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The image is from Young and Freedman's University Physics with Modern Physics, Chapter 29 (Electromagnetic Induction).

The image shows a parallel plate capacitor being charged where the current through the plane surface is the conduction current $i_C$. However there is no conduction current through the bulging surface in between the capacitor's plates. The section is subtitled Generalising Ampere's Law.

What I am stuck with is why is there not a conduction current, $i_C$, in the region between plates and what exactly is the 'bulging surface' and it's function?

Apologies if the answers to these are fairly obvious, I am still getting to grips with this subject and am working through it fairly slowly.

Thanks!

enter image description here

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The picture seems an illustration of the equivalence of the magnetic field generated by the current in the wire ($i_C)$ and the changing of the electric field inside the capacitor ($\frac{\partial E}{\partial t}$).

From the Ampere's Law:

\begin{align} \oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\mathbf{\ell} = \mu_0 \left(\iint_{\Sigma} \mathbf{J} \cdot \mathrm{d}\mathbf{S} + \varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} \right) \\ \end{align}

If the surface $S$ is the plane one, the first integral of the RHS is $i_C$ and the second integral is zero. If it is the bulging surface, the first integral is zero (because there is no current inside the capacitor), but the second integral results in the same magnetic field calculated in the LHS, because there is a changing electric field between the plates.

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What I am stuck with is why is there not a conduction current, iC, in the region between plates

The current between the plates of a capacitor is displacement current, not conduction current. Ideally, there are no charges flowing through the dielectric.

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There can be no conduction between the plates because, by design, there is no conducting medium.

Recalling Maxwell's Laws, the relevant equation to think about is $$ \nabla \times {\bf B} = {\bf J} + \epsilon_0 \frac{\partial{\bf E}}{\partial t},$$ where ${\bf J}$ is the current density. The second term on the rhs is called the displacement current (a name that many people aren't keen on) and was introduced by Maxwell to generalise Ampere's Circuital Law to time dependent cases.

Along the conduction wires the contribution to the current comes from ${\bf J}$ and between the plates, where ${\bf J} = 0$,the current-like term is due to the $\partial{\bf E}/\partial t$ term.

It is only relevant when the current in the wire is changing, so the charge on each plate is changing and, therefore, the electric field between the plates is changing.

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