Maxwell's modification to Ampere's law

[jaydnul]

In my textbook, it uses a sack shape surface to explain why Ampere's law didn't work for a changing electric field between a capacitor. Why did they use this sack shape? Why not just use the same circle around the empty space between the capacitor, where its surface is normal to the electric flux, and calculate using the circumference?

This is sort of what it looks like http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter35/chapter35.html

But in my book, the sack shape isn't a perfect cylinder, so the magnetic field does not look like it would be uniform over the whole surface.

Thanks

 

[Astrum]

The real problem is that the divergence of curl is always zero. If we apply this to Amperes law (without Maxwell's correction) we'll see what happens.

$\nabla \cdot (\nabla \times \mathbf B ) = \mu _0(\nabla \cdot \mathbf J ) = 0$ and we know that $\mathbf J ≠ 0$ when the current is not steady, which shows that without Maxwell's correction, this is not right.

A capacitor is used as a concrete example that Ampere's law will fail under certain circumstances. The surface doesn't really matter, as long as it only covers one half of the capacitor. The point is that $\oint \mathbf B \cdot d \mathbf l = \mu _0 I _{enc}$ just doesn't make sense when charge "piles up".

 

[dauto]

In other words, without Maxwell's term, charge is NOT conserved.