Displacement Current & Maxwell's Equations Displacement Current & Maxwell's Equations A charging capacitor seems to break Ampere’s law if you choose different surfaces for the same loop: take a loop around the wire and you enclose conduction current I; take a bulging soap-film-like surface that passes between the plates and you enclose no conduction current at all. Should the line integral of magnetic field around the loop depend on which surface you imagine? Physics cannot allow that. Maxwell noticed that a changing electric field between the plates creates its own "current-like" effect, which he called displacement current. With this single, elegant correction, Ampere’s law becomes consistent with charge conservation in every situation—steady or time-varying—and electricity and magnetism merge into one theory. The same set of equations also predicts waves of electric and magnetic fields that travel through empty space at a universal speed c, which matches the measured speed of light. In other words, light is an electromagnetic wave. In daily life, phone signals, Wi‑Fi, sunlight, microwave ovens, and X-rays are all the same phenomenon at different frequencies. The key ideas you need: why displacement current must exist, what the four Maxwell equations say (both in words and in math), how a self-sustaining changing E-field generates a B-field and vice versa, why the waves are transverse, why c = 1/ 0 0 , and how energy is shared and transported in EM waves. Keep a mental map: (i) static charges → electric field (Gauss), (ii) no magnetic monopoles (Gauss for B), (iii) changing magnetic flux → electric field (Faraday), and (iv) currents + changing electric flux → magnetic field (Ampere–Maxwell). These four, plus the continuity equation, form the backbone you will repeatedly use in NEET problems. Analogy: A flexible rubber membrane in a water pipe blocks actual water flow across it, yet when you push water from one side, the membrane bulges and the pressure changing in time transmits effects across the gap. Similarly, between capacitor plates, no charges cross, but the changing electric field "transmits" a current-like effect: the displacement current. remember Start with the paradox: for a loop around a charging capacitor circuit, using a surface that cuts the wire gives a nonzero current and a magnetic field. Using a surface that passes between the plates (enclosing no conduction current) would predict zero magnetic field—unless something else counts as "current" there. Maxwell’s insight was to treat changing electric flux as a source of magnetic field on the same footing as conduction current. This guarantees that the circulation of B is independent of the chosen surface and remains tied to a single physical cause: the total current (conduction + displacement) threading the loop. An effective current I d = 0 , d E dt arising from a time-varying electric field (changing electric flux). It produces magnetic fields like a real current but involves no actual transport of charge across the region. Displacement current Current due to motion of charges through a material, e.g., I = nqAv d in a wire. It requires charge carriers and a conducting path. Conduction current Electric flux Measure of the "amount" of electric field passing through a surface: E = E d A . A constant characterizing how electric fields permeate free space: 0 8.854 10 -12 , F/m . Vacuum permittivity Vacuum permeability A constant characterizing magnetic response of free space: 0 = 4 10 -7 , H/m . Ampere–Maxwell law The generalized Ampere’s law: B d l = 0 I cond,enc + 0 0 , d E dt . Continuity equation Mathematical statement of local charge conservation: t + J = 0 . In a charging capacitor, the conduction current in the wires brings charge to each plate. Between the plates there is no conduction path, but the electric field grows as the charge accumulates. This growth means the electric flux through any surface spanning the plates changes with time, and that change produces a magnetic field. For a circular Amperian loop threading the space between plates, the magnetic field depends on the rate of change of electric flux d E/dt , not on literal charge crossing the gap. In steady DC (fully charged capacitor), d E/dt=0 , the displacement current term vanishes, and you recover the original Ampere’s law for magnetostatics. Valid only for steady currents (no time-varying fields). Original Ampere’s law (magnetostatics) This law applies to steady current distributions (magnetostatics) where the magnetic field is time-independent. The electric-flux term that completes Ampere’s law for time-varying fields. Displacement current This current exists in