INTRODUCTION
Electrons exhibit orbital motion and carry an inherent magnetic moment, explaining how iron, nickel, and cobalt become ferromagnetic. Their atoms align at the atomic level when exposed to an external magnetic field, revealing magnetic properties. magnetism in physics fundamentals
Hans Oersted changed science in 1819 when his lecture demonstration showed a live compass needle deflecting near a current-carrying wire — proving moving electric charges generate magnetic fields, sparking the entire field of modern electromagnetism.
From MRI machines to particle accelerators and Van Allen belts, the Lorentz force law governs charged particles through magnetic fields, while Maxwell’s equations ultimately show electricity and magnetism as one inseparable, unified force in nature. electromagnetism as a fundamental force of nature
MAGNETIC POLES
Every magnet carries two inseparable north pole and south pole regions — a fundamental truth I find endlessly fascinating. These field lines emerge from North, arc outward, and converge back toward South, revealing the force geometry visually.
Permanent magnets demonstrate this phenomenon most clearly: iron filings scattered nearby instantly rotate and align, tracing invisible field lines with striking precision. This attractive behavior toward ferromagnetic materials reflects magnetism as a deep physical phenomenon rooted in motion of electric charges.
What makes poles genuinely remarkable is their location dependency — magnetic north never coincides with geographic North, and the magnetic South pole sits near Earth’s rotational axis, making compass navigation a carefully connected, historically navigational science.
Rule of Action (Law of Poles)
The north pole and south pole of any magnet define the regions where forces are strongest — and what I’ve consistently observed is that attractive behavior between north-south pairs isn’t random; it’s structurally governed. Field lines always emerge from N and enter at S, encoding directionality into every interaction.
Most everyday objects — toys, doorbells, hangers, even elevators and computer devices — rely on repulsive and attractive dipoles working in opposition. What’s fascinating is that two dipoles, three dipoles, or even four dipoles arranged in proximity don’t cancel — they compete, producing multipoles like quadrupole, sextupole, hexapole, and octupole configurations that are labeled but rarely discussed.
You cannot isolate an individual magnetic pole — break a magnet into pieces and each fragment retains two poles. Down to the subatomic level, protons and neutrons carry magnetic dipole moments. Theoretical physicists have long predicted magnetic monopoles, yet none remain verified — the unverified single-pole object stays one of physics’ most elegantly isolated puzzles.
Compass
Long before mankind understood electricity or the physics of moving charges, the compass needle was already silently doing its job — pointing travellers toward the geographic North Pole with unsettling reliability, guided entirely by Earth’s invisible magnetic field.
What most overlook is that Earth behaves like a giant dipole magnet, where the magnetic south pole sits near the geographic North Pole — so the compass’s north pole is actually attracted toward it, aligning with the local field direction in real time.
The compass doesn’t just point north; it reveals orientation across any surface, responding to magnetic strength that varies from roughly 45 ± 15 μT — a modest force, yet enough to rotate a magnetized needle and anchor every navigation system ever built on it.
Magnetic Field
A magnetic field is a vector field measured in teslas, observed wherever moving charges or current flow. Its field lines always form closed loops and never intersect — unlike electrostatic forces, which originate from static charges and govern electric field behavior.
Earth’s magnetic field behaves as a giant dipole magnet with magnetic strength of 45 ± 15 μT at the surface. Its orientation shifts because the magnetic axis tilts roughly 10° from the rotational axis, placing the geographic North Pole near a south magnetic pole.
For a moving charged particle, B exerts force perpendicular to both velocity and field, with magnitude proportional to charge and speed. A static magnetic field does no work since Power P = F · v = 0 — direction changes, but kinetic energy stays constant.
Geomagnetism / Earth’s Magnetic Field
Earth behaves like a giant bar magnet, generating a magnetic field that extends far into space. This vector field, measured in teslas, shields life from cosmic rays and dangerous charged particle streams constantly bombarding our atmosphere.
The geographic North Pole aligns near Earth’s south magnetic pole — a fact most overlook. A compass needle’s north pole is attracted to this region, confirming the dipole magnet nature of Earth’s field lines converging and diverging across hemispheres.
Earth’s magnetic orientation isn’t permanent — it flips at long time intervals, confirmed through ancient rock records. Physicists studying magnetic properties of iron oxides and magnetite understand these naturally occurring reversals, reshaping how we interpret historically documented navigational anomalies worldwide.
