Displacement Current

What the Figure Shows

The figure has three panels, (a), (b), and (c), each showing the same parallel-plate capacitor C — two vertical plates, the left one with a column of + charges, the right with − charges. Short horizontal arrows between the plates indicate a uniform electric field pointing left to right. A horizontal wire enters from the left, carrying a time-dependent current $i(t)$ (shown by an arrow). A point P is marked at the top-left of the plate gap, and a point M just inside the gap.

• Panel (a): A thin vertical ellipse (the Amperian loop) of radius $r$ encircles the wire near the plates. Point P lies on this loop. The loop is perpendicular to the wire and centered on it.
• Panel (b): The same loop now forms the mouth of a pot-shaped closed surface that bulges outward and dips its bottom between the capacitor plates, without touching the wire.
• Panel (c): The loop is the rim of a tiffin-box-shaped surface (flat-bottomed). Its flat circular bottom S lies between the capacitor plates, perpendicular to the electric field. The label S marks this flat bottom.

The Physical Idea

The figure illustrates a contradiction that arises when applying Ampere’s circuital law to a charging capacitor. The law states:

where $i(t)$ is the conduction current through the surface bounded by the loop.

• In panel (a), the surface is the circular loop itself. The conduction current $i(t)$ passes through it, so the law gives a nonzero magnetic field at P.
• In panels (b) and (c), the same loop is the rim of different surfaces. Neither surface is pierced by the conduction current (the wire does not go through them). Applying Ampere’s law to these surfaces would give zero magnetic field at P — a contradiction.

Maxwell resolved this by proposing that a changing electric field also produces a magnetic field. The missing term is the displacement current:

where $\Phi_E$ is the electric flux through the surface. For the flat bottom S in panel (c), the electric field between the plates is uniform and perpendicular to S, so the flux is:

Thus, $\varepsilon_0 \frac{d\Phi_E}{dt} = \frac{dQ}{dt} = i(t)$, exactly the conduction current. Adding this term makes the total current the same for any surface bounded by the loop, removing the contradiction.

Key Formula Developed

The generalised Ampere-Maxwell law is:

$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( i_c + \varepsilon_0 \frac{d\Phi_E}{dt} \right)$ …

Figure 8.2 is a two‑panel diagram that shows the electric and magnetic fields inside a charging parallel‑plate capacitor. It is the key visual for understanding how a changing electric field acts as a source of magnetic field — the core idea of displacement current.

Panel (a) – Side view

• The capacitor has two parallel plates: the left plate carries positive charges ($+$), the right plate negative charges ($-$).
• A time‑varying conduction current $i(t)$ enters from the left, so the charge on the plates is increasing with time.
• Point P is located just outside the capacitor (on the wire side), and point M is inside the gap between the plates.
• Between the plates, the electric field $\mathbf{E}$ is drawn as a bold horizontal arrow pointing to the right — from the positive plate toward the negative plate. This field is uniform in the region between the plates (neglecting edge effects).
• At point M, a short vector $\mathbf{B}$ is shown perpendicular to $\mathbf{E}$. This magnetic field is circular around the axis of the capacitor (into/out of the page in this side view). The diagram emphasises that even though there is no conduction current inside the gap, a magnetic field exists there — it is produced by the changing electric field.

Panel (b) – Cross‑sectional (end‑on) view

• This is a view looking along the axis of the capacitor, from the left plate toward the right plate.
• The electric field $\mathbf{E}$ is represented by a disc filled with $\times$ symbols — meaning $\mathbf{E}$ points into the page (away from the viewer). A radial arrow of length $r$ is labelled $\mathbf{E}$, showing that the field is uniform across the disc.
• The magnetic field $\mathbf{B}$ is drawn as circular arrows that are tangent to concentric dotted circles of radii $r$ and $R$ (labelled $r$, $R$). The arrows indicate that $\mathbf{B}$ circulates around the axis — it has no radial component.
• Bold $\mathbf{B}$ labels are placed at several points on the circles, confirming that the magnitude of $\mathbf{B}$ is constant on a given circle but varies with radius.

Physical idea

The figure teaches that a time‑varying electric field produces a magnetic field, just as a conduction current does. Inside the capacitor, there is no moving charge (conduction current $i_c = 0$), yet a magnetic field is measured at point M. This magnetic field arises from the displacement current

where $\Phi_E$ is the electric flux through a surface between the plates. The total current that acts as a source of $\mathbf{B}$ is

Key formula developed with this figure

The Ampere‑Maxwell law (generalised Ampere’s law) is

• $\oint \mathbf{B} \cdot d\mathbf{l}$: line integral of magnetic field around a closed loop.
• $\mu_0$: permeability of free space ($4\pi \times 10^{-7} \, \text{T·m/A}$). …