[{"data":1,"prerenderedAt":1297},["ShallowReactive",2],{"subject:electricity-and-magnetism":3,"course-wordcounts":75,"nav:electricity-and-magnetism":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F02.electricity-and-magnetism\u002Findex.md","Electricity & Magnetism","How charge produces fields and potential, how circuits store and transfer energy,\nand how coupled electric and magnetic fields propagate as light.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Two fields, one theory. The subject builds from charge and the electric field it\ncreates — through potential and Gauss's law, which turn spatial fields into\ncalculations you can actually do — to capacitance, current, and resistance, where\nthose fields meet circuits. Magnetism enters through moving charge: currents make\nmagnetic fields (Ampère), and changing magnetic fields make electric ones\n(Faraday), the pairing that runs every motor and generator. Maxwell's equations\ngather all of it into four statements and reveal a consequence hiding in plain\nsight — self-propagating electromagnetic waves, which are light. From there the\nnotes reach polarization, the electromagnetic spectrum, and the geometrical optics\nof reflection and refraction. The arc follows Tipler & Mosca, Parts IV–V, each\ntopic resting on the fields defined before it.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,28,30,34,36,38,42,44,48,50,54],{"p":20},"Electricity and magnetism begin with a single quantity — \u003Cstrong>charge\u003C\u002Fstrong> —\nand the field it fills space with. Every force, potential, and current in the\nsubject is bookkeeping on that field and how it changes.\n",{"fig":22,"n":23,"caption":24,"large":25},"em-dipole-field","001","A dipole's field: lines stream out of the positive charge and terminate on\nthe negative one, never crossing.\n",true,{"p":27},"A charge sources an \u003Cstrong>electric field\u003C\u002Fstrong>, and a field exerts a force on\nany charge placed in it. Draw the field as lines — dense where it is strong,\nalways leaving \u003Cem>+\u003C\u002Fem> and landing on \u003Cem>−\u003C\u002Fem> — and most of electrostatics\nbecomes a picture you can read.\n",{"p":29},"Two ideas make fields tractable. \u003Cstrong>Potential\u003C\u002Fstrong> collapses a vector\nfield to a single scalar you can add up; \u003Cstrong>flux\u003C\u002Fstrong>, through Gauss's law,\nturns a surface integral into the charge enclosed. Symmetry then does the rest.\n",{"fig":31,"n":32,"caption":33},"em-capacitor","002","A parallel-plate capacitor: opposite charges on two plates make a uniform\nfield in the gap.\n",{"p":35},"Store charge on conductors and you get \u003Cstrong>capacitance\u003C\u002Fstrong>; let it flow and\nyou get \u003Cstrong>current\u003C\u002Fstrong>, \u003Cstrong>resistance\u003C\u002Fstrong>, and the circuits that move\nenergy from one place to another.\n",{"p":37},"Moving charge is the bridge to magnetism. A current sets up a\n\u003Cstrong>magnetic field\u003C\u002Fstrong> that circulates around it, and Ampère's law ties the\nfield to the current the same way Gauss's law ties \u003Cem>E\u003C\u002Fem> to charge.\n",{"fig":39,"n":40,"caption":41},"em-current-loop","003","A current loop threads a field through its centre — a magnetic dipole, the\natom of magnetism.\n",{"p":43},"The two fields are not independent. A \u003Cstrong>changing magnetic flux\u003C\u002Fstrong> drives an\nelectric field around a loop — Faraday's law of induction — which is how every\ngenerator, transformer, and inductor works.\n",{"fig":45,"n":46,"caption":47},"em-induction","004","Induction: a magnet moving through a coil changes the flux and drives a\ncurrent, opposing the change.\n",{"p":49},"Maxwell added the missing symmetry — a changing electric field is itself a\nsource of magnetic field — and the four equations closed on themselves. A\nfield could now sustain a field, with no charges in sight.\n",{"fig":51,"n":52,"caption":53},"em-wave","005","An electromagnetic wave: \u003Cem>E\u003C\u002Fem> and \u003Cem>B\u003C\u002Fem> oscillate perpendicular and in\nphase, travelling at \u003Cem>c\u003C\u002Fem>.\n",{"p":55},"That self-sustaining ripple is \u003Cstrong>light\u003C\u002Fstrong>. Electromagnetic waves carry\nenergy and momentum, come in every wavelength, and — through reflection and\nrefraction — hand the subject over to optics.\n","physics","Electric and magnetic interactions begin with charge, current, and fields, then\nacquire a unified form in Maxwell's equations. Potential and flux turn spatial\nfields into tractable calculations; capacitance, resistance, and inductance connect\nthem to circuits; induction and displacement current lead to electromagnetic waves,\npolarization, and geometrical optics. Follows Tipler & Mosca, Parts IV–V.\n",false,"md",{},"\u002Felectricity-and-magnetism",[],"---\ntitle: Electricity & Magnetism\nstatus: available\ncategory: physics\nblurb: |\n  How charge produces fields and potential, how circuits store and transfer energy,\n  and how coupled electric and magnetic fields propagate as light.\ndescription: |\n  Electric and magnetic interactions begin with charge, current, and fields, then\n  acquire a unified form in Maxwell's equations. Potential and flux turn spatial\n  fields into tractable calculations; capacitance, resistance, and inductance connect\n  them to circuits; induction and displacement current lead to electromagnetic waves,\n  polarization, and geometrical optics. Follows Tipler & Mosca, Parts IV–V.\nbrief:\n  - p: |\n      Electricity and magnetism begin with a single quantity — \u003Cstrong>charge\u003C\u002Fstrong> —\n      and the field it fills space with. Every force, potential, and current in the\n      subject is bookkeeping on that field and how it changes.\n  - fig: em-dipole-field\n    n: \"001\"\n    caption: |\n      A dipole's field: lines stream out of the positive charge and terminate on\n      the negative one, never crossing.\n    large: true\n  - p: |\n      A charge sources an \u003Cstrong>electric field\u003C\u002Fstrong>, and a field exerts a force on\n      any charge placed in it. Draw the field as lines — dense where it is strong,\n      always leaving \u003Cem>+\u003C\u002Fem> and landing on \u003Cem>−\u003C\u002Fem> — and most of electrostatics\n      becomes a picture you can read.