# Maxwell's Equations

Physics I · Principles of Electromagnetism and Optics · https://tryals.app/learn/physics-i/maxwells-equations

Four equations govern all electrical, magnetic, and optical phenomena, revealing that light **is** electromagnetism.

| Equation | Name & Meaning |
| --- | --- |
| $\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}$ | **Gauss's Law**: Charges create electric fields. |
| $\nabla\cdot\vec{B} = 0$ | **No Monopoles**: Magnetic field lines form closed loops. |
| $\nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t}$ | **Faraday's Law**: Changing magnetic fields induce electric fields. |
| $\nabla\times\vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$ | **Ampere-Maxwell**: Currents and changing electric fields create magnetic fields. |

**Displacement current** ($\varepsilon_0\frac{\partial\vec{E}}{\\partial t}$) was added by Maxwell to Ampere's law, making wave propagation possible in vacuum.

## Electromagnetic Waves

Maxwell's equations predict self-sustaining waves travelling at $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00\times 10^8\;\text{m/s}$.

| Property | Expression |
| --- | --- |
| Speed | $c = f\lambda$ |
| $E/B$ ratio | $E = cB$ |
| Energy density | $u = \varepsilon_0 E^{2} = B^{2}/\mu_0$ |
| Poynting vector | $\vec{S} = \frac{1}{\mu_0}\vec{E}\times\vec{B}$ |

**Common pitfall**: An electromagnetic wave needs no medium. The $\vec{E}$ and $\vec{B}$ fields are **in phase**, mutually perpendicular, and both perpendicular to travel direction. Do not draw them out of phase.

## Practice questions

7 of this lesson's 15 practice questions, with answers. The full set is in the app.

### 1. Sort each Maxwell equation by whether it involves divergence or curl.

**Answer:**

- Divergence (∇$\cdot)$: $\nabla \cdot E$ = $\rho$/$\varepsilon_{0}$, $\nabla \cdot B$ = 0, Gauss's law for E
- Curl (∇$\times)$: $\nabla \times E$ = −$\partial B/\partial t$, $\nabla \times B$ = $\mu_{0}J$ + $\mu_{0} \varepsilon_{0}$ $\partial E/\partial t$, Faraday's law

**Why:** Two equations use divergence (flux laws); two use curl (circulation laws).

Page: https://tryals.app/practice/physics-i/maxwells-equations/sort-each-maxwell-equation-by-whether-it-involves-divergence-or-curl

### 2. Match each of Maxwell’s equations to the physical statement it encodes.

**Answer:**

- Gauss’s law for $E$ → Charges are sources of electric field
- Gauss’s law for $B$ → There are no magnetic monopoles
- Faraday’s law → Changing $B$ creates a circulating $E$
- Ampère–Maxwell law → Currents and changing $E$ create $B$

**Why:** Read together, the two circulation laws form a self-sustaining loop: a changing $B$ makes an $E$, whose change makes a $B$…, an electromagnetic wave. Maxwell’s addition of the "changing $E$" term is what turned four laws of circuits and magnets into a theory of light.

Page: https://tryals.app/practice/physics-i/maxwells-equations/match-each-of-maxwells-equations-to-the-physical-statement-it

### 3. Maxwell added the displacement current term to Ampère’s law, and light fell out of the equations. Why was that term the decisive addition?

A. Because it made the four equations symmetric in appearance
B. Because it corrected an arithmetic error in the original law
C. Because a changing electric field can then create a magnetic one
D. Because it introduced the constant $c$ into the equations for the first time

**Answer:** C. Because a changing electric field can then create a magnetic one

**Why:** Without the term, a field in a vacuum has nothing to regenerate it. With it, each field feeds the other, and the propagation speed computes to the measured speed of light from two electrostatic constants.

Page: https://tryals.app/practice/physics-i/maxwells-equations/maxwell-added-the-displacement-current-term-to-amperes-law-and

### 4. Order the historical development of electromagnetism leading to Maxwell's equations.

**Answer:**

1. Coulomb's law for electric force (1785)
2. Ampere's law for magnetic force from currents (1826)
3. Faraday's law of induction (1831)
4. Maxwell adds displacement current (1865)
5. Hertz detects electromagnetic waves (1887)

**Why:** Maxwell unified electricity and magnetism; Hertz confirmed EM waves.

Page: https://tryals.app/practice/physics-i/maxwells-equations/order-the-historical-development-of-electromagnetism-leading-to

### 5. Light crosses the 150 million km from the Sun to Earth. **Estimate** how many minutes that takes.

**Answer:** 8.3 (within ±1.5)

**Why:** About 500 s ≈ 8.3 minutes: you always see the Sun as it was eight minutes ago. Maxwell’s equations predicted this finite speed from two lab-bench constants, $\varepsilon_0$ and $\mu_0$, the moment physics realized light **is** electromagnetism.

Page: https://tryals.app/practice/physics-i/maxwells-equations/light-crosses-the-150-million-km-from-the-sun-to-earth-estimate-how

### 6. Order the reasoning by which Maxwell’s equations predict electromagnetic waves.

**Answer:**

1. In empty space there are no charges or currents
2. A changing magnetic field induces an electric field (Faraday)
3. A changing electric field induces a magnetic field (Maxwell’s term)
4. Each field regenerates the other in a self-sustaining cycle
5. The cycle propagates at speed $c = 1/\sqrt{\mu_0\varepsilon_0}$

**Why:** With sources removed, the two circulation laws chase each other’s tails: $\partial B/\partial t \to E$, $\partial E/\partial t \to B$. The propagation speed computed from the constants matched the measured speed of light, one of the great unifications in science.

Page: https://tryals.app/practice/physics-i/maxwells-equations/order-the-reasoning-by-which-maxwells-equations-predict

### 7. Complete the following passage by choosing the correct terms from the word bank.

**Answer:** Maxwell added the **displacement** current term to Ampere's law, completing the set of four equations that unify electricity and **magnetism**.

**Why:** The displacement current J_$d = \varepsilon_{0}$ $\partial E/\partial t$ completes Ampere-Maxwell law.

Page: https://tryals.app/practice/physics-i/maxwells-equations/complete-the-following-passage-by-choosing-the-correct-terms-from-the