"Strange contradiction" that Maxwell found and resolved

[Swamp Thing]

In "The Strange Story of the Quantum", Banesh Hoffmann writes:

Not content with translating Faraday’s ideas into mathematical form. Maxwell went on to develop the mathematical consequences of the theory and to extend its realm. Soon he came to a contradiction. Evidently all was not well with the theory, but what the remedy might be was not easy to determine. Various scientists sought for a cure, among them Maxwell himself. So refined and mathematical had the theory of electricity and magnetism become by now that when Max well arrived at a cure by sheer intuition based upon most unreliable analogies, he produced a group of equations differing but slightly in external form from the old equations. But not only did the new equations remove the contradiction, they also carried a significant new implication. They required that there should exist such, things as electromagnetic waves, that these waves should move with the speed of light, and that they should have all the other major known physical properties of light.

What was that contradiction?

 

[Baluncore]

What was that contradiction?

I would guess it had to do with Newtons hypothetical aether, and free space having an intrinsic impedance. That concept was needed before EM wave propagation could be considered.

 

[Ibix]

Wikipedia suggests that what he did was add the dependence of the magnetic field on the time variation of the electric field, which is definitely a necessary component of the derivation of the EM wave equation.

I'm not sure what the contradiction would be from leaving that term out.

 

[Hill]

Perhaps the "contradiction" is what is explained here in the section "Maxwell’s Correction to the Laws of Electricity and Magnetism":

 

[Baluncore]

See the introduction paragraph of this wiki page.
https://en.wikipedia.org/wiki/Displacement_current

 

[Ibix]

See the introduction paragraph of this wiki page.
https://en.wikipedia.org/wiki/Displacement_current

The "Necessity" section of that page agrees with @Hill's suggestion.

 

[phyzguy]

The Feynman lectures also explain it quite well. Without the term that Maxwell added (called Maxwell's displacement current), the equations are not self consistent.

 

[Meir Achuz]

The equation,
$$\nabla\times{\bf H}=4\pi{\bf j}/c$$,
contradicts the continuity equation because,
$$\nabla\cdot(\nabla\times{\bf H})=(4\pi/c)\nabla\cdot{\bf j}=-4(\pi/c)\partial_t\rho\neq 0$$.