\n  - p: |\n      Two ideas make fields tractable. \u003Cstrong>Potential\u003C\u002Fstrong> collapses a vector\n      field to a single scalar you can add up; \u003Cstrong>flux\u003C\u002Fstrong>, through Gauss's law,\n      turns a surface integral into the charge enclosed. Symmetry then does the rest.\n  - fig: em-capacitor\n    n: \"002\"\n    caption: |\n      A parallel-plate capacitor: opposite charges on two plates make a uniform\n      field in the gap.\n  - p: |\n      Store charge on conductors and you get \u003Cstrong>capacitance\u003C\u002Fstrong>; let it flow and\n      you get \u003Cstrong>current\u003C\u002Fstrong>, \u003Cstrong>resistance\u003C\u002Fstrong>, and the circuits that move\n      energy from one place to another.\n  - p: |\n      Moving charge is the bridge to magnetism. A current sets up a\n      \u003Cstrong>magnetic field\u003C\u002Fstrong> that circulates around it, and Ampère's law ties the\n      field to the current the same way Gauss's law ties \u003Cem>E\u003C\u002Fem> to charge.\n  - fig: em-current-loop\n    n: \"003\"\n    caption: |\n      A current loop threads a field through its centre — a magnetic dipole, the\n      atom of magnetism.\n  - p: |\n      The two fields are not independent. A \u003Cstrong>changing magnetic flux\u003C\u002Fstrong> drives an\n      electric field around a loop — Faraday's law of induction — which is how every\n      generator, transformer, and inductor works.\n  - fig: em-induction\n    n: \"004\"\n    caption: |\n      Induction: a magnet moving through a coil changes the flux and drives a\n      current, opposing the change.\n  - p: |\n      Maxwell added the missing symmetry — a changing electric field is itself a\n      source of magnetic field — and the four equations closed on themselves. A\n      field could now sustain a field, with no charges in sight.\n  - fig: em-wave\n    n: \"005\"\n    caption: |\n      An electromagnetic wave: \u003Cem>E\u003C\u002Fem> and \u003Cem>B\u003C\u002Fem> oscillate perpendicular and in\n      phase, travelling at \u003Cem>c\u003C\u002Fem>.\n  - p: |\n      That self-sustaining ripple is \u003Cstrong>light\u003C\u002Fstrong>. Electromagnetic waves carry\n      energy and momentum, come in every wavelength, and — through reflection and\n      refraction — hand the subject over to optics.\n---\n\nTwo fields, one theory. The subject builds from charge and the electric field it\ncreates — through potential and Gauss's law, which turn spatial fields into\ncalculations you can actually do — to capacitance, current, and resistance, where\nthose fields meet circuits. Magnetism enters through moving charge: currents make\nmagnetic fields (Ampère), and changing magnetic fields make electric ones\n(Faraday), the pairing that runs every motor and generator. Maxwell's equations\ngather all of it into four statements and reveal a consequence hiding in plain\nsight — self-propagating electromagnetic waves, which are light. From there the\nnotes reach polarization, the electromagnetic spectrum, and the geometrical optics\nof reflection and refraction. The arc follows Tipler & Mosca, Parts IV–V, each\ntopic resting on the fields defined before it.\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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Fields",1,"electric-fields",[993,998,1003,1009,1015],{"title":994,"path":995,"lessonNumber":990,"topics":996,"summary":997},"Charge and Conductors","\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors",[989],"Rub two objects together and one pulls electrons from the other; nothing is created, only moved. We define what electric charge is — conserved, additive, and quantized in units of $e$ — and why a conductor's mobile carriers rearrange until its interior field vanishes and its surface sits at one potential. We follow charge through contact, induction, and grounding, treat the field-free cavity that turns a conductor into a shield, and mark where finite conductivity and leakage set the limits of the electrostatic picture.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"Coulomb's Law","\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law",[989],"Two charges at rest push or pull along the line joining them, and the whole of electrostatics is assembled by adding up such pairs. We measure that force — its inverse-square falloff, its linear dependence on each charge, the sign that says attract or repel — and write it as a vector so direction survives superposition. We work the magnitude and component forms on real numbers, check them against limiting cases and dimensions, and fix the point-charge approximation to source sizes small against every separation.\n",{"title":1004,"path":1005,"lessonNumber":1006,"topics":1007,"summary":1008},"Electric Field and Force","\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force",3,[989],"Rather than ask how one charge reaches across empty space to another, we credit the source with a field that fills the space and let a second charge respond to whatever field sits at its own location. Electric field is force per unit positive test charge, $\\vec E=kq\\hat r\u002Fr^2$ for a point source, and source fields add before any receiving charge is placed. We compute those fields and the force $\\vec F=q\\vec E$ they exert, then follow a charge along its parabolic path through a uniform field and into nonuniform fields where the dynamics turn position-dependent.\n",{"title":1010,"path":1011,"lessonNumber":1012,"topics":1013,"summary":1014},"Electric Field Maps","\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps",4,[989],"A field is a vector at every point of space, and the quickest way to grasp one is to draw it. We build the two standard pictures — continuous field lines tangent to $\\vec E$, and scaled vector arrows — and read direction, magnitude, and the location of nulls straight off them. We fix what a line drawing can and cannot say: density encodes magnitude only under a stated seeding rule, and integral curves never cross at a regular point. From there we work the topology near sources, sinks, and conductor surfaces, and state the step-size and interpolation checks a numerical map must pass.