Magnetic Materials
Not all materials respond to magnetic fields the same way — and that asymmetry is where the real physics lives. Ferromagnetic substances like transition metals, alloys, and rare earths such as gadolinium and dysprosium contain strongly aligned magnetic domains, enabling permanent magnets and magnetite (Fe₃O₄) to hold their magnetic state without external reinforcement.
Paramagnetic and diamagnetic behaviors sit at opposite ends of a spectrum. Weakly attracted molecules and compounds — including molecular oxygen (O₂) — exhibit only ever weak, temporary responses, while diamagnetic elements like carbon, mercury, and water are weakly repelled, aligning opposite to the applied field with equally weak, fleeting effects.
The superdiamagnetic extreme belongs to superconductors: below a very low temperature (T ≲ 10 K or T ≲ 100 K), these materials enter a superconducting state where fields cannot penetrate beyond a thin surface layer — the Meissner effect — producing strongly repelled, magnetic levitation that no ordinary material can replicate.
Ferromagnetic Materials
At the molecular level, ferromagnetic materials reveal something most textbooks understate — it is not merely magnetism but a fierce internal competition where prevails one dominant alignment force. Domains lock together, surrendering individual identity for collective magnetic order that defines the ferromagnetic character entirely.
What gives iron, nickel, cobalt their reputation is that upper hand — the quantum-mechanical tendency where molecular interactions tip the material toward spontaneous magnetization. Once you’ve handled a magnetized steel rod firsthand, you sense this internal level of coordination that no paramagnetic substance ever demonstrates.
Every ferromagnetic substance carries a hidden threshold — the Curie point — beyond which thermal energy wins the competition. That boundary, invisible yet absolute, decides whether the material retains magnetic alignment or collapses into disorder. Understanding this prevails as the most overlooked concept in applied magnetism courses today.
Paramagnetic Materials
Paramagnetic behavior emerges where weakly attracted atoms carry unpaired electrons — each acting as a tiny magnet. Unlike ferromagnets, these domains never self-organize; instead, thermal agitation perpetually disrupts alignment, making magnetization inherently conditional and reversible.
Apply an external magnetic field and something remarkable occurs: electron spins partially orient along field lines, producing induced magnetization proportional to field strength. Remove that applied field, and the random motion restores disorder — no remnant magnetic state survives independently.
Susceptibility defines how readily a paramagnetic material responds — governed by the Curie Law: χ = C/T, where higher temperature actively weakens response. Aluminum, platinum, and oxygen exemplify this class, each demonstrating measurable yet modest magnetic behavior under controlled conditions.
Diamagnetic Materials
Unlike paramagnetic materials that weakly align with fields, diamagnetic substances are actively repelled from any nearby external magnet. Their paired electrons generate opposing moments, resisting flux. Magnetic levitation of bismuth demonstrates this subtle repulsion beautifully.
Most metals exhibit non-magnetic behavior through this mechanism. Interestingly, austenitic steels display diamagnetic tendencies, unlike ferritic steels. Even water is diamagnetic — a fact confirmed when researchers suspended water droplets using strongly focused superconducting coil arrangements.
Diamagnetic effects aren’t limited to exotic minerals alone; copper, bismuth, and organic tissues also qualify. Though weak, the response obeys quantum mechanical rules tied to spinning electrons, vanishing entirely once the external field is removed.
Superdiamagnetic Materials
Unlike diamagnetic materials that weakly align against external fields, superdiamagnetic substances achieve near-total magnetic flux expulsion. This extraordinary behavior in superconducting compounds represents a magnetic response so intense that magnets levitate effortlessly above their surfaces.
Beyond simple repulsion, superdiamagnetic levitation enables high-speed trains to steer and accelerate without friction. Unlike permanent magnets or ferromagnetic materials, these compounds maintain magnetic containment through quantum flux pinning — a mechanism I find profoundly elegant.
Superdiamagnetic research intersects medicine, where magnetic resonance imaging reveals brain activity patterns impossible to detect otherwise. The quantum mechanical basis of magnetic spins represents a frontier consistently surprising even experienced researchers studying these extraordinary materials.
