{"product_id":"aharonov-casher-effect-physics-t-shirt","title":"Aharonov–Casher Effect – Topological Phase Physics T-Shirt","description":"\u003cp\u003eNo electric charge. Still a quantum phase.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn 1984 Yakir Aharonov and Aharon Casher showed that a neutral particle carrying a magnetic moment acquires a quantum phase when it circles a line of electric charge. It is the electromagnetic dual of the Aharonov–Bohm effect: magnetic moment around charge, rather than charge around magnetic flux. Neutron interferometry first observed it in 1989.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e|φ_AC| = |(1\/(ħc²)) ∮ (E × μ) · dl|.\u003c\/strong\u003e\u003c\/p\u003e\u003cp\u003eFor a magnetic moment parallel to the charge line: \u003cstrong\u003e|φ_AC| = |μλ|\/(ħc²ε₀).\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eφ_AC: Aharonov–Casher phase (rad).\u003c\/li\u003e\n\u003cli\u003eμ: particle magnetic dipole moment (J T⁻¹).\u003c\/li\u003e\n\u003cli\u003eE: electric field (V m⁻¹).\u003c\/li\u003e\n\u003cli\u003edl: path element (m); ∮ denotes a closed-loop integral.\u003c\/li\u003e\n\u003cli\u003eλ: linear charge density (C m⁻¹).\u003c\/li\u003e\n\u003cli\u003eħ: reduced Planck constant (J s).\u003c\/li\u003e\n\u003cli\u003ec: speed of light (m s⁻¹).\u003c\/li\u003e\n\u003cli\u003eε₀: vacuum permittivity (F m⁻¹).\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e No classical force acts in the ideal configuration. The phase is strictly topological and independent of path shape when the magnetic moment is parallel to the line charge. The overall sign depends on the orientation convention; the equations above give the magnitude. The printed equation uses the approved E × μ convention in SI.\u003c\/p\u003e\u003ch3\u003eSources\u003c\/h3\u003e\u003cul\u003e\n\u003cli\u003eY. Aharonov \u0026amp; A. Casher, “Topological Quantum Effects for Neutral Particles,” Phys. Rev. Lett. 53, 319–321 (1984), doi:10.1103\/PhysRevLett.53.319.\u003c\/li\u003e\n\u003cli\u003eA. Cimmino et al., “Observation of the topological Aharonov–Casher phase shift by neutron interferometry,” Phys. Rev. Lett. 63, 380–383 (1989), doi:10.1103\/PhysRevLett.63.380.\u003c\/li\u003e\n\u003c\/ul\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eApproved physics print block:\u003c\/p\u003e\u003cp\u003e“AHARONOV–CASHER EFFECT”\u003c\/p\u003e\u003cp\u003e“φ = (1\/ħc²) ∮ (E × μ) · dl”\u003c\/p\u003e\u003cp\u003e“μ in J\/T · E in V\/m · φ in rad”\u003c\/p\u003e\u003cp\u003e“A neutral particle with a magnetic moment, circling a line of electric charge, picks up a quantum phase — the mirror image of the Aharonov–Bohm effect.”\u003c\/p\u003e\u003cp\u003ePrinted front and back. Physics. Personified.\u003c\/p\u003e","brand":"Pixelated Physics","offers":[{"title":"S \/ White","offer_id":51242842554471,"sku":"11075349135266546477","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":51242842587239,"sku":"23057787332600506573","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":51242842620007,"sku":"48383859304835443370","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":51242842652775,"sku":"12361849164810313246","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":51242842685543,"sku":"20377137835017112066","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":51242842718311,"sku":"91505629225109496944","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":51242842751079,"sku":"50822881128765908004","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":51242842783847,"sku":"14090653922205575286","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":51242842816615,"sku":"18786098850058726891","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":51242842849383,"sku":"20090753183648790179","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":51242842882151,"sku":"17249877112039915158","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":51242842914919,"sku":"28013681821440932599","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":51242842947687,"sku":"17046347756525426143","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":51242842980455,"sku":"24352279649415480616","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":51242843013223,"sku":"31676409887850223942","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":51242843045991,"sku":"20973168820394188660","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/aharonov-casher-effect-physics-t-shirt_front_mc28_black.png?v=1793975497","url":"https:\/\/pixelatedphysics.com\/fr\/products\/aharonov-casher-effect-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}