\n",{"title":1016,"path":1017,"lessonNumber":1018,"topics":1019,"summary":1020},"Electric Dipoles","\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles",5,[989],"Most neutral matter carries no net charge yet still responds to an electric field, because its positive and negative charge sit slightly apart. That separation is a dipole, moment $\\vec p=q\\vec d$ pointing from the negative to the positive charge, and it is the leading term in how any neutral distribution looks from far away. We derive the torque $\\vec p\\times\\vec E$ and energy $-\\vec p\\cdot\\vec E$ a uniform field imposes, the net force a field gradient adds, and the axial and equatorial $1\u002Fr^3$ fields the pair produces, then measure how far out the point-dipole approximation still holds.\n",{"module":1022,"moduleNumber":16,"slug":1023,"lessons":1024},"Continuous Charge Distributions","continuous-charge-distributions",[1025,1030],{"title":1026,"path":1027,"lessonNumber":990,"topics":1028,"summary":1029},"Continuous Charge Fields","\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields",[1022],"A charged rod, ring, or disk is not a point, yet its field is still nothing but Coulomb's law added up over the charge it carries. We replace the discrete sum by an integral, with $\\d q=\\lambda\\d\\ell$, $\\sigma\\d A$, or $\\rho\\d V$, so the real work becomes geometry: writing the vector from each source element to the field point, and letting symmetry cancel the components that must cancel before any integral is attempted. We carry the line, ring, and disk fields through in full, then check each result against its near field, its far field, and its dimensions.\n",{"title":1031,"path":1032,"lessonNumber":16,"topics":1033,"summary":1034},"Gauss's Law and Conductors","\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors",[1022],"Adding up Coulomb's law over a whole distribution is laborious; Gauss's law trades that sum for a single statement, that the flux of $\\vec E$ out of any closed surface counts the charge inside, $\\oint\\vec E\\cdot\\d\\vec A=Q_{\\rm enc}\u002F\\varepsilon_0$. The law is always true, but it hands over the field only when the source is symmetric enough to pull $E$ outside the integral. We apply it to spheres, lines, and sheets, then turn it on conductors, where the zero interior field drives every excess charge to the surface and fixes the normal-field jump $\\sigma\u002F\\varepsilon_0$, the charge induced on a cavity wall, and electrostatic shielding.\n",{"module":1036,"moduleNumber":1006,"slug":1037,"lessons":1038},"Electric Potential","electric-potential",[1039,1044,1049,1054,1059],{"title":1040,"path":1041,"lessonNumber":990,"topics":1042,"summary":1043},"Point-Charge Potential","\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential",[1036],"The electrostatic force is conservative, so the work it does between two points\ndepends only on the endpoints. That lets us trade the vector field for a single\nscalar attached to each point, the electric potential, the potential energy a unit\ncharge would have there. We build potential from the work integral, fix the usual\nreference at infinity, and add point sources as scalars, $V=k\\sum_i q_i\u002Fr_i$,\navoiding the vector bookkeeping the field demands. Signed charges, the reference\nchoice, equipotential motion, and far-field expansions each give an independent\ncheck on a result.\n",{"title":1045,"path":1046,"lessonNumber":16,"topics":1047,"summary":1048},"Potential Gradients and Equipotentials","\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials",[1036],"Given the potential everywhere, how do we recover the field? The field is the\nnegative gradient, $\\vec E=-\\nabla V$: it points down the steepest local drop in\npotential, and its magnitude is set by how fast $V$ changes, not by the shape of a\ncontour. We read off components with directional derivatives, reconstruct fields\nfrom measured potential grids using centered differences, and use closed-loop\nintegrals and grid refinement to test whether a reconstructed field is physically\nconsistent.\n",{"title":1050,"path":1051,"lessonNumber":1006,"topics":1052,"summary":1053},"Electrostatic Energy and Pressure","\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure",[1036],"Assembling a charge configuration takes work, and that work is stored, but where\nis it kept and how much is there? We total it two ways: as a sum over the charges,\n$U=\\tfrac12\\sum_i q_iV_i$, and as an integral over the field itself,\n$u_E=\\tfrac12\\varepsilon_0E^2$, energy the field carries in every region it fills.\nDifferentiating the stored energy at fixed charge or at fixed voltage recovers the\nmechanical force on a conductor, and at a charged surface the same field scale\nappears as an outward electrostatic pressure. We work the parallel-plate case in\nfull and mark where curvature and fringing make the pressure nonuniform.\n",{"title":1055,"path":1056,"lessonNumber":1012,"topics":1057,"summary":1058},"Laplace Boundary Problems","\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems",[1036],"Often the charges are not given, only the conductors and the voltages held on\nthem, and the potential in the empty space between has to be found. There $V$ obeys\nLaplace's equation $\\nabla^2V=0$, and the boundary data alone determine a unique solution.\nWe solve it two ways: separation of variables into boundary-matched modes, whose\nhigher spatial frequencies die away with depth into the domain, and finite-difference\nrelaxation for boundaries no analytic mode fits. Residual norms, boundary error, and\nflux balance tell us when the computed potential and its field can be trusted.\n",{"title":1060,"path":1061,"lessonNumber":1018,"topics":1062,"summary":1063},"Continuous Charge Potentials","\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials",[1036],"When charge is spread over a line, a surface, or a volume, the sum over point\nsources becomes an integral, $V(\\vec r)=k\\int \\d q\u002F|\\vec r-\\vec r'|$. Because\npotential is a scalar, this integral sidesteps the component algebra the field\nwould force, until the field is actually wanted through $\\vec E=-\\nabla V$. We set\nup the right density element for each geometry, choose a workable reference, handle\nthe integrable singularities that arise when the observation point sits on the\ncharge, and check every result against symmetry, dimensions, and the far-field\nmultipole limit.