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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. 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Hold the magnet still and move the wire instead, at the same relative speed, and exactly the same current flows. Yet the textbook stories differ. If the magnet moves, a changing magnetic field makes an electric field that drives the charges. If the wire moves, there is no electric field in the magnet's frame, and the magnetic force on the moving charges does the job. A rod of length L moving at speed v across a field B gets the same emf either way.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eε = vBL\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eε: induced emf (volt, V)\u003c\/li\u003e\n\u003cli\u003ev: relative speed of rod and magnet, perpendicular to the field (m\/s)\u003c\/li\u003e\n\u003cli\u003eB: magnetic flux density (tesla, T)\u003c\/li\u003e\n\u003cli\u003eL: length of the rod across the field (m)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eEinstein began “Zur Elektrodynamik bewegter Körper” (Annalen der Physik 17, 891, 1905) with this asymmetry: the observed effect depends only on the relative motion, but the theory of the day explained the two cases in different ways. Special relativity removes the asymmetry: electric and magnetic fields are two views of one electromagnetic field, and how they split depends on the observer's frame.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e ε = vBL is the simple case of a straight rod moving perpendicular to a uniform field at speeds much less than light. It is not two different effects: only the description changes from frame to frame, not the current you measure.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eMagnetoconda twists through a copper ring as a cyan-white loop of current glows in his hand. Printed text: \u003cstrong\u003eMAGNETOCONDA\u003c\/strong\u003e with the line \u003cstrong\u003eε = vBL\u003c\/strong\u003e. 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When the magnet was switched on, the plane of polarisation turned. It was the first experimental link between light and magnetism. He reported it in the Nineteenth Series of his Experimental Researches in Electricity (Phil. Trans. R. Soc. 136, 1, 1846). The rotation angle grows with the field and with the path length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eβ = VBd\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eβ: rotation angle of the plane of polarisation (rad)\u003c\/li\u003e\n\u003cli\u003eV: Verdet constant of the material (rad T⁻¹ m⁻¹)\u003c\/li\u003e\n\u003cli\u003eB: magnetic flux density along the beam (tesla, T)\u003c\/li\u003e\n\u003cli\u003ed: path length through the material (m)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eFaraday rotation is non-reciprocal: the rotation direction is set by the field, not by the beam's direction of travel. So light sent back through the glass rotates further, not back to where it started. 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The photon frees an electron from the photocathode. It is then accelerated onto the first dynode, where it knocks out several secondary electrons. Each of those is accelerated onto the next dynode, and so on. With δ secondaries per electron at each of N stages, the gain multiplies.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = δ\u003csup\u003eN\u003c\/sup\u003e, so ln G = N ln δ\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eG: overall electron gain\u003c\/li\u003e\n\u003cli\u003eδ: secondary-emission ratio per dynode\u003c\/li\u003e\n\u003cli\u003eN: number of dynodes\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eWith δ = 4 and N = 10, G = 4¹⁰ = 1,048,576, about 10⁶ electrons from one. 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