AP Physics C E&M: Displacement Current

Displacement current is the ingenious conceptual fix that completed one of the greatest syntheses in physics: Maxwell's equations. Without it, Ampere's law fails for one of the most common circuit elements—a charging capacitor—and the prediction of self-sustaining electromagnetic waves becomes impossible. Mastering this topic is crucial for understanding how Maxwell unified electricity and magnetism into a single, coherent theory.

The Hole in Ampere's Original Law

The original Ampere's law, which you likely learned first, relates the magnetic field integrated around a closed loop (its circulation) to the total real current passing through any surface bounded by that loop. Mathematically, it's written as $\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{enc}$.

This law works perfectly for steady currents, like those in a long straight wire. However, it contains a critical ambiguity: which surface? For a given closed loop, an infinite number of different surfaces can be bounded by it. Ampere's law insists that the enclosed current $I_{enc}$ must be the same for any surface we choose, or the law is mathematically inconsistent. For steady currents in continuous wires, this is always true because current cannot pile up or disappear.

The problem becomes starkly clear with a charging capacitor. Consider a loop around a wire leading to a capacitor plate. If we choose a flat, "bulging" surface that cuts through the wire ($S_1$), a real conduction current $I$ passes through it. But if we choose a "pot-shaped" surface ($S_2$) that passes between the capacitor plates, no real current passes through it—the gap is an insulator. The original Ampere's law gives two different answers for the same loop, which is a fatal inconsistency. This paradox revealed that Ampere's law, as originally formulated, was incomplete for situations involving time-varying electric fields.

Maxwell's Conceptual Leap: Displacement Current

James Clerk Maxwell resolved this paradox by recognizing a profound symmetry. Just as a changing magnetic field can induce an electric field (Faraday's law), a changing electric field should, in a sense, act like a current and induce a magnetic field. He proposed adding a new term to Ampere's law, called the displacement current $I_d$.

The displacement current is defined as: $$I_d={ϵ}_0\frac{d{\Phi }_E}{dt}$$ where ${ϵ}_0$ is the permittivity of free space and $\frac{d{\Phi }_E}{dt}$ is the rate of change of the electric flux ${\Phi }_E$ through the surface in question. Electric flux is defined as ${\Phi }_E=\int \vec{E}\cdot d\vec{A}$.

Now, let's revisit the capacitor. While no real current flows between the plates, the electric field between them is changing as charge builds up. The changing electric field produces a changing electric flux. For surface $S_2$, this displacement current, ${ϵ}_0(d{\Phi }_E/dt)$, is exactly equal in magnitude to the real conduction current $I$ flowing into the plate on surface $S_1$. Thus, for any surface bounded by our loop, the sum "conduction current plus displacement current" is constant. Maxwell restored consistency by amending Ampere's law to include this term.

The Ampere-Maxwell Law

The complete, corrected law is known as the Ampere-Maxwell law: $$\oint \vec{B}\cdot d\vec{l}={\mu }_0(I_{enc}+{ϵ}_0\frac{d{\Phi }_E}{dt})$$ Here, $I_{enc}$ is the enclosed conduction current, and ${ϵ}_0(d{\Phi }_E/dt)$ is the enclosed displacement current. This law states that magnetic fields are produced by both the flow of real charge and by changing electric fields.

This correction was not just a mathematical trick; it had monumental physical consequences. When combined with Faraday's law, the Ampere-Maxwell law allows for self-sustaining electromagnetic waves. A changing E-field creates a B-field (via displacement current), and that changing B-field creates an E-field (via Faraday's law), propagating through space at the speed of light, $c=1/\sqrt{{\mu }_0{ϵ}_0}$. Displacement current is the crucial link that made the unified theory of electromagnetism possible.

Application: The Magnetic Field in a Charging Capacitor

Let's apply the Ampere-Maxwell law to find the magnetic field between the plates of a parallel-plate capacitor that is charging with a constant current $I$. Assume the plates are circular with radius $R$.

We use a circular Amperian loop of radius $r$ centered on and parallel to the plates, located between them.