\n",{"module":1065,"moduleNumber":1012,"slug":1066,"lessons":1067},"Capacitance","capacitance",[1068,1073,1078,1083],{"title":1069,"path":1070,"lessonNumber":990,"topics":1071,"summary":1072},"Capacitance Fundamentals","\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals",[1065],"How much charge must you separate onto two conductors to hold a given voltage between\nthem? That ratio, $C=Q\u002F\\Delta V$, is fixed by the conductor geometry and the medium,\nnot by how much charge is presently stored. We compute it from the field for the\nparallel-plate, isolated-sphere, concentric-sphere, and coaxial geometries, trace how\nsurface charge and boundary conditions set each result, and see where fringing,\nguarding, and stray coupling separate the ideal formula from what a bridge measures.\n",{"title":1074,"path":1075,"lessonNumber":16,"topics":1076,"summary":1077},"Capacitor Networks","\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks",[1065],"Wire several capacitors together and the source sees one equivalent capacitance — but\nwhich? The answer comes not from how the symbols are drawn but from which conductors\nshare a node: parallel branches hold a common voltage and add, $C_{\\rm eq}=\\sum_iC_i$,\nwhile series branches share a common charge and add reciprocally. We derive both rules\nfrom charge conservation on the floating internal node, then extend the node-charge\nmethod to unequal, precharged, and stray-coupled branches and carry a worked reduction\nthrough to the charge and voltage on every element.\n",{"title":1079,"path":1080,"lessonNumber":1006,"topics":1081,"summary":1082},"Capacitor Energy and Force","\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force",[1065],"Charging a capacitor takes work, because every increment of charge is pushed through\nthe voltage the earlier charge already established. We total that work three\nequivalent ways, $U=Q^2\u002F(2C)=Q\\Delta V\u002F2=C(\\Delta V)^2\u002F2$, locate it in the field as\na density $u=\\tfrac12\\epsilon_0E^2$, then let the plates move. Differentiating the\nstored energy at fixed charge, or the coenergy at fixed voltage, gives the mechanical\nforce; the two boundaries differ only by the work the source supplies. We work the\nparallel-plate attraction and its electrostatic pressure in full, and follow the same\ngradient into pull-in, tilt, comb drives, and traceable force calibration.\n",{"title":1084,"path":1085,"lessonNumber":1012,"topics":1086,"summary":1087},"Dielectric Polarization and Breakdown","\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown",[1065],"Slide a dielectric between the plates and the capacitance rises — but why, and how\nhard can you drive it before the insulator fails? Bound charge answers the first:\npolarization $\\vec P$ sets up surface and volume charge that partly cancels the\napplied field, so $\\vec D=\\varepsilon_0\\vec E+\\vec P$ separates what the circuit\ncontrols from what the material contributes. We follow the field across layered\ndielectrics and interfaces, tie permittivity and loss to their frequency dependence,\nand treat dielectric strength as a measured, geometry-dependent limit rather than one\nmaterial number.\n",{"module":1089,"moduleNumber":1018,"slug":1090,"lessons":1091},"Direct-Current Circuits","direct-current-circuits",[1092,1097,1102],{"title":1093,"path":1094,"lessonNumber":990,"topics":1095,"summary":1096},"Current and Resistance","\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance",[1089],"What does it mean, physically, for charge to flow, and what sets how hard a wire resists that flow? Current counts charge crossing a surface, $I=\\int\\vec J\\cdot\\d\\vec A$, and traces back to a slow drift of many carriers, $\\vec J=nq\\vec v_d$. We establish when the linear law $V=IR$ actually holds, how resistivity and geometry combine into bulk resistance, why real sources sag under load through their internal resistance, and how the three power forms $P=IV=I^2R=V^2\u002FR$ tie electrical work to heating and component ratings.\n",{"title":1098,"path":1099,"lessonNumber":16,"topics":1100,"summary":1101},"Kirchhoff Network Analysis","\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis",[1089],"Once a circuit has more than one loop, no amount of series-parallel folding will reduce it — you need the two conservation laws written as equations. Kirchhoff's junction law is charge conservation at a node; his loop law is energy conservation around a closed path. We turn a labelled network into a linear system in node voltages or mesh currents, fix the sign conventions so a negative answer just means a reversed arrow, and use power balance as an independent check that the algebra describes the circuit that was actually built.\n",{"title":1103,"path":1104,"lessonNumber":1006,"topics":1105,"summary":1106},"RC Transients","\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients",[1089],"How does a circuit get from one steady state to the next when a capacitor refuses to change its voltage all at once? Because a jump would demand infinite current, an RC circuit slides between states exponentially, with a single time constant $\\tau=RC$ that sets the whole schedule: charging fills as $1-e^{-t\u002F\\tau}$, discharge empties as $e^{-t\u002F\\tau}$. We solve the first-order loop equation, read the response off three numbers — the switch-instant voltage, the final dc voltage, and the Thevenin resistance the capacitor sees — and mark where source and probe resistance shift $\\tau$ or where a second storage element hides a mode a one-$\\tau$ fit misses.\n",{"module":1108,"moduleNumber":1109,"slug":1110,"lessons":1111},"Magnetic Field",6,"magnetic-field",[1112,1117,1122,1127,1132],{"title":1113,"path":1114,"lessonNumber":990,"topics":1115,"summary":1116},"Magnetic Trajectories","\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories",[1108],"A charged particle in a magnetic field never speeds up or slows down, yet its path curves relentlessly. We work out why: the magnetic force is always perpendicular to velocity, so it does no work and bends the transverse motion into a circle of radius $r=mv_\\perp\u002F(|q|B)$ while leaving the parallel motion untouched, producing a helix. We derive the cyclotron frequency, show why it is independent of speed until relativity intervenes, and turn the geometry around: a measured curvature reads back a particle's momentum, which is how tracking detectors weigh what they cannot see.