Magnetism & Matter Magnetism & Matter Magnets feel familiar: an iron pin jumps, a compass settles, a speaker cone vibrates. Under these effects sits a clean idea—magnetic dipoles. A bar magnet is treated like a tiny current loop (or a bundle of loops), with a dipole moment vector from its south to north end. Surrounding matter responds to an applied magnetic field by building its own dipoles: some oppose the field (diamagnetism), some slightly align with it (paramagnetism), and some align strongly by forming domains (ferromagnetism). This alignment is summarized by magnetization (the net dipole moment per unit volume). In simple materials, magnetization is proportional to the applied magnetic field intensity; in ferromagnets, it is not—history matters, giving hysteresis. A bar magnet’s field far away looks exactly like the field of a short magnetic dipole: strongest along its axis, weaker on the perpendicular bisector, both falling off sharply with distance as 1/ r 3 . The Earth itself acts like a giant dipole: its field points at a downward angle (dip) in the magnetic meridian; this field resolves into a horizontal part that guides compasses and a vertical part that pushes the needle to tilt. Understanding how matter amplifies, distorts, or resists a field, and how a dipole sits, turns, and oscillates in Earth’s field, ties together the chapter: from neutral points near a magnet to choosing materials for transformer cores and permanent magnets. Think of a bar magnet as a tiny, well-organized crowd of current loops. Far enough away, all their twists blur into one simple dipole with moment m, just like a very short solenoid. remember Magnetic Dipole A system that produces a magnetic field similar to a tiny current loop or a short bar magnet, characterized by a magnetic dipole moment vector. Magnetic Dipole Moment (m) A vector that measures the strength and orientation of a dipole; for a current loop, m = N I A (direction by right-hand rule normal to the loop). Magnetization (M) Net magnetic dipole moment per unit volume of a material. Units: A/m. Indicates how strongly the material is magnetized. Magnetic Field Intensity (H) Auxiliary field that represents the applied field from free currents; inside materials, B = μ0(H + M). Units: A/m. Magnetic Susceptibility (χm) Dimensionless proportionality constant linking magnetization and field intensity in linear, isotropic media: M = χm H. Ratio of material permeability to vacuum permeability: μ = μr μ0; for linear media, μr = 1 + χm. Relative Permeability (μr) In matter, two fields are central: H (from free currents) and B (the total magnetic field). Magnetization M arises from bound currents within the material. In linear, isotropic media, M aligns with H and scales as M = χm H, making B = μ0(H + M) = μ0(1 + χm)H = μ H. Diamagnets have small negative χm, paramagnets small positive χm, while ferromagnets have large, nonlinear, history-dependent M that can persist even after H is removed (remanence). Relations linking B, H, M, susceptibility, and permeability; valid for linear, isotropic materials. Fields inside materials This set of equations applies to linear, isotropic materials where the magnetic response is proportional to the applied field strength. tip Units: use SI. B in T (tesla), H and M in A/m, μ0 = 4 × 10 -7 T·m/A. Keep vectors in mind: directions of H, M, and B align only in simple linear, isotropic cases. A bar magnet can be modeled as an equivalent current loop (or a tightly wound short solenoid). The net result is a dipole moment equal to current times area (for a loop) or magnetization times volume (for a uniformly magnetized bar). This is why distant field lines of a short solenoid and a bar magnet are indistinguishable. Two consistent ways to compute a dipole’s strength. Dipole moment models This definition applies to calculating the total magnetic strength generated by current-carrying loops, solenoids, or bulk materials that ar Field of a short magnetic dipole: along the dipole axis (axial line), the field is stronger; on the perpendicular bisector (equatorial line), it is weaker by a factor of 2. Both vary as 1/ r 3 , so they drop rapidly with distance. Directions