Types of Magnets
Naturally occurring magnetite and iron oxides first exposed nature’s core magnetic properties to early observers. Permanent magnets, made from ferromagnetic materials including Fe₃O₄, sustain their magnetic state indefinitely — a fundamental category underlying modern scientific inquiry.
Electromagnets — driven by flowing electric current from a power source — offer precise control over magnetic force. Electric motors, generators, cars, and refrigerators rely on this switchable category, where field strength responds directly to applied current.
Ferromagnetic materials like Heusler alloy and pyrite expose how microscopic current loops within matter define precise type classification. The alignment of atomic magnetic dipoles separates strongly attracted substances from those exhibiting repulsive or neutral behavior.
Brief History of Magnetism / Electromagnetism
Long before nineteenth century physicists theorized formally, ancient Greeks discovered iron compounds displaying mysterious attraction. This curiosity, much like a lightning bolt of insight, introduced basic ideas foundational to modern everyday life and scientific review.
Historically, the field evolved dramatically when Ampère in 1820 connected electricity to magnetism. Later, Faraday’s effect redefined circuit behavior, proving metals and producing forces weren’t merely dangerous curiosities but time-dependent phenomena worth serious scientific investigation. electromagnetic circuit laws and theory
By 2007, Albert Fert and Peter Grunberg earned the Nobel Prize in Physics for giant magnetoresistance. From MRI to the humble refrigerator magnet, magnetism evolved from ancient mystery into time-independent, safety-critical infrastructure defining technological civilization.
Contemporary Applications of Magnetism
Modern mass spectrometers exploit uniform magnetic field zones to separate ions by q/m ratios, enabling precise isotope detection. Cyclotrons accelerate protons and alpha-particles using cyclotron frequency principles, thus revolutionizing cancer treatment through targeted particle therapy.
The electric motor and DC motor convert magnetic torque into mechanical rotation, while Hall effect sensors detect drift in semiconductors. Velocity selectors filter charged beams across applications, relying on electric fields balancing magnetic force precisely. electromagnetic fields and digital signal processing
Faraday’s Law converts magnetic flux into induced current within solenoid cores. Self-induction sustains LC circuit resonance for wireless charging, while mutual induction powers transformers. Earth’s magnetic bottle traps solar particles, producing the Aurora Borealis naturally.
Magnetic Force (Lorentz Force Law)
The total force on a moving charged particle combines electric force and magnetic influence simultaneously. When velocity and B are perpendicular, the resulting push defines direction of circular motion, with magnitude tied directly to speed.
Magnetic action never does work on any charged particle because the force always remains perpendicular to motion. The cross product v × B governs direction, and sign of charge flips the entire force vector completely.
In crossed fields, condition E = vB ensures zero force, enabling a velocity selector that steers electron beams precisely. This q/m ratio from energy balance transformed how physicists interpret mass and electric potential difference remarkably.
Motion of Charged Particles in Magnetic Fields
When positive charges move through a B field, the right-hand rule governs deflection. The resulting circular motion keeps speed constant—confirming zero power transfers, since the force always acts perpendicular to field, bending not accelerating.
When v⊥ and v‖ coexist simultaneously, trajectories form a helical path. The parallel component drives linear drift while the perpendicular component maintains circular motion, building a helix—geometry rarely appreciated outside specialized plasma confinement studies.
Adding E alongside B reshapes trajectories fundamentally. At specific velocity, forces cancel, yielding straight line motion. Otherwise, crossed electric and B fields produce cycloid paths—an outcome that consistently surprises experienced practitioners studying electromagnetic dynamics.
3D Motion of Charged Particles
Charged particles entering a magnetic field at any angle experience centripetal force on the perpendicular velocity component, defining the circular radius r = mv/(qB), while the parallel component remains unaffected, together creating helical 3D trajectories.
The period T = 2πm/(qB) and f = qB/(2πm) are both independent of velocity — a faster particle simply carves a larger circle yet always completes each revolution in identical time, underpinning the cyclotron resonance principle.
Speed’s constant magnitude remains constant throughout all helical travel, yet the direction continuously changes as the particle steadily revolves around the field axis. The pitch — axial advance per cycle — grows with the parallel velocity component.
Motion in Combined B and E Fields
When charge q moves through crossed fields, the cross product in F = q(E + v × B) defines overall behavior precisely. The electric term acts regardless of speed, while qvB maintains a constant interplay with magnetic forces steadily curving particle trajectories.
