Chapter 568 Witness the Miracle! (Part 2)

Students who’ve been alive since before the Common Era should all know this.

Way back, people discovered that both electric charges and magnets exert forces on each other.

But at first, nobody connected these two kinds of effects.

Until someone noticed that stones struck by lightning would sometimes become magnetic—then people started guessing there might be some relationship between electricity and magnetism.

What happened next is pretty straightforward.

Oster found that electricity could create magnetism, Faraday found that magnetism could produce electricity.

Finally, everyone realized electricity and magnetism are inseparable, started making generators using magnets, and making electromagnets with electric currents.

But as mentioned earlier,

Faraday discovered electromagnetic induction and mapped out magnetic field lines with iron filings,

But it was Newman and Weber—yes, sitting here in our classroom today—who actually summarized the law of electromagnetic induction.

They named the formula after Faraday, in honor of his contributions.

Their derivations are a tad complicated, with Newman’s vector potential An and Weber’s vector equation Aw involved, so let’s skip the gory details for now.

Long story short:

The final form of Faraday’s Law of Electromagnetic Induction is this:

1. E = nΔΦ/t

(1) If the change in magnetic flux is caused by an area change, ΔΦ = BΔS, so E = nBΔS/t;

(2) If the change in flux is caused by a change in the magnetic field, ΔΦ = ΔBS, so E = nΔBS/t;

(3) If both area and magnetic field change, then use the definition, ΔΦ = |Φfinal - Φinitial|,

2. When a conductor rod cuts through magnetic field lines: E = BLv

3. When the rod rotates around one end, cutting magnetic field lines: E =

4. When a wire loop rotates around an axis perpendicular to B: E =

Looking at these formulas, do you recall the existential dread that high school physics inspired?

Ahem...

Xu Yun, taking this as a starting point, wrote another formula that made Faraday’s scalp tingle:

▽×(▽×E) = ▽(▽·E) - (▽·▽)E = ▽(▽·E) - ▽2E

▽2T = ?2T/?X2 + ?2T/?y2 + ?2T/?z2.

That’s right.

Smart students probably already noticed:

The first mini formula is a triple product formula—curl of curl of the electric field E. The second is the Laplacian of the electric field.

This "curl"—the name—was coined by Wheat (Maxwell) in 1871.

But the concept showed up as early as 1839 in building optical field theory—it just hadn’t gotten a formal name yet.

Honestly,

With Faraday’s math background, he probably wouldn’t get this formula instantly. He’d need to do some serious analytic calculations.

But since some less mathematically gifted classmates are about to start sobbing over failed grades, let’s pretend Faraday is possessed by Gauss here...

When Xu Yun wrote out this formula, Weber—the actual math whiz in the room—noticed something.

He stared at the formula for half a minute, then suddenly his eyes lit up.

He flattened his left hand, made a fist with his right, and smacked his palm hard:

"So this is... the value you get by taking the gradient of the electric field’s divergence, then subtracting the electric field’s Laplacian?"

Xu Yun gave him a big thumbs up. No wonder people later said: if Weber hadn’t gone into electromagnetism, maybe math history would have gotten another great master.

That lightning-fast thinking—even in later generations, very few would match it.

In the formula above:

▽(▽·E) means the gradient of the divergence of E, (▽·▽)E is another way to write ▽2E—the Laplacian. That brings us to the Laplacian operator.

If you assume the temperature at any point (x, y, z) in space is T(x, y, z), then this temperature function T(x, y, z) is a scalar function and you can take its gradient ▽T.

Because the gradient is a vector—it has direction, pointing to where the change is fastest—you can then take its divergence, ▽·.

If you expand the ▽ operator and use the vector coordinate multiplication rule, you can express the divergence of the gradient (that is, ▽2T) of the temperature function T(x, y, z).

Very simple, very intuitive.

Alright, let’s pause the math derivation here. (Condensed a lot—if there’s any part you can’t understand, leave a message, I’ll try to explain.)

Xu Yun then looked at Wheat and said:

"Maxwell, here’s another assignment—use the Laplacian operator to express the wave equation we found before."

At this point, Wheat was already entranced by Xu Yun’s formula. As soon as he heard this, he grabbed his pen and began scribbling furiously.

But for some reason,

He felt this formula weirdly familiar...

He even got this bizarre, subtle, can’t-describe feeling:

Seeing Xu Yun write this formula,

It was as if he’d just witnessed his girlfriend holding someone else’s hand, making out passionately right in front of him.....

Oh, right—he doesn’t have a girlfriend. Never mind.

And on the other side,

If Xu Yun knew what Wheat was thinking, his face would probably be pretty weird too.

Because, in a certain sense...

He really did cuck Wheat here:

The formula he wrote—was none other than an expression of Maxwell’s equations using the Laplacian operator...

Too bad Wheat won’t ask, Xu Yun won’t say; this whole thing will just become a mystery no one ever knows about.