\n",{"title":1118,"path":1119,"lessonNumber":16,"topics":1120,"summary":1121},"Hall Effect","\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect",[1108],"Current tells you charge is moving, but not whether the movers are positive or negative, nor how many there are. A magnetic field settles both questions. Push current through a strip in a transverse field and the carriers pile up on one edge until a transverse electric field just balances the magnetic deflection; the sign of the resulting Hall voltage names the carrier's charge and its size counts the carriers per volume. We derive the balance $q\\vec E+q\\vec v_d\\times\\vec B=0$, read off $V_H=IB\u002F(nqt)$, and see why field-and-current reversal is what separates the real Hall signal from the offsets that mimic it.\n",{"title":1123,"path":1124,"lessonNumber":1006,"topics":1125,"summary":1126},"Magnetic Force on Conductors","\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors",[1108],"A magnet pushes on a current-carrying wire even though the wire is electrically neutral. The reason is that each moving carrier feels the Lorentz force, and those microscopic pushes add up to a force the wire's supports must hold. We sum them into $\\d\\vec F=I\\,\\d\\vec\\ell\\times\\vec B$, collapse it to $\\vec F=I\\vec L\\times\\vec B$ for a straight segment in a uniform field, and see exactly when that shortcut fails and the full path integral is needed. The same law runs backward as a measurement: a force-versus-current slope weighs a magnetic field against a known length.\n",{"title":1128,"path":1129,"lessonNumber":1012,"topics":1130,"summary":1131},"Magnetic Dipoles","\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles",[1108],"A compass needle turns to point north; a current loop in a field does the same thing, and for the same reason. Both are magnetic dipoles, and a uniform field cannot push a dipole anywhere, only twist it. We package a loop's response into one vector, the magnetic moment $\\vec\\mu=IA\\hat n$, from which torque $\\vec\\tau=\\vec\\mu\\times\\vec B$ and orientation energy $U=-\\vec\\mu\\cdot\\vec B$ both follow. Stable alignment sits at the energy minimum, a field gradient is what it takes to produce a net force $\\vec F=\\nabla(\\vec\\mu\\cdot\\vec B)$, and the same moment reappears whenever anything from an electron to a planet acts magnetic.\n",{"title":1133,"path":1134,"lessonNumber":1018,"topics":1135,"summary":1136},"Mass Spectrometry","\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry",[1108],"To weigh a single atom you cannot use a scale, so you use a magnetic field instead. A charged ion of unknown mass bends in a field by an amount that depends on its momentum and charge, so if every ion enters with the same velocity, its landing position reads off its mass-to-charge ratio directly. We build the instrument in two stages: crossed electric and magnetic fields that pass only ions with $v=E\u002FB$, and a magnetic sector that bends the survivors along $r=mv\u002F(|q|B)$. Then we ask what blurs a spectral line and how reference ions turn a position into a calibrated mass.\n",{"module":1138,"moduleNumber":1139,"slug":1140,"lessons":1141},"Magnetic Sources",7,"magnetic-sources",[1142,1147,1152,1157,1162,1167],{"title":1143,"path":1144,"lessonNumber":990,"topics":1145,"summary":1146},"Moving-Charge Fields","\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields",[1138],"Every magnetic field comes from charge in motion, and the simplest source is a single point charge drifting past. We work out the field it produces — normal to both the velocity and the line of sight, falling off as the inverse square — and read off why it vanishes straight ahead of the charge and peaks broadside. Summing many such charges is the bridge to steady currents, valid while speeds stay far below $c$ and the motion changes little during the time its field takes to propagate outward.\n",{"title":1148,"path":1149,"lessonNumber":16,"topics":1150,"summary":1151},"Biot–Savart Law","\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law",[1138],"A steady current is a continuous stream of current elements, and the Biot–Savart law hands each one a magnetic contribution — a right-hand cross product that falls off as the inverse square of distance. Summing the contributions along a conductor is a vector line integral, which we carry out for the straight wire to get the endpoint-angle formula. The infinite-wire field $B=\\mu_0 I\u002F2\\pi s$ falls out as the limit where both ends recede, and we mark how fast a finite wire departs from it and when a thin-filament model is safe.\n",{"title":1153,"path":1154,"lessonNumber":1006,"topics":1155,"summary":1156},"Circular Current Loops","\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops",[1138],"A ring of current is the simplest source with a well-defined magnetic axis, and it is the building block of every coil and electromagnet. Symmetry kills the transverse Biot–Savart contributions along that axis and leaves a single clean integral; we evaluate it to get $B_z=\\mu_0 I R^2\u002F[2(R^2+z^2)^{3\u002F2}]$, read off the centre field $\\mu_0 I\u002F2R$, and watch it fall into the $1\u002Fz^3$ tail of a magnetic dipole far away. Stacking turns just adds their axial contributions, which is what makes a solenoid out of a pile of loops.\n",{"title":1158,"path":1159,"lessonNumber":1012,"topics":1160,"summary":1161},"Ampère’s Law","\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law",[1138],"When a current arrangement is symmetric enough, the Biot–Savart integral is overkill: Ampère's law, $\\oint_C\\vec B\\cdot\\d\\vec\\ell=\\mu_0 I_{\\rm enc}$, gets the field from a single line of reasoning about how much current a loop encloses. We see why the law holds for any steady current, then use cylindrical, planar, and toroidal symmetry to turn the circulation into simple algebra — the field inside and outside a wire, an infinite sheet, a solenoid, and a toroid. We also mark the catch: without symmetry the law still holds but no longer hands you the field pointwise.