fundamental fact of charge conservation.

It… was wrong.