Ampère-Maxwell Law and Displacement Current

Hello! Welcome back to your study of electrodynamics.

In our last lesson, we analyzed RLC circuits using the "lumped element" model, where the behavior is described by ordinary differential equations. We noted that this model is an idealization. Today, we confront a fundamental inconsistency in one of the laws we've been using, Ampère's Law, when dealing with situations where electric fields and charges change over time.

This lesson addresses the learning outcome: Formulate the Ampère-Maxwell law by introducing the concept of displacement current. We will uncover the paradox that led James Clerk Maxwell to his brilliant insight, introduce the new concept of "displacement current," and show how its inclusion completes Ampère's Law. This is the final piece we need to assemble the full, majestic structure of Maxwell's equations.

1. The Problem with Ampère's Law

So far, we have used Ampère's Law in the form $\oint B \cdot d l = μ_{0} I_{enc}$ . This law successfully describes the magnetic field produced by steady currents. However, it fails when currents are not steady. This failure manifests in two ways: as a logical paradox in a simple circuit, and as a mathematical contradiction with a fundamental principle.

The Capacitor Paradox

Let's consider a simple circuit where a current $I$ is charging a parallel-plate capacitor. According to Ampère's law, the line integral of $B$ around a closed loop (an "Amperian loop") depends on the current piercing the surface bounded by that loop.

The paradox arises when we consider two different surfaces bounded by the same loop, as illustrated in the video below.

Lecture 26 Maxwell Equations - The Full Story

Professor Carlson provides a clear and intuitive explanation of the paradox that arises when applying Ampere's Law to a charging capacitor. He uses the analogy of a soap film on a bubble wand to represent the Amperian surface.

Please watch from 07:43 to 11:17. Focus on why choosing surface S1 (which intercepts the wire) gives a non-zero magnetic field, while choosing surface S2 (which passes between the capacitor plates) gives a zero magnetic field, leading to a contradiction.

As the video shows, we have a contradiction:

For a flat surface S1 bounded by the loop, the enclosed current is $I$ , so $\oint B \cdot d l = μ_{0} I$ .
For a "bag-shaped" surface S2 bounded by the same loop, no conduction current passes through it, so $I_{enc} = 0$ , implying $\oint B \cdot d l = 0$ .

The magnetic field at the loop cannot be both non-zero and zero. Ampère's Law, as stated, must be incomplete.

The Mathematical Inconsistency

Your appreciation for mathematical rigor will find a more formal argument compelling. The incompleteness of Ampère's Law can be proven by showing it conflicts with the continuity equation, which expresses the conservation of electric charge:

\nabla \cdot J + \frac{\partial ρ}{\partial t} = 0

Here, $J$ is the current density and $ρ$ is the charge density.

The following reading demonstrates this conflict using the differential form of Ampère's Law, $\nabla \times B = μ_{0} J$ .

Displacement current. Maxwell's equations.

These lecture notes from MIT by Scott Hughes provide a concise and rigorous derivation showing the inconsistency between Ampere's Law and the continuity equation.

Please read Section 19.1, 'Inconsistent equations' (the first page). The core of the argument relies on a vector calculus identity you've seen before: the divergence of a curl is always zero, ( abla \cdot ( abla imes \mathbf{A}) = 0).

The argument is as follows:

Take the divergence of both sides of Ampère's Law: $\nabla \cdot (\nabla \times B) = \nabla \cdot (μ_{0} J)$ .
The left side is identically zero.
This implies $\nabla \cdot J = 0$ .
But the continuity equation states $\nabla \cdot J = - \partial ρ / \partial t$ .
Therefore, the original Ampère's Law is only valid when $\partial ρ / \partial t = 0$ , i.e., for situations where charge density is not changing—static charges and steady currents. It fails for dynamic situations like a charging capacitor.

2. Maxwell's Solution: The Displacement Current

Maxwell resolved this inconsistency with a stroke of genius. Inspired by the symmetry of Faraday's Law (a changing magnetic field creates an electric field), he postulated that a changing electric field must create a magnetic field.

