Self/Mutual Inductance & Eddy Currents Think of a loop of wire as a control freak that hates change in its magnetic environment. Magnetic field lines pass through the loop like streams of water. If the flow of water is steady, the wire is happy and does nothing. However, if you try to change that flow by pushing a magnet closer (increasing flow) or pulling it away (decreasing flow), the wire reacts. To stop this change, the wire pushes its own electrons to create a counter-flow magnetic field. This attempt to maintain the status quo is induced electricity: no change, no electricity; fast change, large electricity. Self/Mutual Inductance & Eddy Currents Inductance is the magnetic inertia of circuits. When current in a coil changes, the coil builds a magnetic field whose flux links with the same coil and with any nearby coil. A changing flux induces an emf that always opposes the change that created it. Self-inductance L measures how strongly a coil’s own current links magnetic flux with itself, while mutual inductance M measures how the current in a primary coil links flux with a secondary coil. The quality of this magnetic handshake is quantified by the coefficient of coupling k with 0 k 1 . Magnetic energy is stored in space where the field exists, not inside wires; for an inductor carrying current I , that energy is U= 1 2 LI 2 . Whenever bulk metal sits in a changing magnetic field, swirling loops of charges form inside it; these are eddy currents. They can waste energy as heat, but the same effect is useful in induction cooktops, electric braking of trains, and metal detectors. This lesson builds from Faraday’s law to practical formulas for L , M , k , and energy, shows how to avoid common sign and unit traps, and links these ideas to LC oscillations and AC power where inductors routinely appear. The stubborn thermostat analogy: a house (the coil) spends energy only when the temperature (flux) is changing. Once steady, the heater or AC (induced emf) shuts off. remember Core vocabulary Property of a circuit that opposes change in its own current by inducing an emf in itself; for an N-turn coil, N = LI and = -L , dI dt . Self-inductance (L) Property of two circuits where changing current in the primary induces emf in the secondary: 2 = -M , dI 1 dt . Mutual inductance (M) Coefficient of coupling (k) Measures how effectively two coils share flux. Defined by M=k L 1L 2 with 0 k 1 . The induced emf that opposes the change in current that produced it (Lenz’s law). In an inductor of inductance L, L=-L , dI dt . Back emf Eddy currents Closed-loop currents induced within bulk conductors due to changing magnetic flux, often causing heating. Energy per unit volume in a magnetic field: u B= B 2 2 0 (for linear, non-magnetic media). Magnetic energy density We use the right-handed Cartesian sign convention. Emf symbols carry the Lenz minus by definition; when computing magnitudes, take absolute values. For multi-turn coils, N is the flux linkage; this prevents the common mistake of writing =LI for an N-turn coil. Induced emf is proportional to the time rate of change of magnetic flux through the circuit. Faraday-Lenz law This law quantifies the induced EMF in a coil based on the rate at which the magnetic flux linkage changes. Flux linkage is proportional to current; L depends on geometry and medium. Self-inductance definition Opposes change in current through the coil. Back emf of an inductor This formula accurately calculates the self-inductance of a long, air-core solenoid, assuming uniform magnetic field distribution and neglig Determines the induced EMF in a secondary coil resulting from the time rate of change of current in a coupled primary coil. Mutual induction Emf in secondary due to current change in primary. Solenoid inductance Long solenoid approximation: n=N/l is turns per unit length. In AC circuits, this value determines the opposition to current flow based on the circuit's frequency and inductance. Self-inductance grows with the square of the number of turns because doubling turns both doubles the field for a given current and doubles the flux linkage. Adding a ferromagnetic core multiplies L by the relative permeability r until saturation limits further growth. tip Applicability: L= 0n 2Al holds for a long solenoid with uniform field inside and negligible fringing. For short coils or non-uniform fields, use energy or finite-element methods instead. neet-alert N-turn coil trap: Use N =LI , not =LI . The factor N is the most frequent lost mark in flux-linkage questions. Energy stored in an inductor Ideal inductor, resistance negligible L is constant (no core saturation) U = 1 2 L I 2 Energy required to establish current I in an inductor; released if current falls to zero. Linear, time-invariant coils Flux from primary linking secondary is a fraction k of primary flux M = k L 1 L 2 Coefficient of coupling relation Links mutual inductance to self-inductances via coupling factor k; ensures M L 1L 2 . The coupling factor k depends on coil separation, relative orientation, core material, and leakage flux. Tight, coaxial coils on a common ferromagnetic core can approach k 1 . Widely separated or misaligned coils have small k . Because 0 k 1 , mutual inductance never exceeds L 1L 2 . neet-alert Never report k > 1 or a negative k. In NEET problems, a negative sign may appear only from mutual orientation in induced emf expressions, not in the definition of k. Magnetic energy exists in the field, not inside the copper. Concentrating flux with a high- r core increases stored energy at the same current. In pulsed circuits, beware of large 1 2 LI 2 ; interrupting current suddenly