matter: on the axial line, the field is along the dipole moment; on the equatorial line, it is opposite to the dipole moment. Magnitudes for r dipole size; directions: axis (along m), equator (opposite m). Dipole field (short dipole) This approximation is valid when the observation point is much farther away from the dipole than the dipole's size (r >> L). Point/dipole approximation: observation distance r is much larger than magnet length (r ≫ l). Uniform magnetization or equivalent small current loop. Free space (vacuum) outside the magnet. Magnetic field of a short dipole on axis and equatorial line B axis = 0 4 2 m r 3 , B equator = 0 4 m r 3 Scalar magnetic potential of a dipole (outside sources). Relate potential gradient to field (in current-free region). Evaluate along the dipole axis ( r parallel to m ). Evaluate on the perpendicular bisector ( r m $). Dipole formulas hold only when the observation point is far compared to magnet size. Very near the poles, the short-dipole model breaks down and real geometry matters. For NEET, assume r ≫ l unless the question provides actual pole separations or dimensions. Earth’s magnetic field behaves like a big, slowly changing dipole. At any location, the field vector lies in the magnetic meridian (a vertical north–south plane). It makes an angle of dip (inclination) with the horizontal and can be resolved into a horizontal component, which steers compass needles, and a vertical component, which tilts them. Another angle, declination, is the small horizontal angle between true geographic north and magnetic north. The vertical plane passing through the magnetic north–south direction at a place; Earth’s field lies in this plane. Magnetic Meridian Angle between Earth’s magnetic field and the horizontal in the magnetic meridian; δ = 0° at the magnetic equator, 90° at magnetic poles. Angle of Dip (δ) The angle between geographic (true) north and magnetic north, measured in the horizontal plane. Angle of Declination (θ) Components of Earth’s total field (B) in the magnetic meridian: BH lies horizontal, BV vertical downward in the northern hemisphere. Horizontal (BH) and Vertical (BV) Components Earth’s field components Resolved components of Earth’s field in the magnetic meridian. Right-angled triangle formed by components in meridian plane. By definition of dip δ with respect to horizontal. Direct component relations. Useful derived relation connecting components and dip. B H = B , B V = B , = B V B H Earth’s field B is uniform locally and lies in the magnetic meridian. Resolve B into orthogonal horizontal and vertical directions. BH = B cos δ and BV = B sin δ Resolve Earth’s field B in the magnetic meridian: BH = B cos δ, BV = B sin δ; tan δ = BV/BH. Typical Earth field strengths are 25–65 microtesla. At the magnetic equator, the dip is zero (field horizontal), so BH equals B and BV is zero. At magnetic poles, the dip is 90°, BH is zero, and BV equals B. In India, values lie in between: a non-zero dip and a sizable horizontal component that governs compass directions. B = 50 10 -6 T = 60 Compute horizontal component. Compute vertical component. easy BH and BV At a place, Earth’s field magnitude is B = 50 µT and dip δ = 60°. Find BH and BV. Use BH = B cos δ and BV = B sin δ. tesla Do not confuse dip (δ, tilt in the vertical plane) with declination (θ, horizontal angle between true and magnetic north). Also, if your calculator expects radians, convert degrees before using sin or cos. neet-alert Neutral points appear where the magnet’s field cancels Earth’s horizontal component. For a short bar magnet with its axis along BH, neutral points lie on the equatorial line because the magnet’s equatorial field opposes BH. The cancelation condition gives a neat 1/3-power distance law, revealing how sensitive neutral points are to both the magnet’s moment and local BH. medium Rearrange for r 3 . Numerical substitution. Cube root. m = 0.80 A , m 2 B H = 3.0 10 -5 T 0 4 = 10 -7 T , m/A At neutral point on equator: magnet field opposes BH and cancels it. Use Beq = (μ0/4π) m / r 3 = BH. A short bar magnet (dipole moment m = 0.80 A· m 2 ) is oriented along the magnetic meridian. If BH = 3.0 × 