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Shielda

Debye Shielding Physics

\[\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^{2}}}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Shielda glows with charge, but a swarm of opposite charges mutes her beyond arm's length.

The Physics

Put a charge in a plasma and opposite charges crowd round it, hiding it from far away. Its potential falls off exponentially over the Debye length. A gas counts as a plasma only if it is much bigger than the Debye length and holds many particles in a Debye sphere.

λ_D = √(ε₀ k_B T_e / (n_e e²)), φ(r) = q e^(−r/λ_D) / (4πε₀ r), N_D = (4/3)π n_e λ_D³ ≫ 1

  • λ_D: Debye length (m)
  • ε₀: vacuum permittivity, 8.8541878128 × 10⁻¹² F/m
  • k_B: Boltzmann constant (J/K)
  • T_e: electron temperature (K)
  • n_e: electron number density (m⁻³)
  • e: elementary charge, 1.602176634 × 10⁻¹⁹ C (exact)
  • φ: electric potential (V)
  • q: test charge (C)
  • r: distance (m)
  • N_D: electrons in a Debye sphere

In the solar corona (n_e ~ 10¹⁵ per m³, T_e ~ 100 eV), λ_D ≈ 2 mm. Peter Debye and Erich Hückel worked out the same screening in salt solutions in 1923.

On the shirt

Front: Shielda, the pixel-art character, with the title. Back: the Debye length, the screened potential, the plasma criterion, the solar-corona example and the 1923 electrolyte credit. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Ölçek için: M beden 20 in (50.80 cm) genişliğindedir, yaklaşık 3.143 × 1034 Planck uzunluğu (lP = 1.616255 × 10−35 m). Mezura daha hızlıdır.

Measurements in inches, ±1.5 in tolerance.

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Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Put a charge in a plasma and I crowd round it. Opposite charges gather close and hide it from anyone far away.

The hiding happens over the Debye length, λ_D = √(ε₀k_BT_e/(n_e e²)). Beyond it the charge's potential dies off exponentially. A hot, thin gas only behaves as a plasma if it is much bigger than λ_D and holds many electrons inside a Debye sphere.

In the solar corona, with about 10¹⁵ electrons per cubic metre at about 100 eV, λ_D is about 2 mm. Peter Debye and Erich Hückel worked out the same screening for ions in salt solutions in 1923.

Equations

\[\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^{2}}}\]
\[\phi(r) = \frac{q\,e^{-r/\lambda_D}}{4\pi\varepsilon_0 r}\]
\[N_D = \frac{4}{3}\pi n_e \lambda_D^{3} \gg 1\]
SymbolMeaningUnit
\(\lambda_D\) Debye length \(\mathrm{m}\)
\(\varepsilon_0\) vacuum permittivity, 8.854 187 8 × 10⁻¹² F/m \(\mathrm{F/m}\)
\(k_B\) Boltzmann constant, 1.380 649 × 10⁻²³ J/K (exact) \(\mathrm{J/K}\)
\(T_e\) electron temperature (1 eV ≈ 11 604.5 K) \(\mathrm{K}\)
\(n_e\) electron number density \(\mathrm{m⁻³}\)
\(e\) elementary charge, 1.602 176 634 × 10⁻¹⁹ C (exact) \(\mathrm{C}\)
\(\phi\) electric potential \(\mathrm{V}\)
\(q\) test charge \(\mathrm{C}\)
\(r\) distance from the charge \(\mathrm{m}\)
\(N_D\) number of electrons in a Debye sphere \(\mathrm{1}\)

Sources

  1. Chen (2016) Introduction to Plasma Physics and Controlled Fusion, 3rd ed., Springer, sec. 1.4
  2. Bittencourt (2004) Fundamentals of Plasma Physics, 3rd ed., Springer, ch. 1

Field notes

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