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Fresnel equations 2026-10-14

The Fresnel equations give reflection and transmission amplitudes for both transverse-electric and transverse-magnetic polarization of an electromagnetic wave at a planar interface. They follow by applying interface conditions for electromagnetic fields to incident, reflected and refracted plane electromagnetic waves.

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 36 / 1 / a / Solution Created 2026-10-03 Updated 2026-10-14

Use the real magnetic field

B

in the coupled equations; its leading periodic approximation will be

Re (B e^{iω t})

. In the conducting incompressible flow, the magnetohydrodynamic momentum equation, resistive induction equation and solenoidal magnetic-field constraint are

ρ [\partial_{t} u + (u \cdot \nabla) u] \partial_{t} B \nabla \cdot u = - \nabla p + ρ ν \nabla^{2} u + j \times B, = \nabla \times (u \times B) + η \nabla^{2} B, = 0, \nabla \cdot B = 0, j = μ_{0}^{- 1} \nabla \times B .

Here

p

j

is electric current density, and

η

and

ν

are magnetic diffusivity and kinematic viscosity. The moving-conductor Ohm law gives

E = - u \times B + η \nabla \times B

.

In the insulating exterior, neglecting displacement current, the magnetic field is curl-free away from the wire and solenoidal. The imposed electric current density supplies

\nabla \times B_{ext} = μ_{0} J cos (ω t) δ (x) δ (y + b) e_{z}, \nabla \cdot B_{ext} = 0.

The exterior electric field also satisfies Faraday's law. At the rigid wall, impose the no-slip boundary condition

u = 0

. At finite conductivity, the interface conditions for electromagnetic fields give continuous normal magnetic field, continuous tangential magnetic field for equal permeability and no prescribed singular surface current, and continuous tangential electric field. The normal electric current density is zero at an insulating wall; it is automatic for the present two-dimensional fields with current only along

z

. The induced fields decay far from the wire and wall, and initial conditions, or selection of the periodic state after transients, complete the specification. In the limiting perfect conductor approximation a surface current can emerge, permitting a tangential-field jump.

The fully coupled solution need not remain monochromatic: a harmonic magnetic field produces both a mean and a twice-frequency Lorentz force density, and the resulting velocity can generate further harmonics. The following parts consistently use the leading magnetic response with fluid motion neglected.

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