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Field Lines Hoodie

START POSITIVE | Pixelated Physics

\[\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Four charges sit on the corners of a square: plus, minus, plus, minus. Total charge zero. Total dipole moment zero too. What is left is a quadrupole. The exact field of an electric quadrupole, +q −q +q −q. Every line leaves a positive charge, ends on a negative one. Net charge zero. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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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.

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The physics

Every electric charge produces a field pointing away from it if positive, or toward it if negative, weakening as one over distance squared. The total field of several charges is simply the sum of their fields, and here four alternating charges add up to a quadrupole pattern.

Four charges sit on the corners of a square: plus, minus, plus, minus. Total charge zero. Total dipole moment zero too. What is left is a quadrupole.

Every line in the art was traced through the exact field, the plain sum of four Coulomb terms on the back, stepped along E with a Runge–Kutta integrator. Fourteen lines leave each positive charge at equal angles.

Gauss’s law, ∇·E = ρ/ε₀, does the bookkeeping. Field lines begin only on positive charge and end only on negative charge. With the books balanced, no line here needs to go to infinity, though a few leave the frame before they close.

The faint cross is the equipotential V = 0. By symmetry it is exactly the two axes: every point on them is equally far from a plus and a minus of the same size.

From far away, a quadrupole’s field falls off as 1/r⁴, two powers faster than a single charge. Up close, every line starts positive.

Equations

\[\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}\]
\[\mathbf{E}(\mathbf{r}) = \sum_i \frac{k\,q_i\,(\mathbf{r}-\mathbf{r}_i)}{|\mathbf{r}-\mathbf{r}_i|^3},\qquad k = \frac{1}{4\pi\varepsilon_0}\]
SymbolMeaningUnit
\(\mathbf{E}\) electric field \(\mathrm{V/m}\)
\(\rho\) electric charge density \(\mathrm{C/m^3}\)
\(\varepsilon_0\) vacuum permittivity \(\mathrm{F/m}\)
\(k\) Coulomb constant, 1/(4\pi\varepsilon_0) \(\mathrm{N·m^2·C^{-2}}\)
\(q_i\) charge of the i-th point charge (here +q, −q, +q, −q) \(\mathrm{C}\)
\(\mathbf{r}\) point where the field is evaluated \(\mathrm{m}\)
\(\mathbf{r}_i\) position of the i-th charge \(\mathrm{m}\)

Sources

  1. Coulomb (1785) Premier mémoire sur l’électricité et le magnétisme, Histoire de l’Académie Royale des Sciences
  2. Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 1 and Ch. 4: Multipoles
  3. Griffiths, Introduction to Electrodynamics, Ch. 2: Electrostatics
  4. Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics

Source Coulomb, C.-A. (1785). Premier mémoire sur l’électricité et le magnétisme. Histoire de l’Académie Royale des Sciences (année 1785), 569–577.

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