How is there current through a capacitor? Or in free space??

1. 17.1

Equation 17.4 equates two volume integrals that are over the same volume. The next claim deserves a brief pause for review:

the functions being integrated must be equal

But what if the current density changes over the volume of integration?
What if the charge density isn’t uniform but instead “lumpy”?

Does this seed doubt in your mind? Make your own answer then click for another one.

Details

So, what IS this volume, specifically?

It is not specified! This is the key. The equations prior to 17.4 work for any volume, and, to be precise, over the surface of the chosen “region of interest”. As long as you choose the same volumetric region for the terms, you’re good.

Since you can choose an arbitrary volume, suppose you choose a tiny (read: infinitesimal) volume at a certain position with this larger “region of interest”. The equation still holds for this tiny d_v_ volume, and it holds for any other certain position without regard for any specifics.

Therefore: the integrands must be equal.

Example 17.1

What is the closest SI prefix to 10-19?

Details

atto is 10-18, so 0.34 as.

2. 17.2

2.1. Problem #1

Battle between the continuity equation and the first version of Ampere’s Law.

  1. Ampere’s Law \(\vec{\nabla} \times \vec{B} = \mu_0 \cdot \vec{J}\)

  2. Take divergence of both sides \(\vec{\nabla} \bullet (\;)\)

  3. Recall that the divergence of the curl is zero, and we have:

\[\begin{align} \vec{\nabla} \bullet \left(\vec{\nabla} \times \vec{B}\right) &= \vec{\nabla} \bullet \left(\mu_0 \cdot \vec{J}\right) \\ 0 &= \mu_0 \vec{\nabla} \bullet \vec{J} \\ 0 &= \vec{\nabla} \bullet \vec{J} \\ \end{align}\]

The continuity equation (a.k.a. “conservation of charge”) says that:

\[\vec{\nabla} \bullet \vec{J} = - \frac{\partial \rho_v}{\partial t}\]

substituting, we get

\[- \frac{\partial \rho_v}{\partial t} = 0\]

In words, this says

The charge density never changes.

with the “never” because we have no constraints on the value of t, so it must work for all time.

Does that seem right? It should sound obviously bogus because it is an easy experiemnt to show that charge can and does move over time. The problem isn’t the math, it is in the assumptions that all of the equations are correct. Pick your poison:

  • Ampere’s Law

  • divergence of the curl

  • continuity equation

The div-curl is backed up by a mathematical proof and so is true, leaving us

  • Ampere’s Law

  • continuity equation

The continuity equation comes directly from conservation of charge, which sounds obvious but wasn’t convincinly proven until Michael Faraday did so in 1843, though William Watson and Benjamin Franklin proposed the idea in 1746-7.

So Ampere’s “Law” is incorrect by process of elimination. (?!?)

2.2. Problem #2

Magnetic fields in a capacitor makes no sense because there is no moving charge between the plates, right?