\n",{"title":1163,"path":1164,"lessonNumber":1018,"topics":1165,"summary":1166},"Gauss’s Law for Magnetism","\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism",[1138],"Electric field lines start and end on charges; magnetic field lines do neither, because no one has ever found an isolated magnetic pole. That single experimental fact is Gauss's law for magnetism: the flux of $\\vec B$ through any closed surface is zero, $\\oint\\vec B\\cdot\\d\\vec A=0$, or in differential form $\\nabla\\cdot\\vec B=0$. We work through what it says — every field line that enters a closed surface must leave it, so field lines close on themselves — and, just as important, what it does not say, since flux through an open surface is generally nonzero.\n",{"title":1168,"path":1169,"lessonNumber":1109,"topics":1170,"summary":1171},"Magnetic Materials","\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials",[1138],"Put matter in a magnetic field and its atoms respond, each acting as a tiny current loop; the aligned moments per unit volume are the magnetization $\\vec M$, whose bound currents add to the field. Separating what we control (the free current) from what the material supplies leads to $\\vec H$ and the relation $\\vec B=\\mu_0(\\vec H+\\vec M)$. We sort materials into diamagnets, paramagnets, and ferromagnets by how $\\vec M$ answers, follow a ferromagnet around its hysteresis loop, and see why the loop's area is the energy dissipated per cycle and why a sample's shape changes the field it actually feels.\n",{"module":1173,"moduleNumber":1174,"slug":1175,"lessons":1176},"Electromagnetic Induction",8,"electromagnetic-induction",[1177,1182,1187,1192,1197,1202,1207,1212],{"title":1178,"path":1179,"lessonNumber":990,"topics":1180,"summary":1181},"Magnetic Flux","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux",[1173],"A magnetic field threading a loop collapses to one signed number, the flux, and every induced voltage in this module turns out to be a rate of change of that number — so defining the flux and its sign comes first. We define it as the surface integral of $\\vec B$ over an oriented surface, reduce it to $BA\\cos\\theta$ for a uniform field on a flat loop, and carry the flux linkage $N\\Phi_B$ of a coil. The chosen normal fixes the sign; reversing it flips the sign without touching the field. Nonuniform fields and curved surfaces force the integral, so we also build the numerical estimate and the checks that separate a reliable value from a nominal field-times-area product.\n",{"title":1183,"path":1184,"lessonNumber":16,"topics":1185,"summary":1186},"Faraday's Law","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law",[1173],"Move a magnet toward a coil, or ramp the current in a nearby circuit, and a voltage appears with no battery in sight. Faraday's law names the cause: the emf around a loop equals minus the rate of change of the magnetic flux through it, so any change of field, area, orientation, or position that alters the flux drives an emf. We separate the emf, which lives around the boundary whether or not current can flow, from the current that follows only when the path is closed; fix the single sign convention that ties flux to loop orientation; and read the emf off rotating coils and off flux sampled at discrete times.\n",{"title":1188,"path":1189,"lessonNumber":1006,"topics":1190,"summary":1191},"Lenz's Law","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law",[1173],"The minus sign in Faraday's law is not decoration: it decides which way the induced current flows, and it always chooses the direction that fights the change that produced it. Lenz's law reads that sign off energy conservation — a current that aided the change would be free energy — and turns it into a repeatable procedure. We fix a surface normal and a positive loop direction so the sign is calculable, then work through approaching magnets, expanding loops, coupled coils, and rotating generators, using mechanical work and Joule heating as an independent check on every direction we draw.\n",{"title":1193,"path":1194,"lessonNumber":1012,"topics":1195,"summary":1196},"Motional EMF","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf",[1173],"Push a wire through a magnetic field and its free charges feel a sideways magnetic force that piles them up at the ends — a battery made of motion. Motional emf is that effect: the work per unit charge a moving conductor supplies is the line integral of $\\vec v\\times\\vec B$ along it, which for a rod moving perpendicular to both its length and the field collapses to $B\\ell v$. We chase where the energy comes from — the hand or motor fighting the magnetic drag, never the magnetic force itself — solve the sliding-rail circuit from both flux and carrier forces, and carry the idea into rotating rods, homopolar disks, generators, and the back emf of a motor.\n",{"title":1198,"path":1199,"lessonNumber":1018,"topics":1200,"summary":1201},"Eddy Currents","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents",[1173],"A wire carries current along one path; a solid block of metal offers a continuum of them, and any changing flux threading that block sets charge circulating in closed loops it chooses for itself. We ask what those eddy currents do — where they heat, where they drag, and how Lenz's law fixes their direction — and why the same circulation is a feature in an induction furnace and a loss to be suppressed in a transformer core. From a representative-loop estimate we get the scaling (heating grows with the square of frequency and flux rate) and the two design levers, lamination and resistivity, that break the paths a solid conductor would otherwise hand the current.\n",{"title":1203,"path":1204,"lessonNumber":1109,"topics":1205,"summary":1206},"Self-Inductance","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance",[1173],"A coil resists changes to its own current. Drive current through it and the flux it produces threads its own turns; change that current and Faraday's law turns the coil against the source with a back emf $\\mathcal E_L=-L\\,\\d I\u002F\\d t$. We define self-inductance as the flux linkage per ampere fixed by winding and core geometry, derive the long-solenoid value $L=\\mu_0 N^2A\u002F\\ell$, and follow the consequence that dominates circuits: because a finite voltage can only sustain a finite $\\d I\u002F\\d t$, an inductor's current cannot jump — which is why opening a switch on a live coil throws a spark.