Let's return to the capacitor. While no charge flows through the gap, the electric field $E$ between the plates increases as charge accumulates. This changing electric field is the missing piece.

The following video segment shows how this new idea resolves the paradox.

Lecture 26 Maxwell Equations - The Full Story

Professor Carlson now derives the form of Maxwell's new term by calculating the rate of change of electric flux inside the capacitor and showing it's directly proportional to the conduction current.

Please watch from 11:17 to 15:15. Follow the derivation that connects the charging current I = dQ/dt to the changing electric flux rac{d\Phi_E}{dt}.

The derivation shows that the current $I$ charging the capacitor is related to the electric flux $Φ_{E} = \int E \cdot d A$ between the plates by:

I = \frac{d Q}{d t} = ϵ_{0} \frac{d Φ _{E}}{d t}

Maxwell proposed that this term, which he called the displacement current, should be added to Ampère's Law. It's not a current of moving charges, but a changing electric field that acts as a source for $B$ as if it were a current.

Displacement Current: $I_{d} = ϵ_{0} \frac{d Φ _{E}}{d t}$

This leads to the corrected law, now known as the Ampère-Maxwell Law:

\oint_{C} B \cdot d l = μ_{0} (I_{enc} + I_{d}) = μ_{0} (I_{enc} + ϵ_{0} \frac{d Φ _{E}}{d t})

With this addition, the paradox is resolved:

For surface S1, we enclose the conduction current $I_{enc} = I$ , and the displacement current is zero.
For surface S2, the conduction current is zero, but we enclose the displacement current $I_{d} = ϵ_{0} \frac{d Φ _{E}}{d t} = I$ .

In both cases, the right-hand side is $μ_{0} I$ , yielding a consistent result for the magnetic field.

Displacement Current & Maxwell's Equations — This diagram summarizes the concept for a charging capacitor. The conduction current $I_c$ flowing into the plate is equal to the displacement current $I_D$ "flowing" out of the volume between the plates, ensuring the continuity of the total generalized current.

3. Verification and Calculation

To solidify this concept, let's watch a masterclass by Walter Lewin, who demonstrates the power and consistency of the Ampère-Maxwell law by explicitly calculating the magnetic field around a charging capacitor.

8.02x - Module 07.01 - Discplacement Current.

Walter Lewin's lecture provides a classic, problem-based demonstration of the Ampère-Maxwell law. He shows that you get the same answer regardless of which surface you choose and then calculates the field inside the capacitor where only displacement current exists.

First, watch from 08:14 to 17:00. This part shows the two ways of calculating the B-field for a point P2 (outside the radius of the plates but in the gap), confirming that the flat surface (using displacement current) and the 'bag' surface (using conduction current) give the identical result. Then, watch from 17:00 to 23:37. This part calculates the B-field for a point P3 inside the capacitor gap (r < R). This is a crucial calculation as it depends entirely on the displacement current.

The key results from the video are:

Outside the capacitor gap ( $r > R$ ): $B = \frac{μ _{0} I}{2 π r}$ . This is the same as the field around a long straight wire, which makes sense as the capacitor gap is enclosed.
Inside the capacitor gap ( $r \leq R$ ): $B = \frac{μ _{0} I r}{2 π R ^{2}}$ . The field is zero at the center and increases linearly with radius $r$ . This result is only obtainable using the displacement current.

Test your understanding!

A fat wire of radius $a$ carries a constant current $I$ , uniformly distributed. It has a narrow gap of width $w ≪ a$ , forming a parallel-plate capacitor. Find the magnetic field $B$ in the gap at a distance $s < a$ from the axis. (This is Problem 7.34 from Griffiths, featured in one of your resources).

Show answer

Inside the gap, there is no conduction current ( $I_{enc} = 0$ ). The magnetic field is generated solely by the displacement current. We apply the Ampère-Maxwell law to a circular loop of radius $s$ .

