• Saddlebane – Earnshaw’s Theorem physics T-shirt
  • Saddlebane – Earnshaw’s Theorem physics T-shirt
  • Saddlebane – Earnshaw’s Theorem physics T-shirt
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SaddlebaneEarnshaw’s Theorem

Saddlebane – Earnshaw’s Theorem Physics T-Shirt

$29.99 USD
$29.99 USD
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Shipping calculated at checkout. Free US shipping · $10 flat shipping outside the US.
Color
White · Black
Size
S–5XL
Made
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Can static charges trap another charge in a stable spot? Earnshaw’s theorem says no. In empty space the electrostatic potential has no minimum, only saddles, so a charge balanced at one point always has a direction to slip away. That is why ion traps use oscillating fields and levitation needs diamagnets, superconductors or feedback. Pixel-art unisex tee; equation, symbols and sources in the physics panel below.

Color
Size

Size guideS–5XL · inches

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

Measurements in inches, ±1.5 in tolerance.

ShippingFree US

Free US shipping · $10 flat shipping outside the US. Printed to order for you.

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

Try to hold a charge or a magnet still in mid-air using only fixed charges and permanent magnets. Somewhere it will always slip out.

In empty space the potential obeys Laplace's equation, so its curvatures in three directions add to zero. If it curves up one way it must curve down another. Every equilibrium is a saddle, never a bowl. Samuel Earnshaw proved it in 1842.

The loopholes are real: diamagnets, superconductors, feedback and spinning tops can levitate. Saddlebane gets none of them. Balanced. Never stable.

Equations

\[\nabla^{2}\Phi = \frac{\partial^{2}\Phi}{\partial x^{2}} + \frac{\partial^{2}\Phi}{\partial y^{2}} + \frac{\partial^{2}\Phi}{\partial z^{2}} = 0\quad\text{(charge-free region)}\]
\[U = q\Phi\;\Rightarrow\;\text{no point where all three curvatures of } U \text{ are positive}\]
SymbolMeaningUnit
\(\Phi\) electrostatic potential (or magnetic scalar potential) from fixed sources \(\mathrm{V}\)
\(x, y, z\) Cartesian coordinates \(\mathrm{m}\)
\(U\) potential energy of the test charge \(\mathrm{J}\)
\(q\) charge of the test particle \(\mathrm{C}\)

Sources

  1. Feynman Lectures on Physics Vol. II Ch. 5: Application of Gauss' Law (no stable equilibrium in an electrostatic field)
  2. Earnshaw (1842) On the nature of the molecular forces which regulate the constitution of the luminiferous ether, Trans. Camb. Phil. Soc. 7, 97 (citation only)
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