# Maxwell’s Equations and Electromagnetic Waves

Physics II · Electromagnetism · https://tryals.app/learn/physics-ii/maxwells-equations-and-electromagnetic-waves

## The Missing Term

Ampère's law as inherited was inconsistent. Apply it to a charging capacitor: a loop round the wire encloses current through one surface, and nothing at all through a surface bulging between the plates. Same loop, two answers.

Maxwell's fix was the **displacement current**, a term proportional to the rate of change of electric flux:

$$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}$$

Between the plates no charge flows, but $\mathbf{E}$ grows, and the new term supplies exactly the missing amount. The four equations then read:

| Equation | Statement |
|---|---|
| $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ | Charge sources the electric field |
| $\nabla\cdot\mathbf{B} = 0$ | No magnetic monopoles |
| $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ | Changing $\mathbf{B}$ drives $\mathbf{E}$ |
| $\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\partial\mathbf{E}/\partial t$ | Current and changing $\mathbf{E}$ drive $\mathbf{B}$ |

The last two are now symmetric, and that symmetry has a consequence. In empty space, taking the curl of the third and substituting the fourth gives a wave equation:

$$\nabla^2\mathbf{E} = \mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}, \qquad c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$$

Putting in the measured constants gives $3.00 \times 10^8$ m/s, the speed of light, arrived at from electrical measurements that had nothing to do with optics. That is what identified light as an electromagnetic wave.

Plane waves are **transverse**, with $\mathbf{E} \perp \mathbf{B} \perp$ propagation, $E = cB$, and energy flow given by the **Poynting vector** $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$.

> **Common pitfall:** calling the displacement current a flow of charge. Nothing moves between the capacitor plates. It is a changing *electric field* that sources a magnetic field exactly as a real current would, the name is historical, and it misleads.

## Practice questions

8 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Why was Ampere’s law inconsistent before Maxwell added the displacement current?

A. Ampère’s law gave different answers for two surfaces sharing one loop
B. Faraday’s law of induction predicted magnetic field lines must terminate
C. The divergence of the electric field was zero inside non-vacuum media
D. It violated conservation of charge for time-dependent electric fields

**Answer:** A. Ampère’s law gave different answers for two surfaces sharing one loop

**Why:** For a charging capacitor, a flat surface catches the wire’s current while a bulging one catches none, yet both share the same boundary loop. The displacement current supplies the missing contribution, since $\mathbf{E}$ is changing between the plates.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/why-was-amperes-law-inconsistent-before-maxwell-added-the

### 2. The displacement current involves an actual flow of charge between capacitor plates.

**Answer:** False

**Why:** False, nothing crosses the gap. The term is a changing *electric field* that sources a magnetic field exactly as a current would. The name is historical and actively misleading.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/the-displacement-current-involves-an-actual-flow-of-charge-between

### 3. An electromagnetic wave in vacuum has an electric field amplitude of $6.0 \times 10^2$ V/m. Set the magnetic field amplitude, in units of $10^{-6}$ T, using $c = 3.0 \times 10^8$ m/s.

**Answer:** 2 (within ±0.3)

**Why:** $B = E/c = 6.0\times10^2/3.0\times10^8 = 2.0\times10^{-6}$ T. The magnetic amplitude is smaller by the enormous factor $c$, which is why the electric force dominates the interaction of light with matter.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/an-electromagnetic-wave-in-vacuum-has-an-electric-field-amplitude-of

### 4. Which statements about electromagnetic plane waves in vacuum are correct?

A. The wave requires a material medium to propagate
B. The electric and magnetic fields are perpendicular to each other
C. The field amplitudes are related by E equals c times B
D. Both fields are perpendicular to the direction of propagation

**Answer:** B. The electric and magnetic fields are perpendicular to each other; C. The field amplitudes are related by E equals c times B; D. Both fields are perpendicular to the direction of propagation

**Why:** The wave is transverse with mutually perpendicular fields and $E = cB$. It needs no medium, the wave equation was derived in vacuum, which is what killed the luminiferous aether.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/which-statements-about-electromagnetic-plane-waves-in-vacuum-are

### 5. Faraday's law shows that a changing magnetic field drives an electric field, whilst Maxwell's addition shows that a changing electric field drives a magnetic field. What fundamental consequence follows from this mutual coupling in completely empty space?

A. A continuous flow of physical charge is induced in the void
B. Both electric and magnetic fields vanish instantly in vacuum
C. Fields propagate as self-sustaining waves without charges
D. Electric flux converts directly into static magnetic poles

**Answer:** C. Fields propagate as self-sustaining waves without charges

**Why:** Mutual induction allows fields to regenerate one another through empty space rather than decaying, creating propagating radiation. Treating displacement current as real charge transport or assuming fields require matter to exist overlooks how Maxwell unifies optics with electromagnetism.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/faradays-law-shows-that-a-changing-magnetic-field-drives-an-electric

### 6. Arrange these steps in the logical order by which Maxwell’s equations predict light.

**Answer:**

1. Add the displacement current to make Ampère’s law consistent
2. Note that a changing E drives B and a changing B drives E
3. Combine the two curl equations into a wave equation
4. Identify the resulting speed with the measured speed of light

**Why:** The displacement current comes first, purely to fix an inconsistency. Its symmetry with Faraday’s law then permits combining the two into a wave equation, whose speed turns out to match light, a prediction, not an assumption.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/arrange-these-steps-in-the-logical-order-by-which-maxwells-equations

### 7. An antenna radiates into space. Which quantity describes how fast that energy crosses unit area?

A. The rate of energy flow per unit area carried by the field
B. The total electrostatic energy density stored in the field
C. The propagation velocity of the transverse wavefront in vacuum
D. The displacement current density generated between the charges

**Answer:** A. The rate of energy flow per unit area carried by the field

**Why:** $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$ is the energy flux (power per unit area), pointing along the direction of wave travel. Energy density is a scalar volume density, while wave speed describes phase propagation rather than energy flux.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/an-antenna-radiates-into-space-which-quantity-describes-how-fast

### 8. Maxwell’s equations predicted the speed of light from constants measured in purely electrical experiments.

**Answer:** True

**Why:** True, $\varepsilon_0$ and $\mu_0$ were fixed by capacitance and current-force measurements, yet $1/\sqrt{\mu_0\varepsilon_0}$ came out at the measured speed of light. This is one of the great unifications in physics.

Page: https://tryals.app/practice/physics-ii/maxwells-equations-and-electromagnetic-waves/maxwells-equations-predicted-the-speed-of-light-from-constants