\n",{"title":1208,"path":1209,"lessonNumber":1139,"topics":1210,"summary":1211},"Magnetic Energy","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy",[1173],"Building current in a coil means working against its back emf, and that work does not vanish — it sits in the magnetic field as recoverable energy $U_B=\\tfrac12LI^2$, spread through space at density $u_B=B^2\u002F(2\\mu_0)$. We derive both forms, show they agree for a solenoid, and read a force out of the same energy: an armature is pulled toward higher inductance, and $B^2\u002F(2\\mu_0)$ doubles as a magnetic pressure. The lesson closes on the accounting a real switching event demands, where recoverable energy, copper heating, core loss, and clamp dissipation must balance a single ledger.\n",{"title":1213,"path":1214,"lessonNumber":1174,"topics":1215,"summary":1216},"RL Circuits","\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits",[1173],"Put a resistor and an inductor in series and the current cannot switch on or off at will: it climbs to $V_0\u002FR$ and falls away exponentially on a single time scale $\\tau=L\u002FR$ set by how much flux the coil hoards against how fast the resistor bleeds it. We solve the turn-on and turn-off, then confront the practical sting — because the coil's current refuses to stop instantly, breaking its path throws up a large voltage, which is why real inductive circuits carry freewheel diodes and clamps that trade voltage stress against how quickly the current dies.\n",{"module":1218,"moduleNumber":1219,"slug":1220,"lessons":1221},"Alternating Current",9,"alternating-current",[1222,1227,1232,1237,1242],{"title":1223,"path":1224,"lessonNumber":990,"topics":1225,"summary":1226},"AC Fundamentals","\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals",[1218],"A wall socket delivers a voltage that averages to zero over each cycle, yet it still heats a filament and runs a motor. The resolution is that dissipation follows the mean of the square, not the mean, so we define the root-mean-square value that makes an alternating source the equal of a DC one for resistive heating. We show a sinusoid's RMS is its peak divided by $\\sqrt2$, work out the average power an ideal resistor draws when its current stays in phase with the applied voltage, and separate the peak, average, and RMS descriptions that a single number cannot combine.\n",{"title":1228,"path":1229,"lessonNumber":16,"topics":1230,"summary":1231},"Reactance","\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance",[1218],"A resistor obeys Ohm's law instant by instant, but a capacitor responds to how fast its voltage changes and an inductor to how fast its current changes. Under a steady sinusoid that rate-dependence collapses to a fixed quarter-cycle phase shift and a frequency-dependent amplitude ratio, the reactance. We derive $X_C=1\u002F(\\omega C)$ and $X_L=\\omega L$, adopt phasors to turn the defining derivatives into multiplication by $j\\omega$ so a single complex impedance carries amplitude and phase together, and track the energy an ideal reactance stores and returns without dissipating it. Real windings and dielectrics add loss, leakage, and self-resonance that bound where the ideal formulas hold.\n",{"title":1233,"path":1234,"lessonNumber":1006,"topics":1235,"summary":1236},"RLC Resonance","\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance",[1218],"Put a resistor, inductor, and capacitor in one loop and their reactances work against each other: inductive reactance grows with frequency while capacitive reactance shrinks, and at one frequency they cancel exactly. There the branch looks purely resistive, the current peaks, and the inductor and capacitor voltages can swing far above the source. We locate that resonance at $\\omega_0=1\u002F\\sqrt{LC}$, measure how sharp the peak is with the quality factor $Q=\\omega_0L\u002FR$, tie its half-power bandwidth $R\u002FL$ to the ringdown of the unforced circuit, and read the same poles off as bandpass and peaked filters at the R, L, or C terminals.\n",{"title":1238,"path":1239,"lessonNumber":1012,"topics":1240,"summary":1241},"AC Power","\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power",[1218],"Multiply an AC load's RMS voltage by its RMS current and you get an answer in volt-amperes that the wiring must carry, but not in general the watts the load consumes. The phase between voltage and current splits that product into a part that does net work and a part that merely sloshes energy back and forth. We derive the average power $P=V_{\\rm rms}I_{\\rm rms}\\cos\\phi$, package amplitude and phase into complex power $S=P+jQ$ so that real, reactive, and apparent power form one right triangle, and see why a harmonic-rich current forces the time-domain definition $P=\\langle vi\\rangle$ in place of a single phase angle.\n",{"title":1243,"path":1244,"lessonNumber":1018,"topics":1245,"summary":1246},"Transformers","\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers",[1218],"Two coils sharing an iron core exchange no charge, yet a changing current in one drives a voltage in the other, and the ratio of their turns sets how voltage and current trade off between the windings. That lets a transformer step a voltage up or down, isolate two circuits, and make a load look larger or smaller to the source by the square of the turns ratio. We build the ideal ratio element from Faraday's law and the dot convention, derive the reflected-impedance rule, then add the winding resistance, leakage, magnetizing current, and core loss that turn the ideal ratios into real regulation, efficiency, and a bounded voltage-frequency range.\n",{"module":1248,"moduleNumber":1249,"slug":1250,"lessons":1251},"Maxwell’s Equations and Electromagnetic Waves",10,"maxwell-electromagnetic-waves",[1252,1257,1262,1267,1272],{"title":1253,"path":1254,"lessonNumber":990,"topics":1255,"summary":1256},"Displacement Current","\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current",[1248],"Ampère's law asks for the current through a surface bounded by a loop, but a charging capacitor breaks it: slide the surface off the wire and into the gap and the enclosed conduction current drops to zero, while the magnetic field around the loop plainly does not. Maxwell's repair is to count a changing electric flux as itself a source of magnetic circulation. We derive the displacement-current term $\\varepsilon_0\\,\\d\\Phi_E\u002F\\d t$, show that charge continuity demands it, compute the magnetic field it produces inside a charging capacitor, and see how it closes the Ampère–Maxwell law so that electric and magnetic fields can sustain one another as a wave.