The velocity selector appears when F = 0: electric and magnetic forces balance precisely. Specific v passes straight while others deflect. This detection principle drives mass spectrometers, where particle source separation reveals charge-to-mass ratios despite natural resistance from field interactions.
In 3D combined field scenarios, motion divides: parallel drift traces field direction while perpendicular forces curve the path. The axis orientation determines helical pitch. Real devices like spectrometers employ F = qv × B to filter particles by velocity.
Magnetic Flux and Gauss’s Law for Magnetism
Unlike electric flux, magnetic flux carries a deeper constraint — no magnetic charges exist, so every field line that enters a closed surface must exit it. The ∫ B · dA integral always returns zero, a fact that reshapes how physicists model source behavior.
What makes this law quietly profound is its connection to dipoles: break any magnet into pieces, and dipoles persist down to the subatomic level. The surface integral confirms that flux cannot accumulate — field lines form closed loops, never terminating on isolated poles.
Practitioners often overlook that Gauss’s Law for magnetism is not merely valid mathematically — it encodes conservation at a fundamental level. Strength, density, and symmetry of B field distributions all obey this silent rule governing every contour in nature.
Force on a Current-Carrying Wire
What fascinates me about a long straight wire carrying current I is how measurably the net magnetic force emerges. F = BIL sin(angle θ) ties B field magnitude directly to wire length and strength.
Using vector form F = IL × B field, direction follows a right-hand rule — curled fingers align with currents, curl toward field lines, while thumb shows force. Maximum force occurs when angle θ = 90°.
Battery-powered setups I’ve observed in labs make this vivid — current I inside a B field generates torque, causing coils to rotate. Force is proportional to strength, wire length, and I; ammeter designs confirm this.
Magnetic Torque on a Current Loop / DC Motor
In a uniform magnetic field, a rectangular current loop faces net torque, not simple translation. Segments cancel the net force, yet F = iLB on parallel straight wire sides drives steady rotation about the axis.
The DC motor converts rotational torque directly into mechanical work. Current-carrying coils of wire wound around a rotor align with magnetic poles; direct current commutation sustains revolving motion, converting electromagnetic energy into continuous rotational output.
Technological applications like elevators and hard drives exploit this torque principle. Conducting current loops inside a permanent magnet experience torque proportional to flux, where N turns multiply the effect, scaling rotational force with loop geometry.
Magnetic Fields from Currents
Current flows and closed loops of vector field emerge—exhibiting no magnetic charges, unlike electric dipoles. This surpasses mere visual representation; the density of field lines reveals field strength, and they never start, never end.
Symmetry makes Ampère’s theorem genuinely elegant. Circulation of B · ds along a good contour yields the enclosed current—field strength scaling inversely with distance, a contribution that transforms abstract law into measurable, repeatable results.
André Marie Ampère demonstrated that one current becomes a source of magnetic phenomena. Electromagnets—a coil connected to a power source—now enable computer hard drives, navigation systems, and life-saving medical diagnostic tools used worldwide.
Magnetic Force Between Two Parallel Wires
What struck me first: like poles repel, yet this rule of action governs wires too. Same-direction conductors exhibit attractive forces; reversed direction of current produces repulsive forces — a simple empirical result encoding profound electromagnetic truth.
Force per unit length F/L = μ₀I₁I₂/(2πd) is inversely proportional to wire separation. μ₀, the magnetic permeability constant at 4π × 10⁻⁷ T·m/A, governs all interactions — increasing current or reducing distance amplifies this field.
This interaction formally defined the SI ampere. Using the screw rule, one determines field direction around each wire. The source traces back to molecular currents — a profound link between macroscopic conductors and microscopic charge behavior.
Biot-Savart Law
Jean-Baptiste Biot and Felix Savart discovered the magnetic effect of stable electric currents. A metallic wire carrying current I follows dB = (μ₀/4π)(I dl × r̂)/r², where integration over each current element readily yields total magnetic field at any location.
Integration along a circle of radius R yields μ₀I/(2R) at the center. This sharply contrasts with B = μ₀I/(2πr), which is valid for infinite straight wires — a comparison that reveals the physical reason geometry shapes field magnitude.