regions where the electric field is changing over time, such as in the gap between the plates of a charging capacitor Ampere–Maxwell law (integral form) Magnetic field circulation is sourced by conduction current and changing electric flux through any open surface bounded by the loop. This law applies to all regions of space, describing the relationship between the circulation of the magnetic field and the total enclosed c Use the integral form of Ampere–Maxwell law when symmetry allows you to choose an Amperian loop with nearly constant B, and ensure the surface bounded by the loop is any open surface—your result must be independent of which surface you pick if you include both terms on the right. tip With displacement current added, the circulation of B becomes surface-independent: if your surface cuts the wire, I cond,enc = I and the flux term matches; if your surface threads between the plates, I cond,enc = 0 , but d E/dt is nonzero and compensates exactly. This is not a trick—it's the only way to preserve charge conservation and keep magnetic fields well-defined in time-dependent situations. Maxwell’s Four Equations (Vacuum) Electric flux through a closed surface equals enclosed charge over 0 . Gauss’s law (integral) Electric field diverges from charge density. Gauss’s law (differential) Gauss’s law for magnetism (integral) No magnetic monopoles: net magnetic flux through any closed surface is zero. The total electric flux through a closed surface depends only on the net charge enclosed within that specific volume. Magnetic field lines are continuous; no sources or sinks. Gauss’s law for magnetism (differential) Faraday’s law (integral) Changing magnetic flux induces a nonconservative electric field. This law quantifies the induced electromotive force (EMF) generated by a changing magnetic flux through a coil. Faraday’s law (differential) Curl of E is tied to time-changing B. This law describes how a time-varying magnetic field (changing magnetic flux) induces a circulating electric field (Faraday's Law, which is Curl of B is produced by conduction current density and changing E. 10 Ampere–Maxwell (differential) For electromagnetic waves propagating in a vacuum, the maximum amplitudes of the electric and magnetic fields are directly related by the speed of light. Physical meanings in one line each: (i) Gauss (E) says charges are sources/sinks for E; (ii) Gauss (B) says there are no isolated magnetic charges, so B-lines form loops; (iii) Faraday says a changing B creates a curling E (basis of generators and transformers); (iv) Ampere–Maxwell says currents and changing E create a curling B (basis of inductors and EM wave self-sustenance). Together they imply charge conservation and predict waves in vacuum without any material medium. Displacement current means charges actually flow across the capacitor gap. No charges cross the vacuum gap. Displacement current is an effective term arising from changing electric flux; it produces magnetic fields but carries no mass or charge through the gap. Charge conservation must hold locally: any decrease of charge in a region must equal the net outward current flow. Maxwell’s correction encodes this by ensuring that if E changes because charge density changes, the curl of B adjusts so that J precisely balances - / t . The result is the continuity equation, which you can derive directly from Ampere–Maxwell and Gauss’s law (E). Vacuum (use of 0, 0) but result is general Fields are sufficiently smooth to exchange derivatives Continuity equation: J + t = 0 Start with Ampere–Maxwell (differential form). Divergence of a curl is identically zero. Take divergence of both sides. Use Gauss’s law (differential form). Substitute and simplify. Divide by 0 . J + t = 0 neet-alert Surface-choice trap: For a loop around a charging capacitor, if you choose a surface that cuts the wire, include conduction current. If you choose a surface that passes between the plates, include the displacement term 0 0 ,d E/dt . Omitting either makes the magnetic field depend on surface choice, which is unphysical. Changing electric and magnetic fields feed each other in time. A time-varying B creates curling E (Faraday), and a time-varying E creates curling B (Ampere–Maxwell). In free space with no charges and currents, these curls are not "driven" by matter but by the fields themselves, and the mathematics reduces to a wave equation. The solution is a transverse wave where E and B oscillate perpendicular to each other and to the direction of propagation. Wave Equation in Vacuum and Speed of EM Waves Qualitatively, if an electric field begins to change in time, it generates a magnetic field that