{"product_id":"recoila-the-abraham-lorentz-force-physics-t-shirt","title":"Recoila – The Abraham–Lorentz Force Physics T-Shirt","description":"\u003cp\u003eRecoila accelerates, radiates, and her own light shoves her back.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eAn accelerating charge radiates, and the radiation carries away energy and momentum, so the charge feels a recoil. In the non-relativistic limit that radiation-reaction force is proportional to the rate of change of acceleration, the jerk.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF_rad = (μ₀ q² \/ 6πc) da\/dt,   P = μ₀ q² a² \/ (6πc)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eF_rad: radiation-reaction force (N)\u003c\/li\u003e\n\u003cli\u003eμ₀: vacuum permeability (N\/A²)\u003c\/li\u003e\n\u003cli\u003eq: charge (C)\u003c\/li\u003e\n\u003cli\u003ec: speed of light, 299 792 458 m\/s (exact)\u003c\/li\u003e\n\u003cli\u003ea: acceleration (m\/s²)\u003c\/li\u003e\n\u003cli\u003eda\/dt: jerk (m\/s³)\u003c\/li\u003e\n\u003cli\u003eP: radiated (Larmor) power, v ≪ c (W)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eFor an electron the time scale is τ = μ₀e² \/ (6π m_e c) ≈ 6.3 × 10⁻²⁴ s. Abraham (1903), Lorentz (1904) and Dirac (1938) built the theory.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Taken literally the equation has runaway and pre-acceleration solutions. Reduced-order (Landau–Lifshitz) forms tame it; a full classical cure is still debated.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Recoila, the pixel-art character, with the title. Back: the radiation-reaction force, the Larmor power, the electron's 6.3 × 10⁻²⁴ s, and the runaway caveat. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960821715047,"sku":"33830937019711223645","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960821747815,"sku":"32438615516890492360","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960821780583,"sku":"20763584119631044684","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960821813351,"sku":"11378838363462409875","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960821846119,"sku":"16307177476877089921","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960821878887,"sku":"22840254369481563874","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960821911655,"sku":"89507053122514928854","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960821944423,"sku":"11951085126016045791","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960821977191,"sku":"26087856108779571405","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960822009959,"sku":"18073479290198948289","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960822042727,"sku":"26685515705675707679","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960822075495,"sku":"97942949276476937788","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960822108263,"sku":"11455445145299593841","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960822141031,"sku":"30719728964191644399","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960822173799,"sku":"20669112141406594875","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960822206567,"sku":"21022372492795052016","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/abraham-lorentz_front_01_white.png?v=1793628460","url":"https:\/\/pixelatedphysics.com\/tr\/products\/recoila-the-abraham-lorentz-force-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}