Physics · Ch 5 — Electromagnetic Waves

Displacement Current and Maxwell's Correction to Ampere's Circuital Law

5.1.1

Displacement Current and Maxwell's Correction to Ampere's Circuital Law

Faraday's law says a changing magnetic flux through a loop induces an electric field around that loop: $\oint\vec E\cdot d\vec l=-\dfrac{d\Phi_B}{dt}$ . Maxwell asked whether the reverse must also be true, and confirmed mathematically (later verified experimentally by Hertz in 1888) that a changing electric flux through a loop likewise induces a magnetic field around it: $\oint\vec B\cdot d\vec l=\mu_0\epsilon_0\dfrac{d\Phi_E}{dt}$ , called Maxwell's law of induction. To see physically where this extra magnetic field comes from, consider charging a parallel-plate capacitor with a non-conducting medium between the plates, carrying a time-varying conduction current $i_C$ in the connecting wire. Applying the ordinary Ampere's circuital law, $\oint\vec B\cdot d\vec l=\mu_0 i_{enclosed}$ , to an Amperian loop around the wire gives an inconsistency: if the flat surface $S_1$ bounded by the loop is chosen so that the wire pierces it, the enclosed current is $i_C$ and the integral is $\mu_0 i_C$ ; but if instead a balloon-shaped surface $S_2$ with exactly the same boundary loop is chosen so that it bulges through the empty gap between the plates (where no wire is present), the enclosed conduction current is zero, giving $\oint\vec B\cdot d\vec l=0$ . Since Ampere's law for a fixed loop must give the same answer regardless of which surface is stretched across it, this contradiction shows ordinary Ampere's law is incomplete. Maxwell resolved it by noting that while the capacitor charges, the electric flux between its plates, $\Phi_E=EA=q/\epsilon_0$ (from Gauss's law, where $A$ is the plate area), is itself changing with time, and its rate of change $\epsilon_0\,d\Phi_E/dt$ has the same units as a current. He named this quantity the displacement current, $i_d=\epsilon_0\,d\Phi_E/dt=dq/dt$ , and defined it generally as the current that comes into play in any region where the electric field (or electric flux) is changing with time. Adding this displacement current to the conduction current in Ampere's law removes the inconsistency completely: through $S_1$ the enclosed current is entirely conduction current $i_C$ (displacement current there is zero, since there is no changing $E$ -field along the wire), while through $S_2$ the enclosed current is entirely displacement current $i_d$ (equal to $i_C$ , since charge conservation demands the changing charge on the plates matches the current in the wire) -- so both surfaces now agree. The corrected law, known as the Ampere-Maxwell law, is written $\oint\vec B\cdot d\vec l=\mu_0 i_C+\mu_0\epsilon_0\dfrac{d\Phi_E}{dt}=\mu_0(i_C+i_d)$ , and when the current in a circuit is steady (constant), the displacement current is zero and this reduces back to the original Ampere's law. This correction matters far beyond capacitors: in completely empty space, far from any wire, there is no conduction current at all, so ordinary Ampere's law alone would forbid any magnetic field from ever being produced there -- meaning starlight and every other form of electromagnetic radiation reaching Earth across empty space would be impossible. Because Maxwell's correction …

Figure 5.3Applying Ampere's circuital law -- loop enclosing surface S1

What this figure shows. A parallel plate capacitor is shown being charged by a conduction current $i_C$ flowing in the connecting wire, with the plates marked + and -. A circular Amperian loop is drawn around the wire at a point P outside the capacitor, and this loop is shown bounding a flat disc-shaped surface $S_1$ that is pierced by the current-carrying wire. Because the wire actually passes through $S_1$ , applying Ampere's circuital law to this loop-and-surface combination gives a straightforward, non-zero answer for the line integral of the magnetic field: $\oint …

Figure 5.4Applying Ampere's circuital law -- loop enclosing surface S2

What this figure shows. The identical Amperian loop from Figure 5.3, at the same point P, is shown again -- but this time it is drawn as the boundary of a completely different, balloon-shaped surface $S_2$ that bulges out and passes through the empty gap between the capacitor's plates instead of being pierced by the wire. Because Ampere's law only cares about the loop's boundary and not the shape of the surface stretched across it, both surfaces should, in principle, give the same answer. But since no conduction current actually crosses this bulged surface $S_2$ (it slips through the gap between the plates, where there is no wire), a naive application of Ampere's law to $S_2$ gives zero -- directly contradicting the non-zero answer obtained from $S_1$ , and exposi …

Figure 5.5Applying Gauss's law between the plates of the capacitor

What this figure shows. The parallel plate capacitor is redrawn focusing on the region strictly between its plates, where the conduction current $i_C$ in the external wire is shown arriving at the + plate and leaving the - plate, while inside the gap itself a uniform electric field $\vec E$ is drawn pointing from the positive plate towards the negative plate. The bulged surface $S_2$ from Figure 5.4 is shown passing through this field region at point P, and this is the picture used to compute the changing electric flux through $S_2$ via Gauss's law -- the changing flux $\Phi_E=EA=q/\epsilon_0$ across this same gap is exactly what Maxwell identifies as the source of the missing displacement current …

Figure 5.6Magnetic field produced by conduction and displacement currents

What this figure shows. The complete circuit is shown with the conduction current $i_C$ flowing in the wire on both sides of the capacitor and the displacement current $i_d$ flowing in the gap between the + and - plates in its place, with the electric field $\vec E$ drawn pointing between the plates. Circular magnetic field loops labelled $\vec B$ are drawn both around the current-carrying wire outside the capacitor and around the axis inside the gap, showing that the magnetic field at a point near the capacitor is exactly the same whether it is produced by the conduction current in the wire or by the displacement current between the plates -- visually confirming that $i_d$ be …

Misc Example 5.1Displacement current in a charging AC capacitor

Worked out. A parallel plate capacitor with plate separation 1 mm and plate area 20 cm $^2$ is connected to a 230 V RMS, 50 Hz AC supply, and the task is to find the displacement current at $t=1$ s. The instantaneous voltage is $V=V_{max}\sin(2\pi ft)=230\sqrt2\sin(100\pi t)$ V. Since the plates are closely spaced, the field between them at any instant is $E=V/d$ , so the electric flux is $\Phi_E=EA=VA/d$ , and the displacement current is $i_d=\epsilon_0\,d\Phi_E/dt=(\epsilon_0A/d)\,dV/dt$ . Differentiating the voltage expression gives $dV/dt=230\sqrt2\times100\pi\cos(100\pi t)$ , and substituting $t=1$ s, $\epsilon_0=8.85\times10^{-12}$ F/m, $A=20\times10^{-4}$ m $^2$ and $d=1\times10^{-3}$ m into $i_d=(\epsilon_0A/d)\,dV/dt$ gives a displacement current of magnitude $i_d\approx1.81\ \mu\text{A}$ . The key teaching point is that even though no charge physically crosses the gap between the plates, this changing electric field between t …

Misc ~note-displacement-current-nameWhy is it called "displacement" current?

Worked out. A short historical note on the naming of displacement current. Maxwell himself coined the term "displacement current" while working within the mechanical-ether framework of 19th century physics, in which he pictured the changing electric field as literally displacing some hypothetical elastic medium filling space. That mechanical picture of the ether was later abandoned entirely once relativity and modern electromagnetism took hold, but the name Maxwell gave the quantity stuck by convention even though nothing is physically being displaced in the modern understanding -- it is simply the rate of change of electric flux through a surface, dressed up in the same units as an ordinary current so that it can be added directly to the conduction current in Ampere's law. Students are cautioned not to be misled by the word "displacement" into imagining a …

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