curls around it. If that magnetic field then changes in time, it generates a curling electric field, and so on. These self-sustaining changes move outward through space: an electromagnetic wave. Quantitatively, applying curl to one equation and using vector identities converts the coupled first-order time equations into second-order wave equations for E and B. The constant multiplying the second time derivative has the form 0 0 , which sets the wave speed. Start with Faraday and Ampere–Maxwell in vacuum. Take curl of Faraday’s law. Use vector identity and substitute Ampere–Maxwell. In vacuum with no charge, Gauss’s law gives zero divergence. Obtain the wave equation for E. Similarly for B, using B =0. Differentiate Ampere–Maxwell in time and substitute. Wave equation for B. 2 E = 0 0 , 2 E t 2 , 2 B = 0 0 , 2 B t 2 Vacuum: = 0, J = 0 Fields are sufficiently smooth; use vector identity ( A ) = ( A ) - 2 A Wave equations: 2 E = 0 0 , 2 E t 2 , 2 B = 0 0 , 2 B t 2 Wave speed in vacuum 11 Universal EM wave speed predicted by Maxwell’s equations. This constant speed determines how fast self-sustaining electromagnetic waves propagate through a vacuum. Numerical check with constants from NCERT: 0 = 4 10 -7 , H/m and 0 = 8.854 10 -12 , F/m . Then c = 1/ 0 0 1/ (4 10 -7 )(8.854 10 -12 ) 3.00 10 8 , m/s . This matches the measured speed of light, confirming that light is an electromagnetic wave. In a medium, replace 0, 0 by , to get v = 1/ , which is always less than c for ordinary materials. remember For a plane EM wave in vacuum: E ⟂ B, both ⟂ direction of propagation; E and B are in phase; and S = 1 0 , E B points along propagation, carrying energy. Field amplitude relation In vacuum, the ratio E 0/B 0 equals c. 12 Plane wave form (example) Wave traveling in +x, with E along +y and B along +z. Here k=2 / and =2 f . 13 Use this to calculate the average power transmitted by the wave per unit area, given the energy density. E and B (arbitrary units) Field value Two sine curves of the same phase: E along +y and B along +z, both zero at x=0,t=0, peaking together. B has smaller amplitude scaled by 1/c. Phase start E0 λ/4 E and B maxima λ/2 Zero crossing dependent E(x) at fixed t B(x) at fixed t derived Position x custom B is in phase with E, on the perpendicular axis, scaled by 1/c. 2D PLOT Electromagnetic wave: E vs position E = E0 sin(k x) E0 Amplitude Wavenumber Energy in EM waves is stored in both fields. In vacuum, the instantaneous energy density in the electric field is u e = 1 2 0 E 2 , and in the magnetic field u b = B 2 2 0 . For a plane wave in vacuum, u e = u b at every point and time, so total energy density is u = u e + u b = 0 E 2 = B 2 0 . Energy flows with the Poynting vector S = 1 0 E B , and its time average over a cycle gives the intensity I , the power transported per unit area. Energy density in the electric field: u e = 1 2 0 E 2 . In a plane EM wave, the magnetic part equals it on average and instantaneously in vacuum. Energy stored in a capacitor. Capacitance and potential difference for uniform field. Substitute and simplify. Divide by the volume between plates. u e = 1 2 , 0 E 2 Parallel-plate capacitor, uniform field between plates Vacuum or air (use 0), linear medium u e = 1 2 , 0 E 2 Magnetic energy density In a plane EM wave in vacuum, u b = u e instantaneously. 14 This density quantifies the energy stored in the magnetic field component of the electromagnetic wave. The Poynting vector magnitude for a plane wave is S = 1 0 EB . Using B = E/c gives S = 0 c E 2 . For a sinusoidal wave, the time average over one cycle is I = S = 1 2 0 c E 0 2 = 1 2 c 0 B 0 2 . This ties field amplitudes directly to measurable power flow per unit area (intensity). In exams, watch whether E means peak, rms, or instantaneous; the average intensity uses E 0 (peak) in the 1 2 0 c E 0 2 form, or E rms in the equivalent I = 0 c E rms 2 . Intensity of a plane EM wave (average) Average over one cycle for sinusoidal waves. 15 tip Boundary conditions and media: In a medium replace 0, 0 with , . The relations B 0=E 0/c and I= 1 2 0 c E 0 2 generalize to v=1/ and I= 1 2 v E 0 2 . Use rms values if the problem states them. Use u e = 1 2 0 E 2 . u e Compute the electric energy density for an electric field of magnitude E = 10 , V/m in vacuum. E = 10 , V/m 0 = 8.854 10 -12 , F/m Energy density in fields, NCERT exemplar style J/ m 3 Insert values with SI units. Compute numerically. easy The result is tiny because the field is weak. In practical EM waves like sunlight at Earth’s surface (intensity ~ 1000 W/m²), the associated field amplitudes are much larger, giving energy densities of the order of a few 10 -6 to 10 -5 , J/m 