10 -5 T, find the distance r on the equatorial line where the net horizontal field is zero. Neutral point distance r on equatorial line A magnetic dipole in a uniform field turns to align with the field. The torque magnitude is τ = m B sinθ, zero when the dipole is parallel or antiparallel to the field. The potential energy U = − m B cosθ is minimum when the dipole aligns (θ = 0°) and maximum when it is anti-aligned (θ = 180°). Small angular displacements produce a restoring torque and simple harmonic motion. Stable equilibrium at θ = 0°, unstable at θ = 180°. Torque and energy of a dipole The potential energy describes the system's stability, reaching a minimum when the dipole aligns with the field and maximum when anti-aligned. For small oscillations (θ small), the restoring torque is approximately −m B θ. With moment of inertia I about the suspension axis, I d 2 θ/dt 2 + m B θ = 0 gives simple harmonic motion. The time period is T = 2π √(I/(m B)). Setting B =$BH lets us extract a magnet’s dipole moment from its oscillation period in Earth’s field. Vibration magnetometer relation With Earth’s field, replace B by BH. This formula provides the period of oscillation for a rigid body pivoted at a point, assuming small angular displacements and negligible dam This relationship demonstrates how the period of oscillation depends only on the length and the local acceleration due to gravity. M = 0.020 kg , L = 0.080 m I = 1 12 M L 2 B H = 3.6 10 -5 T T = 3.0 s Moment of inertia of a uniform rod about center. Substitute numerical values. Compute the ratio. hard A bar magnet (mass 0.020 kg, length 0.080 m) oscillates with small amplitude in Earth’s horizontal field BH = 3.6 × 10 -5 T. Its period is T = 3.0 s. Approximating the magnet as a uniform rod about its center, find its magnetic dipole moment m. Use T = 2π √(I/(m BH)) ⇒ m = (4π 2 I)/( T 2 BH). A· m 2 Materials fall into three broad classes by their response to H. Diamagnets develop an induced magnetization opposite to H (weak repulsion). Paramagnets align slightly with H (weak attraction) and follow Curie’s law (χ ∝ 1/T). Ferromagnets have strong, cooperative alignment of domains, showing large, non-linear M, hysteresis, remanence, and coercivity; they may lose ferromagnetism above the Curie temperature and become paramagnetic. Magnetic Materials Material Type Susceptibility ( χ ) Permeability ( μ r ) Response to B-Field Temp Dependence Dia repels regardless of T , Para attracts via 1/T , and Ferro strongly attracts until the T C limit. Diamagnetic Materials Small and Negative ( -1 < 0 ) Less than Unity ( 0 r < 1 ) Weakly repelled; moves from stronger to weaker field regions Independent of temperature ( T ) Paramagnetic Materials Small and Positive ( 0 < < ) Slightly greater than Unity ( r > 1 ) Weakly attracted; moves from weaker to stronger field regions Follows Curie's Law: 1/T Ferromagnetic Materials Very Large and Positive ( 1 ) Much greater than Unity ( r 1 ) Strongly attracted; moves rapidly toward stronger field regions Follows Curie-Weiss Law: 1/(T - T C) above T C Examples Bi, Cu, H 2O, Si, N 2 (at STP ) Al, Na, Ca, O 2 (at STP ) Fe, Co, Ni, Gd, Fe 3O 4 Transition at Curie Temperature ( T C ) magnetic materials Ferromagnets exhibit hysteresis: when H cycles, B lags behind. The B–H loop encloses a finite area equal to the energy dissipated per unit volume in one cycle (∮ H dB). The intercepts mark remanent flux density (Br) and coercive field (Hc). Soft magnetic materials have narrow loops (low loss, small Hc), ideal for transformer cores. Hard magnetic materials have wide loops (large Hc, high Br), better for permanent magnets. Remanence Br Br Coercivity Hc ±Hc custom Magnetic field intensity H A/m control dependent Area of B–H loop = energy loss per cycle per unit volume. Magnetic flux density B Typical ferromagnetic hysteresis loop showing remanence and coercivity. Transformers run at AC; to minimize heating, choose soft magnetic cores with low coercivity and high permeability (narrow loops). Permanent magnets need high coercivity to resist demagnetizing fields and temperature; alloys like Alnico or rare-earth magnets (NdFeB) are preferred for stability and strong