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Electromagnetic Fields

Plane waves, the Helmholtz equation and the skin effect

Sinusoidal steady state and complex permittivity, the Helmholtz equation with proof, the plane wave and wave impedance, power, lossy materials and skin depth.

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Maxwell’s equations admit solutions that detach from their sources and travel on their own: electromagnetic waves. The simplest and most useful case is the plane wave, which approximates any wave well far from the source that generated it. In this lesson we derive the wave equation in sinusoidal steady state, the properties of the plane wave, the wave impedance and propagation in lossy materials, with the skin effect.

Sinusoidal steady state

As in circuits, with sinusoidal sources at angular frequency \omega phasors are used: each field becomes a complex vector that is a function of position only, and the time derivative becomes multiplication by j\omega. The curl equations, in a source-free material, become

    \[ \nabla\times\mathbf{E} = -j\omega\mu\mathbf{H},\qquad \nabla\times\mathbf{H} = (\sigma + j\omega\varepsilon)\mathbf{E} = j\omega\varepsilon_c\mathbf{E} \]

where \varepsilon_c = \varepsilon - j\sigma/\omega is the complex permittivity: its imaginary part accounts for losses, by conduction or by the friction of dipoles turning around.

The Helmholtz equation

Theorem. In a homogeneous, source-free material the electric field satisfies

    \[ \nabla^2\mathbf{E} + k^2\mathbf{E} = 0,\qquad k^2 = \omega^2\mu\varepsilon_c \]

Proof. Take the curl of the first equation and substitute the second:

    \[ \nabla\times\nabla\times\mathbf{E} = -j\omega\mu\,\nabla\times\mathbf{H} = \omega^2\mu\varepsilon_c\mathbf{E} \]

By the identity \nabla\times\nabla\times\mathbf{E} = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E}, and since without charges \nabla\cdot\mathbf{E} = 0, we get -\nabla^2\mathbf{E} = k^2\mathbf{E}. ∎

The plane wave

Look for a solution that depends only on z, with the electric field along x. The equation becomes \frac{d^2E_x}{dz^2} + k^2E_x = 0, with solution

    \[ E_x(z) = E_0\,e^{-jkz} \]

a wave travelling towards +z (in time, E_0\cos(\omega t - kz)). The first Maxwell equation gives the magnetic field: H_y = E_x/\eta.

Plane wave: electric and magnetic fields perpendicular to each other and to the direction of propagation
The electric field oscillates along x, the magnetic field along y, in phase with each other; the wave propagates along z, in the direction of E × H. The distance between two crests is the wavelength.

The properties, in a lossless material:

  • phase velocity v = \omega/k = 1/\sqrt{\mu\varepsilon}; in vacuum c \approx 3\cdot 10^8 m/s, in a material c/n with n = \sqrt{\mu_r\varepsilon_r};
  • wavelength \lambda = 2\pi/k = v/f;
  • the wave is transverse: \mathbf{E}, \mathbf{H} and the direction of propagation are mutually perpendicular, and \mathbf{E}\times\mathbf{H} points in the direction of propagation;
  • the ratio of the magnitudes is the wave impedance \eta = \sqrt{\mu/\varepsilon}, which in vacuum is \eta_0 = \sqrt{\mu_0/\varepsilon_0} \approx 377 Ω.

Power. The average Poynting vector is

    \[ \langle S\rangle = \frac{|E_0|^2}{2\eta} \]

Example. Sunlight at ground level carries about 1000 W/m². The corresponding electric field has amplitude E_0 = \sqrt{2\eta_0\cdot 1000} \approx 870 V/m.

Lossy materials

With \sigma \ne 0, k is complex. Writing jk = \gamma = \alpha + j\beta, the wave becomes

    \[ E_x(z) = E_0\,e^{-\alpha z}e^{-j\beta z} \]

a wave that propagates while decaying: \alpha is the attenuation constant (in Np/m), \beta the phase constant.

Good conductor. When \sigma \gg \omega\varepsilon, \varepsilon_c \approx -j\sigma/\omega and k^2 \approx -j\omega\mu\sigma. Since \sqrt{-j} = (1 - j)/\sqrt2,

    \[ \alpha = \beta = \frac{1}{\delta},\qquad \delta = \sqrt{\frac{2}{\omega\mu\sigma}} \]

The field, and with it the current, drops by a factor e every skin depth \delta: this is the skin effect.

Material and frequency\delta
copper, 50 Hz9.3 mm
copper, 1 MHz66 µm
copper, 1 GHz2.1 µm
sea water, 1 kHzabout 8 m

The practical consequences are many. At high frequency current flows only in a thin surface layer and a conductor’s resistance grows as \sqrt f: that is why silver-plated conductors are used, as are stranded insulated wires (Litz wire) in power supply inductors, and why high-voltage line cables can have a steel core for mechanical strength, while the current flows mostly in the outer aluminium. A metal shield a few \delta thick blocks waves: a few micrometres of copper are enough to shield a radio-frequency circuit. Finally, submerged submarines communicate only at very low frequencies, because sea water attenuates everything else within a few metres.

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