Unlike Ampere’s loop, the Biot-Savart Law handles arbitrary wire paths without symmetry. Circular field lines diverge from the origin, and I dl × r̂ encodes field direction, making it useful for any current density geometry.
Ampère’s Theorem (Law)
What most students overlook about Ampère’s Law is its elegant dependency on surface geometry. The chosen contour only needs to enclose relevant current; the actual path shape, whether circular or square, never alters final results.
The theorem’s applicability condition hinges on current symmetry. In-page currents and out-of-page currents determine the CCW or clockwise orientation of magnetic field lines. Recognizing static charge distributions helps distinguish truly Ampèrian scenarios from induction-based ones.
Ampère’s Law elegantly extends to complex shape geometries via the superposition principle. For parallel wires carrying same direction currents, individual contour contributions sum cleanly — unlike Biot-Savart’s per-element method, which demands full wire element geometry knowledge.
Solenoid
Most courses present solenoids using the straight solenoid formula B = μ₀nI, rarely unpacking the closed contour logic behind Ampère’s elegant derivation. That rectangular path, traced carefully around the cylinder, exposes why external fields collapse to zero. A-level magnetism and electromagnetism notes
Inside, lines stay perfectly parallel — quite unlike the looping field lines around an isolated wire. Increasing N turns raises ienc proportionally, strengthening the internal field while leaving the surrounding external region essentially undisturbed and field-free.
Solenoids show time-dependent behavior — a change in current shifts induced flux through nearby conducting loops, generating EMF. This rate of change dynamic is why solenoids remain central to electric generator design and transformer engineering worldwide.
Toroidal Coil
A toroidal coil wraps tightly-wound wire into a ring-shaped core. Unlike a long coil, the field remains entirely confined inside. Applying Ampère’s theorem with µ₀Ienc gives B = μ₀NI/(2πr), where r is the radial distance.
Outside the toroid, fields cancel completely, a remarkable contrast to solenoids. The N turns wound around a torus create zero external flux. This self-contained uniform field makes toroids ideal for circuit diagrams and precision devices.
What practitioners appreciate most is how cross-sectional shape barely affects internal field strength. Whether smooth or rectangular, B depends only on r, current, and N, not geometry. This independence distinguishes toroids from all other coil configurations.
Faraday’s Law of Electromagnetic Induction (Part 1)
Michael Faraday established that electromagnetic induction arises purely from relative motion. When a bar magnet approaches a non-conducting loop, changing magnetic flux triggers an induced EMF — the response grows larger and faster as motion intensifies.
Faraday’s law captures this precisely: the negative sign within ε = −dΦ_B/dt naturally encodes opposition. A frame with varying B = B(t), or a conducting rod at velocity v, both produce induction through flux variation.
Joseph Henry and Heinrich Lenz, working in the same century as Faraday, reinforced how conservation of energy governs the induced field. Their experimental validation of ε = vLB for moving conductors confirmed electromagnetic induction’s universality.
The Law
The direction of EMF in any loop isn’t dictated by flux magnitude alone, but by how fast it actually changes. Faster variation reliably generates larger EMF, making time the true variable every practitioner must track.
Opposes — that single word grows into Faraday’s most misunderstood concept. When flux always penetrates a circuit out of page, the response becomes counter-clockwise per Lenz’s rule, designed to resist the original magnetic flux increase naturally.
The induced electric field, produced entirely from changing flux, has no start and no end — unlike the electrostatic field. These similarities and differences clarify why charges follow a tangent path, with work W moved continuously.
Lenz’s Rule
Practitioners miss what Lenz’s Law truly protects—total energy. When net magnetic flux shifts through a loop, opposition emerges, sustaining constant electromagnetic balance. I’ve found this countering behavior foundational in understanding free oscillations during induction.
The interplay between electric energy and magnetic energy during opposition reveals deeper symmetry. ε = −B · dA/dt captures this resistance—the negative sign embodies Lenz’s law itself, not a formatting choice or mathematical accident.
Consider induced magnets demonstrating Φ_B shifts through a conductor — these triggered currents counter that exact change. The system’s natural frequency governs reaction, while ∮E · ds = −dΦ_B/dt extends Lenz’s principle beyond simple circuits.