1. Symmetry: The changing electric field between the plates is uniform and perpendicular to the plates. This symmetry implies the induced magnetic field lines are circles concentric with the axis of the plates.
2. Left-hand side of the law: $\oint \vec{B}\cdot d\vec{l}=B(2\pi r)$, since $B$ is constant in magnitude along the circular path and parallel to $d\vec{l}$.
3. Right-hand side of the law: There is no conduction current between the plates, so $I_{enc}=0$. We must calculate the displacement current term.

• The electric field between the plates is $E=\sigma /{ϵ}_0=Q/({ϵ}_0\pi R^2)$, where $Q$ is the instantaneous charge.
• The electric flux through the flat surface bounded by our Amperian loop is ${\Phi }_E=E\cdot A_{loop}=[Q/({ϵ}_0\pi R^2)]\cdot (\pi r^2)=(Q{r}^2)/({ϵ}_0{R}^2)$.
• Therefore, ${ϵ}_0\frac{d{\Phi }_E}{dt}={ϵ}_0\frac{d}{dt}(\frac{Q{r}^2}{{ϵ}_0{R}^2})=\frac{r^2}{R^2}\frac{dQ}{dt}$.
• Since the charging current is $I=dQ/dt$, the enclosed displacement current is $(r^2/R^2)I$.

1. Apply the Ampere-Maxwell Law:

$$B(2\pi r)={\mu }_0\left(0+\frac{r^2}{R^2}I\right)$$ Solving for $B$ gives: $$B=\frac{{\mu }_{0}I}{2\pi R^2}r\text{(for }r<R\text{)}$$ Inside the region between the plates, the magnetic field increases linearly with distance $r$ from the center. For a point outside the plates ($r>R$), the entire displacement current ${ϵ}_0(d{\Phi }_E/dt)$ equals $I$, yielding $B={\mu }_{0}I/(2\pi r)$, exactly as if the current $I$ flowed continuously through the capacitor. This result confirms the consistency the law was designed to achieve.

Common Pitfalls

1. Thinking displacement current is a flow of charge. This is the most critical misconception. Displacement current is not a current of moving charges. It is a mathematical quantity proportional to the rate of change of electric flux. It has the units of current and plays the role of a "current" in the Ampere-Maxwell law, but it does not involve charge carriers moving through space. In a capacitor's gap, there is no charge flow, only a changing E-field.
2. Applying the displacement current term to static fields. The displacement current term ${ϵ}_0(d{\Phi }_E/dt)$ is only non-zero when the electric field (and hence electric flux) is changing with time. For electrostatic situations, this term is zero, and the Ampere-Maxwell law reduces to the original Ampere's law. Always ask: "Is $\vec{E}$ constant or changing?" before including this term.
3. Forgetting that displacement current exists everywhere a changing E-field exists, not just in capacitors. While the capacitor example is classic, displacement current is a universal concept. It is present in the space around any wire carrying a changing current, and it is the essential mechanism for electromagnetic wave propagation in empty space, where no conduction current can flow.
4. Misidentifying the area for electric flux calculation. When using the Ampere-Maxwell law, the electric flux ${\Phi }_E$ must be calculated through the same open surface used to determine $I_{enc}$. The result for the line integral $\oint \vec{B}\cdot d\vec{l}$ must be independent of the surface choice, but you must be consistent within a single calculation.

Summary

• Displacement current, defined as $I_d={ϵ}_0\frac{d{\Phi }_E}{dt}$, is Maxwell's addition to Ampere's law that resolves inconsistencies with time-varying fields, particularly in circuits with capacitors.
• The complete Ampere-Maxwell law is $\oint \vec{B}\cdot d\vec{l}={\mu }_0(I_{enc}+{ϵ}_0\frac{d{\Phi }_E}{dt})$. Magnetic fields are produced by both conduction currents and by changing electric fields.
• Displacement current is not a flow of real charge; it is a quantity linked to a changing electric flux. In a charging capacitor, it numerically equals the conduction current in the wires, ensuring consistency in the law.
• This conceptual fix was essential for predicting electromagnetic waves. The coupling between a changing E-field producing a B-field (via displacement current) and a changing B-field producing an E-field (via Faraday's law) allows waves to propagate.
• When solving problems, you must calculate the electric flux ${\Phi }_E$ through the same surface bounded by your Amperian loop. The symmetry of the changing electric field dictates the symmetry of the induced magnetic field.