\n",{"title":1258,"path":1259,"lessonNumber":16,"topics":1260,"summary":1261},"Electromagnetic Waves","\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves",[1248],"Once a changing electric flux can drive a magnetic field, the two curl laws feed each other: a disturbance in one regenerates the other, and the pair walks off through empty space with no medium holding it up. We take the curl of Faraday's law, land on a wave equation whose speed is fixed entirely by $\\mu_0$ and $\\varepsilon_0$, and find that $c=1\u002F\\sqrt{\\mu_0\\varepsilon_0}$ falls out of purely electric and magnetic constants. The plane-wave solution then fixes the geometry — $\\vec E$, $\\vec B$, and the propagation direction mutually perpendicular, oscillating in phase, with amplitudes locked at $E=cB$ — a set of independent predictions any real measurement must meet at once.\n",{"title":1263,"path":1264,"lessonNumber":1006,"topics":1265,"summary":1266},"Electromagnetic Momentum","\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum",[1248],"A light beam carries no mass, yet it pushes: shine it on a surface and the surface feels a force. We trace that force back to the fields, which store energy with density $\\varepsilon_0E^2$ and carry it along the Poynting vector $\\vec S=\\vec E\\times\\vec B\u002F\\mu_0$. Because that energy also carries momentum $U\u002Fc$, an absorbed beam presses with $I\u002Fc$ and a mirror with $2I\u002Fc$. We derive the Poynting theorem as local energy conservation, tie intensity to field amplitude, and work the momentum balance carefully enough that oblique incidence, partial reflection, and finite beams all drop out of one accounting.\n",{"title":1268,"path":1269,"lessonNumber":1012,"topics":1270,"summary":1271},"Dipole Radiation","\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation",[1248],"Only accelerating charge radiates, and the simplest accelerator is a charge sloshing back and forth: an oscillating electric dipole. We work out the field it throws off, keeping the part that survives to large distance — the $1\u002Fr$ radiation field whose intensity goes as $\\sin^2\\theta\u002Fr^2$, zero along the dipole axis and strongest broadside. From it follow the $\\omega^4$ scaling of total radiated power, radiation resistance as the feed's view of that escaping power, and, through reciprocity, the fact that a good transmitter receives well in the same directions. The near-zone terms that fall off faster carry no net power, and we mark carefully where each description is allowed to be used.\n",{"title":1273,"path":1274,"lessonNumber":1018,"topics":1275,"summary":1276},"Polarization","\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization",[1248],"A plane wave still leaves one thing free: which way its electric field points as it oscillates. That freedom is polarization, set entirely by the relative amplitude and phase of the two transverse field components — in phase gives a line, equal amplitudes a quarter cycle apart give a circle, everything else an ellipse. We work out how a linear analyzer reads a state through Malus's law $I=I_0\\cos^2\\theta$, why that scan alone cannot tell circular light from unpolarized, and how a quarter-wave plate plus a few analyzer settings recover the full Stokes vector and the degree of polarization.\n",{"module":1278,"moduleNumber":1279,"slug":1280,"lessons":1281},"Geometrical Optics",11,"optics",[1282,1287,1292],{"title":1283,"path":1284,"lessonNumber":990,"topics":1285,"summary":1286},"Reflection and Refraction","\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction",[1278],"Light meeting a boundary between two transparent media splits into a reflected ray and a bent transmitted one, and predicting where those rays go is the whole starting point of geometrical optics. Fixing one convention — every angle measured from the surface normal — we get reflection's equal angles and derive Snell's law $n_1\\sin\\theta_1=n_2\\sin\\theta_2$ from wavefront timing. That single relation, applied once or twice, yields the critical angle and total internal reflection, prism deviation, the lateral shift through a window, apparent depth, and a fiber's acceptance cone; a wavelength-dependent index then adds dispersion. We mark throughout where the ray picture is trustworthy: feature sizes large against the wavelength and clean interface geometry.\n",{"title":1288,"path":1289,"lessonNumber":16,"topics":1290,"summary":1291},"Thin Lenses","\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses",[1278],"A lens gathers the light spreading from one point back onto another, and a single paraxial relation $1\u002Fs+1\u002Fs'=1\u002Ff$ predicts where that image lands and how large it is. We collapse two refractions into one bending plane, read image position and orientation off the three principal rays, and trace focal length back to glass and curvature through the lensmaker equation. Sign conventions carry the physics here — they separate real from virtual images and upright from inverted — so we drill them before chaining lenses in sequence and in contact. The lesson ends on how focal length is actually measured on a bench, and where finite thickness, aperture, and dispersion break the thin-lens picture.\n",{"title":1293,"path":1294,"lessonNumber":1006,"topics":1295,"summary":1296},"Spherical Mirrors","\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors",[1278],"Curve a mirror and it stops merely reflecting an image and starts forming one: the same $1\u002Fs+1\u002Fs'=1\u002Ff$ that governs lenses reappears, now with $f=R\u002F2$ and reflected rays and object sharing one side of the glass. We derive the mirror equation from the reflection geometry of a single paraxial ray, then let signed distances do the sorting — real inverted images on the near branch, virtual upright ones behind the surface — and check the concave, convex, and plane-mirror limits against each other. The second half turns to how focal length is actually measured on a bench, by finite conjugates, distant targets, return imaging, and sagitta, and to the aperture and off-axis aberrations the single paraxial focus cannot capture.\n",1786059306723]