Unit 5: Maxwell's Equations

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5.1 Gauss's Law for Electric Fields

Maxwell Equation 1: Electric charges are sources of electric fields.

$$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$

Electric field lines start on + charges and end on − charges. Net flux through a closed surface is proportional to enclosed charge. Differential form: $\nabla \cdot \vec{E} = \rho/\varepsilon_0$.

Exam Tip The AP exam may ask you to identify which Maxwell equation applies to a given situation. This one: "net electric flux through closed surface = enclosed charge over epsilon-zero."

5.2 Gauss's Law for Magnetic Fields

Maxwell Equation 2: No magnetic monopoles exist.

$$\oint \vec{B} \cdot d\vec{A} = 0$$

B-field lines are always closed loops — they never start or end. Net B-flux through any closed surface = 0. Cut a bar magnet in half → two smaller magnets, each with N and S. Differential form: $\nabla \cdot \vec{B} = 0$.

Common Mistake Through a closed surface, net B-flux = 0. Through an open surface, B-flux can be non-zero — that's what Faraday's Law describes.

5.3 Faraday's Law

Maxwell Equation 3: Changing B creates a circulating E-field.

$$\oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}$$

Time-varying B induces a non-conservative E-field ($\oint\vec{E}\cdot d\vec{\ell} \neq 0$). This drives current in conductors — the principle behind generators. Differential form: $\nabla \times \vec{E} = -\partial\vec{B}/\partial t$.

Practice

Q5.1: Identity the Maxwell Equation

A time-varying B-field through a loop induces circulating E-field: $\oint\vec{E}\cdot d\vec{\ell} = -d\Phi_B/dt$. Which Maxwell equation is this?

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Faraday's Law (Maxwell Equation 3). Describes induction — changing B produces curling E. Principle of transformers and generators.

5.4 The Ampere-Maxwell Law

Maxwell Equation 4: Currents AND changing E-fields create B-fields.

$$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}} + \mu_0\varepsilon_0 \frac{d\Phi_E}{dt}$$

Maxwell's crucial addition is the displacement current term: $I_d = \varepsilon_0\frac{d\Phi_E}{dt}$. Even between capacitor plates (no moving charges), a changing E acts as an effective current producing B-field.

Without this term, Ampere's law is inconsistent for time-varying scenarios. Differential form: $\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$.

Critical The displacement current was Maxwell's great theoretical contribution. It completes the symmetry: changing E → B (this), changing B → E (Faraday). Together they predict EM waves.

Practice

Q5.2: Displacement Current

Parallel-plate capacitor (radius 5.0 cm) charges with $E(t)=1.0\times10^6t\ \text{V/m·s}$. Find displacement current density and B at r=3.0 cm.

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$J_d = \varepsilon_0 dE/dt = 8.85\times10^{-6}\ \text{A/m}^2$. Total $I_d = J_d\cdot\pi R^2 = 6.95\times10^{-8}\ \text{A}$.

Ampere-Maxwell for r

5.5 Electromagnetic Waves

Maxwell's equations predict self-sustaining EM waves — propagating oscillations of E and B requiring no medium.

Speed of light:

$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00 \times 10^8\ \text{m/s}$$

Plane wave structure:

$\vec{E} \perp \vec{B} \perp \vec{v}$ — mutually perpendicular
E and B are in phase: $E = E_0\sin(kx-\omega t)$, $B = B_0\sin(kx-\omega t)$
Amplitude: $E_0 = cB_0$
Propagation direction: $\vec{E} \times \vec{B}$

Poynting vector = energy flux (W/m²): $\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$

Average intensity: $I = \langle S \rangle = \frac{1}{2}c\varepsilon_0 E_0^2 = \frac{E_0^2}{2\mu_0 c}$

Radiation pressure: $p = I/c$ (perfect absorber), $p = 2I/c$ (perfect reflector).

RMS Values: For sinusoidal waves: $E_{\text{rms}} = E_0/\sqrt{2}$, $B_{\text{rms}} = B_0/\sqrt{2}$. Average intensity: $I = \frac{1}{2}c\varepsilon_0 E_0^2 = c\varepsilon_0 E_{\text{rms}}^2$.

Energy density in EM waves: $u = \varepsilon_0 E^2 = B^2/\mu_0$ (instantaneous). Average: $\langle u \rangle = \frac{1}{2}\varepsilon_0 E_0^2 = \varepsilon_0 E_{\text{rms}}^2$.

$$c = \lambda f = 3.00 \times 10^8\ \text{m/s}$$

Practice

Q5.3: EM Wave Direction

An EM wave has $\vec{E}=100\sin(ky+\omega t)\hat{x}\ \text{V/m}$. Find propagation direction and $\vec{B}(y,t)$.

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Argument $(ky+\omega t)$ → plus sign between k and ω means propagation in −y direction.

$\vec{v} \propto \vec{E}\times\vec{B}$: (−ĵ) ∝ î × B̂ → B must be along +k̂ (î×k̂ = −ĵ ✓).

$B_0 = E_0/c = 100/(3\times10^8) = 3.33\times10^{-7}\ \text{T}$. $\vec{B}=3.33\times10^{-7}\sin(ky+\omega t)\hat{z}\ \text{T}$.

Practice

Q5.4: Intensity and Radiation Pressure

EM wave: $E_0=150\ \text{V/m}$. Find average intensity and radiation pressure on a perfect absorber.

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$I = \frac{1}{2}c\varepsilon_0 E_0^2 = 0.5(3\times10^8)(8.85\times10^{-12})(22500) = 29.9\ \text{W/m}^2$.

$p = I/c = 29.9/(3\times10^8) = 1.0\times10^{-7}\ \text{Pa}$. Very small — why solar sails need large area.

Practice

Q5.5: Maxwell Summary

Match each to its description:
(a) $\oint\vec{E}\cdot d\vec{A}=Q_{\text{enc}}/\varepsilon_0$ — (i) No magnetic monopoles
(b) $\oint\vec{B}\cdot d\vec{A}=0$ — (ii) Changing B makes E
(c) $\oint\vec{E}\cdot d\vec{\ell}=-d\Phi_B/dt$ — (iii) Charges make E
(d) $\oint\vec{B}\cdot d\vec{\ell}=\mu_0 I+\mu_0\varepsilon_0 d\Phi_E/dt$ — (iv) Current + changing E makes B

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(a)→(iii): Gauss's Law for E — charges produce electric fields.
(b)→(i): Gauss's Law for B — no magnetic monopoles; B lines are closed loops.
(c)→(ii): Faraday's Law — changing magnetic flux induces circulating E-field.
(d)→(iv): Ampere-Maxwell Law — conduction current AND displacement current produce B-field.

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