{"product_id":"capacitwin-two-capacitor-paradox","title":"Capacitwin – The Two-Capacitor Paradox \/ Half the Energy Lost Physics Sticker","description":"\u003cp\u003eCapacitwin, twin jar-sprites of the Paradox Parlour, share one charge between them, and half of the stored energy vanishes in a spark.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eCharge one capacitor C to voltage V and join it to an identical empty one. Charge is conserved, so each ends at V\/2 with half the charge. But the stored energy is now Q²\/4C instead of Q²\/2C: half the energy is gone. This holds however small the wire resistance is.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔE = ½ · C\u003csub\u003e1\u003c\/sub\u003eC\u003csub\u003e2\u003c\/sub\u003e\/(C\u003csub\u003e1\u003c\/sub\u003e + C\u003csub\u003e2\u003c\/sub\u003e) · (V\u003csub\u003e1\u003c\/sub\u003e − V\u003csub\u003e2\u003c\/sub\u003e)²\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eΔE: energy lost when the capacitors are joined (J)\u003c\/li\u003e\n\u003cli\u003eC\u003csub\u003e1\u003c\/sub\u003e, C\u003csub\u003e2\u003c\/sub\u003e: capacitances (F)\u003c\/li\u003e\n\u003cli\u003eV\u003csub\u003e1\u003c\/sub\u003e, V\u003csub\u003e2\u003c\/sub\u003e: starting voltages (V)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eFor equal capacitors with one empty (C\u003csub\u003e1\u003c\/sub\u003e = C\u003csub\u003e2\u003c\/sub\u003e, V\u003csub\u003e2\u003c\/sub\u003e = 0), ΔE is exactly half the starting energy, which is what the sticker means by “E LOST = ½”. For unequal capacitors the fraction lost is C\u003csub\u003e2\u003c\/sub\u003e\/(C\u003csub\u003e1\u003c\/sub\u003e + C\u003csub\u003e2\u003c\/sub\u003e).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Energy is not destroyed. With resistance R it goes into heat, ∫I²R dt, and the total is independent of R. With R → 0 the lumped circuit model breaks down: the inductance of the loop makes the charge oscillate, and the energy leaves as electromagnetic radiation and is lost in any small resistance. The paradox only appears if you ignore the wires.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eThe twins sit on their two jars with a spark crackling between them. Printed text: \u003cstrong\u003eCAPACITWIN\u003c\/strong\u003e with the line \u003cstrong\u003eE LOST = ½\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50963719028839,"sku":"26896950766285449381","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50963719061607,"sku":"19206169164307717152","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50963719094375,"sku":"32165941973271459840","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50963719127143,"sku":"26313106671113623664","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/13682892903039825191_1200.jpg?v=1793666584","url":"https:\/\/pixelatedphysics.com\/ko\/products\/capacitwin-two-capacitor-paradox","provider":"Pixelated Physics","version":"1.0","type":"link"}