can generate dangerous voltage spikes as the inductor tries to keep current continuous. Uniform field in linear medium without magnetic materials (or use of the medium for linear media). Magnetic energy density The energy density in the magnetic field determines the total stored energy in an inductor or the energy dissipated by eddy currents. Eddy currents: effects and control A bulk conductor exposed to changing magnetic flux develops circulating eddy currents in internal loops. According to Lenz’s law, these currents create magnetic fields that oppose the change. The I²R heating can be large because the conductor offers many wide current paths of low resistance. To reduce loss, machines use laminated iron cores: thin insulated sheets force currents to take narrow paths, increasing resistance and lowering eddy current magnitude. Induction cooktops: AC magnetic field induces surface currents that heat the pan directly. Electromagnetic braking: a metal rotor entering a magnetic field experiences eddy currents that oppose motion; smooth, contactless braking results. Energy meters and speedometers: damping via eddy currents steadies motion. Metal detection and flaw detection: changes in eddy-current response reveal material properties or cracks. Common applications of eddy currents They need changing magnetic flux. Even a stationary metal under a time-varying field develops eddy currents. Eddy currents need a moving conductor. They also allow large eddy loops and heavy heating. Laminations or ferrites are preferred at AC to suppress losses. Solid thick cores are always better because they carry more flux. Magnet near a copper solenoid: field lines arch from N to S and enter the coil where induced current sparks are illustrated. Bar magnet with field lines entering a copper coil; labels show induced current and solenoid. Clean diagram of magnet velocity through loop with induced current arrows. Schematic: a magnet moves through a circular loop; arrows show increasing flux and induced current direction (Lenz’s law). Artistic ring with flowing magnetic field creating induced current. Art depiction of Faraday’s law: changing magnetic flow through a conducting ring produces a bright induced current. Feature Self-inductance Mutual inductance Eddy currents What changes? Current in the same coil Current in a neighboring coil Magnetic flux through a bulk conductor Induced emf =-L , dI dt 2=-M , dI 1 dt Distributed emf loops; no single terminal Key parameter L depends on geometry and core M depends on both coils and k Material conductivity, thickness, frequency Typical effect Opposes change in I; stores magnetic energy Transfers signals/power; transformer principle Heating and magnetic damping; needs control Energy in the magnetic field rises quadratically with current. custom Current I Magnetic energy U control Parabolic increase of U with I; starts at origin and grows as 1/2 L I 2 . No current, no energy 1/2 L I 2 Stored magnetic energy Natural angular frequency of ideal LC circuit: =1/ LC . Angular frequency of LC oscillations = 1 LC Ideal inductor and capacitor No resistance (undamped) Average of a sinusoidal current over one half-cycle: I avg = 2I 0 . Average current over a half-cycle Sinusoidal current i=I 0 t Average taken from 0 to T/2 I avg = 2I 0 Average power for sinusoidal steady state: P avg =V rms I rms . Average power in AC circuit Sinusoidal voltage and current of same frequency P avg =V rms I rms Opposition of a capacitor to AC: X C= 1 C = 1 2 f C . X C = 1 C Ideal capacitor Sinusoidal steady state Capacitive reactance Although this concept centers on inductance, AC questions often combine L with C. Knowing =1/ LC , X C=1/ C , and P avg helps navigate mixed LCR contexts that appear frequently in NEET. A long air-core solenoid has length 0.40 , m , area 2.0 10 -4 , m 2 , and N=800 turns. Find its self-inductance and the induced emf when current increases uniformly from 0 to 2.0 , A in 10 , ms . Take 0=4 10 -7 , H/m . easy H, V L and | | during the ramp l=0.40 m, A=2.0 10 -4 m 2 , N=800 I: 0 2.0 A in 10 ms Use L= 0 N 2 A/l and | |=L ,dI/dt . Use M=k L 1L 2 and | 2|=M ,|dI 1/dt| . H, V L1=4.0 mH, L2=9.0 mH, k=0.80 dI1/dt = 250 A/s M and | 2 | medium Two coaxial coils have L 1=4.0 , mH and L 2=9.0 , mH with coupling k=0.80 . The current in coil 1 changes at dI 1/dt=250 , A/s . Find M and the magnitude of emf induced in coil 2. For series coupling: L eq ( ) =L 1+L 2 2M . hard Two inductors have L 1=2.0 , H and L 2=0.50 , H with mutual inductance M=0.60 , H . They are connected in series (i) aiding and (ii) opposing. Find the equivalent inductance in each case. Check that the values are physically consistent with the bound M L 1L 2 . L eq aiding and L eq opposing; verify bound L1=2.0 H, L2=0.50 H, M=0.60 H Series-coupled inductors: L eq =L 1+L 2 2M . Choose + if coil polarities aid the mutual field, − if they oppose it. tip In power devices, eddy-current loss roughly scales as frequency squared and thickness squared for similar materials. That is why high-frequency transformer cores use ferrites (high resistivity) or very thin laminations to keep heating manageable, while low-frequency devices can use thicker sheets. MEK hook: M = k√(L1L2). If k→0, coils are Magnetically Empty to each other; if k→1, they are Kissing (tightly coupled). Direction matters. The minus sign in Faraday’s law encodes Lenz’s law: the induced current produces a magnetic field that opposes the change in flux. In practice, choose a reference direction for loop traversal and normal vector; a consistent right-hand rule then fixes the algebraic sign of . It resists change in current. In steady DC