remanence. Transformer cores: soft iron, silicon steel, ferrites (low coercivity, high μr, low loss). Permanent magnets: hardened steel, Alnico, rare-earth magnets (high coercivity, high remanence). High-frequency cores: ferrites to reduce eddy currents and losses. Choosing materials Energy loss per unit volume in one full cycle equals the area of the B–H hysteresis loop. Narrow loop → less heating. remember Paramagnets have tiny positive susceptibility and weak attraction that vanishes with heat (χ ∝ 1/T). Ferromagnets show domain-driven, huge, non-linear response with remanence and coercivity. Paramagnets and ferromagnets are both strongly attracted; they behave essentially the same. At the magnetic equator the horizontal component BH is zero. BH is maximum at the magnetic equator (dip δ = 0°, field is horizontal). The vertical component BV is zero there. Short-dipole formulas (B ∝ 1/ r 3 ) require r much larger than the magnet length. Near the poles or for long magnets at close range, use exact expressions or given geometry—do not use the 1/ r 3 law blindly. tip Curie’s law for paramagnets links thermal agitation and alignment: as temperature rises, randomization wins and χ falls inversely with T. Ferromagnets above their Curie temperature behave like paramagnets; more precisely, many follow the Curie–Weiss law χ = C/(T − Tc) close to the transition, where Tc is the Curie temperature. Curie and Curie–Weiss laws C is Curie constant; Tc is Curie temperature. χm is dimensionless. These laws describe the temperature dependence of magnetic susceptibility ( m ) for paramagnetic materials and materials approaching a C = 0.60 K T = 300 K A paramagnetic salt has Curie constant C = 0.60 K. Estimate its susceptibility at T = 300 K. χm at 300 K Direct substitution. Use Curie’s law χm = C/T. easy Tc very large Curie temperature (ferromagnet → paramagnet) Temperature T custom control χm dependent Magnetic susceptibility vs temperature (Curie law) chi = C/(T - Tc) chi 2D PLOT Curie constant Tc Curie temperature Paramagnetic χm decreases ~1/T; ferromagnet’s χm diverges near Tc (Curie–Weiss). Susceptibility χm Set T c > 0 for the Curie–Weiss divergence of a ferromagnet near its transition. A short solenoid carrying current behaves like a bar magnet: both produce the same far-field pattern. The end where field lines emerge is like a magnetic north end; the other is like south. This “equivalent solenoid” picture is useful: count turns and current to get m = N I A, or, if the bar is uniformly magnetized, use m = M V. Field lines of a short solenoid and a bar magnet look identical at distances r ≫ length. Field line comparison of a solenoid and a bar magnet Side-by-side sketch of field lines: (a) short solenoid carrying current I, (b) bar magnet. Arrows show identical dipole-like lines far away. Quick workflow for dipole-in-Earth-field problems Draw directions: mark m, BH, and possible B of magnet at the point (axis vs equator). Pick the correct magnitude formula: Baxis = (μ0/4π)(2m/ r 3 ) or Beq = (μ0/4π)(m/ r 3 ). Decide add vs subtract (vector directions). For neutral points, set magnitudes equal; solve for r with the 1/3 power. For torque/oscillations, use τ = mB sinθ and T = 2π√(I/(mBH)). neet-alert Do not mix SI and CGS. In SI: B in tesla, H in A/m, m in A· m 2 , μ0 = 4π × 10 -7 T·m/A. Many older problems use CGS with different constants—stick to the system used in the question. Direction practice: On the axial line of a dipole, B is along m; on the equatorial line, B is opposite m. If a magnet’s axis is along geographic east with north end pointing east, and BH points geographic north, then at a point on the magnet’s equatorial line, the two fields are perpendicular and cannot cancel—cancellation demands head-to-head opposition along the same line. Magnetic pole language (pole strength, separation) is a historical model. Modern treatments favor dipole moment and magnetization. If a problem gives pole strength–separation data, you can still translate it: the product (pole strength × separation) is proportional to dipole moment for a short magnet. Superposition always applies to magnetic fields. For combined sources (a magnet plus