Induced Electric Field
The induced electric field has no begin or end — unlike magnetic induction, it forms fully closed, non-conservative loops in vacuum. A changing Φ_B ≈ BA cos[φ(t)] creates this field throughout surrounding space, conductor or not.
ε = −A · dB/dt quantifies the induced field precisely. The derivative of flux governs angular frequency ω of oscillations, while RLC circuit analysis reveals how induced E-field magnitude behaves across different geometries and scenarios.
The equation linking surface-integrated flux to E-field circulation is fundamental. The sum of E around any closed path exposes the absence of scalar potential energy, confirming why currents and charge-free regions alike feel its influence.
Self-Induction and Inductance
Self-induction occurs when current in inductor changes, altering field through each winding. In a solenoid, B = μ₀nI scales with turns per length, so every variation induces a back-EMF opposing itself within the same coil.
The SI unit for inductance is the henry. When maximum current flows, self-induced EMF equals zero. Studying charge on capacitor behavior reveals how current in inductor lags, forming the fundamental basis for oscillatory electromagnetic circuits.
Using Euler’s formula and complex exponentials, inductive behavior maps directly onto e^(±iα) representations. When zero current exists, stored energy vanishes entirely, revealing how the magnetic dipole moment framework elegantly captures inductance beyond simple circuit equations.
Electromagnetic Oscillations (LC and RLC Circuits)
In an LC circuit, energy continuously transfers between capacitor and inductor, producing free oscillations. Unlike static stable equilibrium in mechanics, this system oscillates perpetually at resonance frequency ω₀ = 1/√(LC), with no permanent energy loss.
The highest energy state occurs when capacitor is fully charged; maximum current shifts lowest energy entirely to the magnetic field. This oscillating magnetic flux mirrors spring-mass dynamics, completing one cycle per period T = 2π√(LC).
Adding resistance R transforms LC into damped RLC oscillations. Energy dissipates each cycle, ultimately reaching stable equilibrium at zero amplitude. External AC sources drive forced resonance, maximizing current when driving frequency equals natural ω₀ precisely.
LC-Circuit, Free Oscillations
An LC circuit’s free oscillation exposes a fundamental paradox: zero charge on the capacitor signals maximum energy transfer, not absence. The period formula T = 2π√(LC) reveals that inductance and capacitance alone ultimately dictate the rhythm of exchange.
Starting from a fully charged capacitor holding Q_max, the differential equation solution naturally demands the imaginary unit i = √(−1). Euler’s famous formula cos α ± i sin α provides a clean exponential shortcut, while t/T normalizes time into one complete oscillatory cycle.
Verifying that d/dt e^(±iωt) = ±iωe^(±iωt) truly holds is genuinely satisfying — it confirms that both cos α and sin α oscillate at completely identical natural rates. Near maximum charge, energy follows a reduced time pattern, cycling seamlessly between electric and magnetic domains.
Maxwell’s Equations and Electromagnetic Waves
Maxwell’s four equations don’t merely describe fields. They expose the deep unified architecture of physical reality, where Gauss’s Law for electricity ties charge to flux, and Gauss’s Law for magnetism firmly confirms monopoles cannot exist.
The Ampère-Maxwell Law completed the four equations by introducing displacement current. Maxwell then elegantly derived electromagnetic waves traveling at c = 1/√(μ₀ε₀), precisely 3 × 10⁸ m/s, identifying the speed of light from pure mathematics.
Maxwell’s unified view proves ∮B · dA = 0 without exception, ruling out magnetic monopoles. |B|, expressed in Wb/m², oscillates transversely within propagating waves—a relationship first predicted analytically before experimental verification confirmed it.
Relationship Between Electricity and Magnetism
In 1820, Danish physicist Hans Ørsted noticed a compass needle deflect near a current-carrying wire, proving that moving electric charges generate magnetic fields. This revelation stunned the French Academy of Science and transformed physics forever.
Michael Faraday, inspired by Alessandro Volta’s electric battery, demonstrated that a magnetized core linked to a galvanometer registers current only during flux changes, ultimately proving EMF arises not from charges but from dynamic shifting fields. explore magnetism and electromagnetic concepts on HyperPhysics.
Maxwell unified both realms by demonstrating that negative charges always experience reversed Lorentz force directions. His four elegant equations collectively proved that oscillating electric and magnetic fields perpetually generate each other, propagating as electromagnetic waves.
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