3 . Charging capacitor magnetic field R = 5.0 , cm = 0.050 , m r = 2.0 , cm = 0.020 , m I = 2.0 , A 0 = 4 10 -7 , H/m A parallel-plate capacitor of plate radius R=5.0 , cm is being charged by a steady current I=2.0 , A . Find the magnetic field magnitude at a point between the plates at radius r=2.0 , cm from the axis. B(r) between the plates Use Ampere–Maxwell with a circular loop of radius r lying between the plates. The enclosed displacement current equals the fraction of conduction current proportional to area: I d( enclosed ) = I , (r 2/R 2) . medium Symmetry gives tangential B of constant magnitude on the loop. Solve for B. Insert numbers in SI units. Evaluate (T). Inside the plates, the magnetic field grows with r for small r and depends on the current feeding the capacitor. For r > R (outside the plate area), use the full conduction current I (or the full displacement current flux) and recover the usual B = 0 I/(2 r) dependence. T, W/ m 2 Magnetic amplitude in tesla. Compute intensity using peak E. Numerical evaluation in W/ m 2 . hard B 0 and I Use B 0 = E 0/c and I = 1 2 0 c E 0 2 . EM wave amplitudes and intensity A plane EM wave in vacuum has electric field amplitude E 0 = 120 , V/m . Find (i) the magnetic field amplitude B 0 , and (ii) the average intensity I . E 0 = 120 , V/m c = 3.00 10 8 , m/s 0 = 8.854 10 -12 , F/m The intensity 19 W/m² is comparable to moderate illumination. Direct sunlight at noon is about 1000 W/m², corresponding to E 0 870 , V/m and B 0 2.9 10 -6 , T in vacuum. Where each equation applies (quick boundary notes) Gauss (E): Any static or dynamic situation; use closed surfaces. Gauss (B): Always zero net flux; helps rule out magnetic monopoles in NCERT scope. Faraday: Applies to any changing magnetic flux; direction by Lenz’s rule. Ampere–Maxwell: For time-varying fields include displacement term; for steady currents the flux term vanishes. Wave equations: Vacuum or linear media; replace 0, 0 with , in media. Choose a circular Amperian loop of radius r centered on the capacitor axis. If your surface cuts the wire: use conduction current I (or fraction for r < wire radius). If your surface spans between plates: compute I d = 0 , d E/dt using E = EA for uniform field. Set B d l = 0(I + I d) and solve for B(r) given symmetry. Check limits: r 0 gives B 0 ; r R recovers B= 0 I/(2 r) . Charging capacitor: safe step plan control u e dependent Electric field E V/m Quadratic growth of electric energy density with field strength. custom Electric energy density u e J/ m 3 A parabola opening upward since u e E 2 ; doubling E quadruples u e . Maxwell in 4: “GE-GM-F-AM” → Gauss (Electric), Gauss (Magnetism), Faraday, Ampere–Maxwell. Read as: charges source E; no mono‑B; changing B curls E; currents + dE/dt curl B. Even if numerically small, the displacement term is essential whenever d E dt 0 . Omitting it breaks surface independence and can violate charge conservation in the math. In the Ampere–Maxwell law, the displacement current is optional and can be ignored for small fields. Do not mix constants: In vacuum use 0, 0 . In a dielectric, use = r 0 and (if magnetic) = r 0 . Many NEET errors come from inserting 0 where is required, giving wrong intensities or wave speeds. neet-alert Summary (vacuum forms) Equation Integral statement Differential statement Key idea Gauss (E) Flux of E through closed surface = Q enclosed/ε0 div E = ρ/ε0 Charges source/sink electric field Gauss (B) Flux of B through closed surface = 0 div B = 0 No magnetic monopoles Faraday Circulation of E = − dΦ B/dt curl E = − ∂B/∂t Changing B induces E Ampere–Maxwell Circulation of B = μ0 I enclosed + μ0ε0 dΦ E/dt curl B = μ0 J + μ0ε0 ∂E/∂t Currents + changing E induce B Wave speed c = 1/√(μ0ε0) Key exam connections: (i) In symmetric setups (long straight wire; charging capacitor), pick loops/surfaces that make one side of the law simple (B nearly constant along the loop, uniform E between plates). (ii) When moving from vacuum to a medium, remember to replace 0, 0 with , consistently in energy density, intensity, and speed. (iii) For sinusoidal waves, be clear whether E means instantaneous, peak ( E 0 ), or rms ( E rms =E 0/ 2 ). (iv) When asked about direction, use the right-hand rule: k (propagation) is along E B . Glossary recap Electric flux current Effective current I d= 0 , d E/dt due to changing electric flux. Displacement current Ampere–Maxwell law Circulation of B sourced by conduction current and displacement current. Generalized Ampere’s law S = 1 0 , E B gives energy flux (W/m²). Poynting vector In fields: u e= 1 2 0 E 2 , u b= B 2 2 0 . Energy density Wave speed In vacuum c=1/ 0 0 ; in medium v=1/ .