Electric Field: The charge density on the plates is $σ = q / (π a^{2})$ . The electric field is $E = σ / ϵ_{0} = q / (ϵ_{0} π a^{2})$ .
Displacement Current Density: The displacement current density is $J_{d} = ϵ_{0} \frac{\partial E}{\partial t}$ . $J_{d} = ϵ_{0} \frac{\partial}{\partial t} (\frac{q}{ϵ _{0} π a ^{2}}) = \frac{1}{π a ^{2}} \frac{d q}{d t} = \frac{I}{π a ^{2}}$ Notice that the displacement current density inside the gap is uniform and has the same magnitude as the conduction current density in the wire. The total displacement current is $I_{d} = J_{d} (π a^{2}) = I$ .
Apply Ampère-Maxwell Law: For a loop of radius $s < a$ , the enclosed displacement current is $I_{d, enc} = J_{d} \times (π s^{2}) = \frac{I}{π a ^{2}} (π s^{2}) = I \frac{s ^{2}}{a ^{2}}$ . $\oint B \cdot d l = B (2 π s) = μ_{0} I_{d, enc} = μ_{0} I \frac{s ^{2}}{a ^{2}}$
Solve for B: $B = \frac{μ _{0} I s}{2 π a ^{2}}$ This is the same result as the magnetic field inside a fat wire carrying current $I$ . The gap is, in a sense, invisible to the magnetic field.

4. The Differential Form and Conservation of Charge

The Ampère-Maxwell law can also be written in its more powerful differential form. The "inspired guesswork" that the MIT notes mention is a beautiful piece of reasoning.

Displacement current. Maxwell's equations.

Let's revisit the MIT notes to see the formal derivation of the differential form of the Ampère-Maxwell law.

Please read Section 19.2, 'Fixing the inconsistency'. This section shows how to find the necessary correction term by requiring that the final equation be consistent with the continuity equation.

As derived in the notes, the differential form is:

\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t}

We can now verify that this equation inherently respects the conservation of charge. If we take the divergence of both sides:

\nabla \cdot (\nabla \times B) = \nabla \cdot (μ_{0} J) + \nabla \cdot (μ_{0} ϵ_{0} \frac{\partial E}{\partial t})

The left side is zero. We can swap the divergence and time derivative on the right:

0 = μ_{0} \nabla \cdot J + μ_{0} ϵ_{0} \frac{\partial}{\partial t} (\nabla \cdot E)

Now, substituting Gauss's Law, $\nabla \cdot E = ρ / ϵ_{0}$ :

0 = μ_{0} \nabla \cdot J + μ_{0} ϵ_{0} \frac{\partial}{\partial t} (\frac{ρ}{ϵ _{0}}) = μ_{0} (\nabla \cdot J + \frac{\partial ρ}{\partial t})

This shows that the Ampère-Maxwell law is mathematically consistent with the continuity equation $\nabla \cdot J + \partial ρ / \partial t = 0$ . The inclusion of the displacement current was not just a clever trick; it was a necessary step to make the theory of electromagnetism self-consistent.

Conclusion

Today we have completed the foundation of classical electrodynamics by correcting Ampère's Law. This was the final, crucial step in unifying electricity and magnetism.

Key Takeaways:

Ampère's original law, $\oint B \cdot d l = μ_{0} I_{enc}$ , is only valid for steady currents and static charge distributions. It is inconsistent with the conservation of charge for dynamic fields.
Maxwell resolved this by postulating that a changing electric flux creates a magnetic field. This effect is captured by the displacement current, $I_{d} = ϵ_{0} \frac{d Φ _{E}}{d t}$ .
The complete law, the Ampère-Maxwell Law, is given in integral form by $\oint B \cdot d l = μ_{0} (I_{enc} + ϵ_{0} \frac{d Φ _{E}}{d t})$ and in differential form by $\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t}$ .
The displacement current is not a flow of charge, but it acts as a source of magnetic field and ensures the continuity of "current" through elements like capacitors.

Preview of the Next Lesson:
We have now derived all four of the fundamental equations of electromagnetism. In the next lesson, we will assemble and interpret the full set of Maxwell's equations in their integral and differential forms. This set of equations provides a complete description of all classical electromagnetic phenomena and leads to one of the most profound predictions in the history of science: the existence of electromagnetic waves.

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