after transients, ideal inductors behave like short wires (zero voltage drop). An inductor resists current. Ideal inductors consume average power in AC. In a pure inductor, voltage leads current by 90 , so average power over a cycle is zero; energy shuttles to and from the magnetic field. Unit vigilance: 1 , H =1 , Wb/A =1 , V ,s/A . Microhenry and millihenry conversions are frequent traps. neet-alert In laboratories, mutual inductance is measured by driving a known sinusoidal current in the primary and reading the induced emf in the secondary with a lock-in amplifier. The phase confirms Lenz’s law, and the ratio yields M once is known. Transformer action is a practical example of mutual induction. In the ideal case, k 1 and nearly all primary flux links the secondary. Real devices have leakage flux, copper resistance, and core losses (hysteresis and eddy currents), each modeled as separate loss elements. Reducing eddy-current losses Use laminated steel cores with insulation between sheets to increase path resistance. Choose ferrites (high resistivity) for high-frequency applications. Shape the core to minimize unnecessary flux spread. Avoid solid conducting masses near strong time-varying fields. When a metal plate swings between magnet poles, eddy currents damp its motion. The induced currents are stronger when velocity is higher because the flux through a given loop changes faster. This is why eddy-current brakes give smooth, speed-proportional retarding force without contact. Scenario What changes? Math tool Edge condition Current step in inductor dI/dt large at t=0 L=-L ,dI/dt If L→0, response becomes resistive, not inductive Transformer coupling Primary current varies 2=-M ,dI 1/dt If k→1 and leakage→0, ideal transformer behavior Static magnet at rest No change in flux d /dt=0 No induced emf or eddy currents Average power in AC (worked with phasors) Same as earlier derivation; presented as phasor view for reinforcement P avg =V rms I rms Energy language connects geometry to circuit values. Compute L from geometry when possible, then predict transient response or stored energy. Conversely, measure L electrically and infer magnetic design quality ( k , leakage) for coupled coils. Dot convention: a current entering the dotted terminal of coil 1 produces a positive mutual voltage at the dotted terminal of coil 2. If both currents enter dotted ends simultaneously, coupling is aiding; if one enters a dotted end and the other leaves its dotted end, coupling is opposing. Repeated for emphasis: average power depends on the power factor . Dimensional checks are quick sanity tests. [L]=[ H ] = [ Wb/A ] , [M] has the same dimension as L , and magnetic energy U has units of joule. For M=k L 1L 2 , k is dimensionless, guaranteeing consistent units. Half-cycle average of sine (alternate route) I avg = 2I 0 Use symmetry of sine wave Mutual inductance can be negative only as a sign in induced-voltage expressions when coil orientation is chosen oppositely; the parameter M representing magnitude is taken positive, while the sign information is carried by the dot convention or the chosen reference directions. Steady state DC in an ideal inductor: =0 , current constant, magnetic field constant, energy fixed at U= 1 2 LI 2 . remember Power dissipation by eddy currents is fundamentally resistive. Because induced emf scales with the rate of flux change, which is proportional to frequency, the induced current and therefore P f 2 for geometrically similar situations. Thus even modest frequency increases can overheat solid metal parts. u B = B 2 2 Linear, isotropic medium with permeability Energy density in uniform field Coupled-coil transients: if a switch opens in a circuit with significant L, the inductor will generate a large voltage to keep current continuous, which can arc contacts. Practical circuits include snubbers, freewheeling diodes, or RC networks to tame this effect. Mutual inductance depends on geometry and the magnetic circuit. Coaxial solenoids share more flux than side-by-side ones. Introducing a high-permeability core that threads both coils increases L 1, L 2, and, often more dramatically, M , raising k toward unity. For NEET numericals, keep two significant figures unless stated otherwise. Convert mH to H and µH to H carefully. If given turns per unit length and core permeability, compute L first; then use =L ,dI/dt or U= 1 2 LI 2 as required. Half-cycle averaging appears in rectifier and measurement questions; do not confuse with RMS. Capacitive reactance often pairs with inductive effects in mixed LCR settings. A quick check for realism: if two coils are far apart (tiny k), M must be small and the induced emf negligible. If a question yields a large 2 with tiny k , re-examine units or the dI 1/dt you used. Laminations work because each sheet is electrically isolated. The induced currents are then confined to small loops within each sheet, sharply increasing resistance of the path and reducing I²R loss. Ferrites go further: their resistivity is so high that eddy currents are inherently weak even at high frequency. Recap glossary Flux linkage linkage Product N linking a coil; equals LI for linear coils. Self-inductance Proportionality constant between flux linkage and current in the same coil. Proportionality constant linking induced emf in one coil to rate of change of current in another. Mutual inductance Coupling factor Fractional flux linkage between two coils; M=k L 1L 2 . Back emf Induced emf opposing the change that created it. Looping currents in bulk conductors created by changing flux; cause heating. Eddy currents Magnetic energy Stored field energy U= 1 2 LI 2 .