Earth’s field), add vectors: account for magnitudes and directions separately. Neutral points require exact opposition; near-cancellation that is not exact still leaves a small resultant field whose direction is critical for compass behavior. Domains in ferromagnets are tiny regions where atomic moments align. In an unmagnetized piece, domains are randomly oriented and net magnetization is near zero. An external H grows domains aligned with it. On removing H, domain walls do not fully return: remanence remains unless a reverse field (coercivity) or heat/magnetic “shaking” is applied to erase it. In paramagnets, atomic or molecular moments exist but are disordered by thermal agitation. Applying H biases their orientation slightly. As T increases, agitation dominates and susceptibility falls (χ ∝ 1/T). In diamagnets, no permanent moments exist; an applied field induces currents that oppose the change, giving a small negative χ roughly independent of T. In practical measurements, BH is often determined using a tangent galvanometer (tangent law) or a vibration magnetometer (oscillation period). Knowing BH and δ allows complete reconstruction of Earth’s field vector B and helps set up laboratory null experiments with bar magnets. Bar magnets used in meters should be magnetically hard (high Hc) to avoid drift, while transformer cores must be magnetically soft (low Hc) to keep losses minimal. Ferrites are preferred at high frequencies to reduce eddy currents while maintaining decent permeability. A dipole in a non-uniform magnetic field experiences a net force toward stronger field if aligned with it, and toward weaker field if anti-aligned. Approximate relation along an axis: F ≈ m (dB/dx). This is the basis of Stern–Gerlach type separations (qualitatively) and many magnetic trapping techniques. medium Field gradient is constant. Compute the force magnitude. m = 0.050 A , m 2 B(x) = 2.0 10 -3 + 4.0 10 -3 x ( T ) For small dipoles in slowly varying fields, F ≈ m (dB/dx). F at x = 0.10 m (m aligned with +x) A small magnetic dipole (m = 0.050 A· m 2 ) is aligned with a weakly non-uniform field along x: B(x) = 2.0 × 10 -3 + (4.0 × 10 -3 ) x (T), with x in meters. Estimate the force at x = 0.10 m. neet-alert Be careful with the factor of 2: Baxis = (μ0/4π)(2m/ r 3 ), but Beq = (μ0/4π)(m/ r 3 ). Many errors come from using the wrong expression on the wrong line. DIP points Down Into the Plane (think: D → Down): Dip relates to the vertical tilt and sets BV. DEC for Declination is the DEviation on the Compass (horizontal angle from true north). Worked-directions check: If a bar magnet’s north end points geographic north, then along its axis to the north, its field points northward. At a point due north and far away (axial line), the magnet’s B adds to BH. At a point due east on the equatorial line, the magnet’s B points west (opposite to its m direction), typically not collinear with BH—so you must use vector addition, not simple subtraction. Precision tips: Keep 2 significant figures for NEET unless otherwise specified. Write μ0/(4π) = 10 -7 T·m/A to speed substitution. When cube rooting, estimate powers first (10 -3 → 10 -1 ) and then the mantissa to avoid calculator slips. Key terms recap Source whose field far away looks like that of a tiny current loop or bar magnet. Magnetic dipole Dipole moment Strength and orientation of a dipole; for loop, m = N I A. Dipole moment per unit volume of material. Magnetization Field intensity Auxiliary field from free currents; with M gives B via B = μ0(H + M). Magnetic field B measured in tesla. Flux density χm Proportionality linking M and H in linear media. Susceptibility μr μ/μ0; equals 1 + χm for linear isotropic media. Relative permeability Angle of dip Tilt of Earth’s field below horizontal in the meridian. Horizontal angle between true and magnetic north. Declination BH Horizontal component Component of Earth’s field in the horizontal direction. BV Component of Earth’s field in the vertical direction. Vertical component Lagging of B behind H; loop area equals energy loss per cycle. Hysteresis Remanence and coercivity Residual B at H = 0 (Br) and reverse H needed to reduce B to zero (Hc). Br, Hc