{"product_id":"casimira-casimir-effect-pixel-sticker","title":"Casimira – The Casimir Effect Pixel Sticker","description":"\u003cp\u003eCasimira, an armoured knight-mage, presses two upright plates together with a sliver of gap between them while violet vacuum light crackles all around.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003ePut two uncharged, perfectly conducting parallel plates a small distance a apart in vacuum and they attract. Between the plates only electromagnetic modes that fit the boundary conditions are allowed; outside, every mode is. Summing the zero-point energies ½ħω of the modes, and subtracting the free-space value, leaves a finite energy that falls as the plates approach. Hendrik Casimir derived it in 1948.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP = π²ħc \/ (240 a⁴)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eP: attractive pressure, force per unit plate area (Pa)\u003c\/li\u003e\n\u003cli\u003eħ: reduced Planck constant, h\/2π = 1.054 571 817 × 10⁻³⁴ J s\u003c\/li\u003e\n\u003cli\u003ec: speed of light, 299 792 458 m s⁻¹\u003c\/li\u003e\n\u003cli\u003ea: gap between the plates (m)\u003c\/li\u003e\n\u003cli\u003eπ²\/240: numerical factor from summing the allowed modes\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eAt a = 1 µm, P ≈ 1.3 × 10⁻³ Pa (1.3 mPa); halve the gap and the pressure rises sixteen-fold. S. K. Lamoreaux first measured the force accurately in 1997, using a sphere and a flat plate (\u003cem\u003ePhys. Rev. Lett.\u003c\/em\u003e 78, 5); Bressi and colleagues measured it between true parallel plates in 2002 (\u003cem\u003ePhys. Rev. Lett.\u003c\/em\u003e 88, 041804). Source for the theory: H. B. G. Casimir, \u003cem\u003eProc. K. Ned. Akad. Wet.\u003c\/em\u003e 51, 793 (1948).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The formula is for ideal mirrors at zero temperature; real metals, roughness and temperature change it. Whether the force “proves” vacuum energy is debated: R. L. Jaffe (\u003cem\u003ePhys. Rev. D\u003c\/em\u003e 72, 021301, 2005) showed it can be derived as a relativistic van der Waals force between the plates’ charges, without reference to zero-point energy.\u003c\/p\u003e\u003ch3\u003eIn character\u003c\/h3\u003e\u003cp\u003e\u003cem\u003eLeave me nothing but a gap and I will still push. The emptiness outside is fuller than the emptiness within, and that difference is my strength.\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003ePrinted text: \u003cstrong\u003eCASIMIRA\u003c\/strong\u003e with the line \u003cstrong\u003eP = π²ħc\/(240a⁴)\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker in 2″, 3″, 4″ and 6″ sizes. See more \u003ca href=\"\/ko\/collections\/stickers\"\u003ephysicist stickers\u003c\/a\u003e or \u003ca href=\"\/ko\/collections\/quantum-physics-t-shirts\"\u003equantum designs\u003c\/a\u003e.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50654988730471,"sku":"86020910698195047825","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50586289930343,"sku":"25944383438766048735","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50654988763239,"sku":"19092551843076278650","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50654988796007,"sku":"11409284345474330245","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/17135163981716564855_1200.jpg?v=1793668866","url":"https:\/\/pixelatedphysics.com\/ko\/products\/casimira-casimir-effect-pixel-sticker","provider":"Pixelated Physics","version":"1.0","type":"link"}