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The source-free Maxwell equations in a linear medium are

\nabla \cdot D = 0, \nabla \cdot B = 0, \nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times H = \frac{\partial D}{\partial t} .

Here the constitutive relations are

D = ε E

and

B = μ H

Substitution of the sinusoidal plane waves into the two divergence equations gives the transversality of an electromagnetic plane wave,

k \cdot E_{0} = 0, k \cdot B_{0} = 0.

(2)

Faraday's law and the Ampère-Maxwell equation give, respectively,

k \times E_{0} = ω B_{0}, k \times B_{0} = - μ ε ω E_{0} .

(3)

Taking

k \times

the first equation and using transversality yields

- k^{2} E_{0} = - μ ε ω^{2} E_{0} .

(4)

Thus a nonzero plane electromagnetic wave in a linear medium must satisfy

k^{2} = μ ε ω^{2}, B_{0} = \frac{k \times E _{0}}{ω}, v_{ph} = \frac{ω}{k} = \frac{1}{μ ε} .

(5)

In particular,

E_{0}

B_{0}

, and

k

are mutually orthogonal, with

∣ B_{0} ∣ = μ ε ∣ E_{0} ∣

Take the interface to be

x = 0

and the plane of incidence to be the

x y

-plane. The incident, reflected, and transmitted wavevectors may be written

k_{I} k_{R} k_{T} = k_{-} (cos θ_{I}, sin θ_{I}, 0), = k_{-} (- cos θ_{R}, sin θ_{R}, 0), = k_{+} (cos θ_{T}, sin θ_{T}, 0) .

(6)

The phase matching at a planar wave interface requires one common angular frequency and one common tangential wavevector:

k_{-} sin θ_{I} = k_{-} sin θ_{R} = k_{+} sin θ_{T} .

(7)

Since

k_{\pm} = ω ε_{\pm} μ = n_{\pm} ω / c

, this gives the Snell law for electromagnetic waves,

θ_{R} = θ_{I}, n_{-} sin θ_{I} = n_{+} sin θ_{T} .

(8)

For the final polarization, take the signed electric amplitudes along

z

. The dielectric-interface boundary conditions require continuity of tangential

E

and

H

. Since

H = \frac{k \times E}{μ ω},

(9)

their

z

- and

y

-components give

E_{I} + E_{R} = E_{T},

(10)

and

k_{-} cos θ_{I} (E_{I} - E_{R}) = k_{+} cos θ_{T} E_{T} .

(11)

Eliminating

E_{I}

yields

\frac{E _{R}}{E _{T}} = \frac{1}{2} (1 - \frac{k _{+} c o s θ _{T}}{k _{-} c o s θ _{I}}) .

(12)

Using

θ_{R} = θ_{I}

and Snell's law,

\frac{k _{+} c o s θ _{T}}{k _{-} c o s θ _{I}} = \frac{s i n θ _{R} c o s θ _{T}}{s i n θ _{T} c o s θ _{R}} = \frac{t a n θ _{R}}{t a n θ _{T}} .

(13)

Therefore the reflected-to-transmitted transverse-electric amplitude ratio is

\frac{∣ E _{R} ∣}{∣ E _{T} ∣} = \frac{1}{2} (1 - \frac{tan θ _{R}}{tan θ _{T}}),

(14)

with the polarization directions chosen so that the displayed signed factor is nonnegative; otherwise its absolute value gives the magnitude ratio.

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 36D / a / Solution Created 2026-09-24 Updated 2026-09-29

In the absence of free electric charge and electric current, the source-free Maxwell equations in a linear medium are

\nabla \cdot D = 0, \nabla \cdot B = 0, \nabla \times E = - \partial_{t} B, \nabla \times H = \partial_{t} D,

together with the constitutive relations

D = ε E

and

B = μ H

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 36D / b / ii / Solution Created 2026-09-24 Updated 2026-09-29

For each

X \in {I, R, T}

, substitution of a complex plane electromagnetic wave in a linear medium into the source-free Maxwell equations in a linear medium requires

k_{X} \cdot E_{X} = 0, ∣ k_{X} ∣^{2} = μ ε_{X} ω_{X}^{2},

where

ε_{I} = ε_{R} = ε_{-}

and

ε_{T} = ε_{+}

. The accompanying magnetic field is

B (x, t) = ⎩ ⎨ ⎧ X = I, R \sum Im [\frac{k _{X} \times E _{X}}{ω _{X}} e^{i k_{X} \cdot x - i ω_{X} t}], Im [\frac{k _{T} \times E _{T}}{ω _{T}} e^{i k_{T} \cdot x - i ω_{T} t}], x < 0, x > 0.

(2)

The incident and transmitted waves have

k_{I x} > 0

and

k_{T x} > 0

, while the reflected wave has

k_{R x} < 0

. Matching the fields at the stationary interface additionally imposes the phase matching at a planar wave interface conditions

ω_{I} = ω_{R} = ω_{T}

and equality of all tangential wavevector components.

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 36B / a / Solution Created 2026-09-24 Updated 2026-09-25

The source-free Maxwell equations in a linear medium are

\nabla \cdot D = 0, \nabla \cdot B = 0,

\nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times H = \frac{\partial D}{\partial t},

(2)

with

D = ϵ E

and

B = μ H

Take complex plane waves

E = E_{0} e^{i (k \cdot x - ω t)}, B = B_{0} e^{i (k \cdot x - ω t)} .

(3)

The divergence equations give

k \cdot E_{0} = k \cdot B_{0} = 0,

(4)

and Faraday's law gives

B_{0} = \frac{1}{ω} k \times E_{0} .

(5)

Ampère's law gives

k \times \frac{B _{0}}{μ} = - ω ϵ E_{0} .

(6)

Substituting the expression for

B_{0}

and using

k \times (k \times E_{0}) = - ∣ k ∣^{2} E_{0}

yields

\frac{∣ k ∣ ^{2}}{μ ω} E_{0} = ω ϵ E_{0} .

(7)

Hence the plane electromagnetic wave in a linear medium has

ω^{2} = \frac{∣ k ∣ ^{2}}{ϵ μ}, v = \frac{ω}{∣ k ∣} = \frac{1}{ϵ μ},

(8)

and equivalently

B_{0} = ϵ μ k \times E_{0} .

(9)