In short

Drive a ferromagnet (iron, nickel, cobalt, or an alloy) through a full cycle of an externally-applied magnetising field H — up, back to zero, reverse, and back — and the internal magnetic flux density B does not retrace the same curve. It traces a closed loop called a hysteresis loop.

Two numbers on the loop carry almost all the physics:

  • Retentivity (remanence) B_r — the residual flux density that survives when the external field H is reduced back to zero. A non-zero B_r is why a piece of iron, once magnetised, stays magnetised.
  • Coercivity H_c — the reverse external field needed to drive B back down to zero. Large H_c means the magnet is hard to erase.

Energy lost per cycle (to heat in the material):

\boxed{\;W_{\text{loss}} = \oint H\,dB \quad \text{(per unit volume per cycle, joules/m}^3)\;}

which is numerically equal to the area enclosed by the B-H loop. Multiply by the driving frequency f to get power dissipated per cubic metre.

Two material families, opposite uses:

  • Soft magnetic materials (silicon steel, mu-metal, soft ferrites) — narrow loop, tiny area, low coercivity. Magnetise easily, demagnetise easily, waste little energy. Transformer cores at the Tarapur and Kudankulam grids are laminated silicon steel for exactly this reason.
  • Hard magnetic materials (Alnico, ferrite, neodymium-iron-boron) — fat loop, large area, huge coercivity. Very hard to magnetise, very hard to demagnetise. Permanent magnets in Maruti EV traction motors, in every hard disk and BLDC fan, are neodymium-iron-boron (NdFeB), which has the largest known coercivity at room temperature.

Why it matters. A transformer core going through 50 cycles per second for a year racks up 1.5 billion traversals of the loop. Each traversal costs area-under-the-loop joules per cubic metre. A 1% reduction in loop area across India's power-grid cores saves an estimated 2000 GWh per year.

A blacksmith in Moradabad heats a steel chisel in the forge, strikes it a few times near a lodestone lying on the anvil, and then quenches it in oil. The chisel — previously just a piece of plain steel — is now a magnet. It will pick up nails and screws from the workshop floor. It will do this for years. Nobody needs to plug it into anything. Where did the magnetism come from, and why does it stay?

A factory-floor electrician at Tata Power's Trombay plant opens up a 400 kV transformer cabinet. Inside, a massive laminated steel core weighing several tonnes. The core is driven at 50 Hz by the primary winding — three billion field reversals a year. If the core behaved like that chisel, after one cycle it would be permanently magnetised, and the transformer would stop working. It does not. The core is engineered from a completely different kind of iron — a soft magnetic material — that forgets every cycle and starts fresh.

Two ferromagnets, two opposite requirements, one underlying phenomenon: hysteresis. The word is Greek for "lagging behind" — the magnetisation B lags behind the external field H that causes it, and the record of that lag is a loop.

Setting up the B–H plane

To discuss hysteresis you need two magnetic quantities, not one. They are distinct and confusing the first time you meet them, so let us anchor both with a concrete experiment.

Take an iron ring — a torus — and wind N turns of insulated copper wire around it uniformly. Pass a current I through the coil. The coil produces a magnetic field in the absence of the iron:

H = \frac{NI}{\ell}

where \ell is the mean circumference of the ring. This quantity H is the magnetising field (or magnetic field strength) — SI unit ampere per metre (A/m). It depends only on the coil and the current, not on the iron.

Why: for an ideal solenoid or toroid, Ampère's law gives H \cdot \ell = NI, so H = NI/\ell. Importantly, H is defined so that it comes purely from free currents (the currents you put in with wires and batteries) — it does not include the contribution from bound currents in the material.

Now slide the iron inside the coil. The iron's atomic dipoles align with H, and the total magnetic flux density inside the iron is:

\vec{B} = \mu_0(\vec{H} + \vec{M})

where \vec{M} is the magnetisation (dipole moment per unit volume of the material) and \mu_0 = 4\pi \times 10^{-7}\ \text{T}\cdot\text{m/A} is the permeability of free space. In SI, B is measured in tesla (T), H in A/m, and M in A/m.

For a linear material, \vec{M} = \chi_m \vec{H} and so \vec{B} = \mu_0(1 + \chi_m)\vec{H} = \mu_0 \mu_r \vec{H} — the familiar proportional law. For a ferromagnet, the relation is not linear and not single-valued — B depends not just on the current H but on the whole history of how you got there.

That is hysteresis.

Tracing the loop

Start with a perfectly demagnetised piece of soft iron (B = 0, H = 0, M = 0). Slowly increase the coil current so H grows. Plot B versus H and watch what the graph does.

The hysteresis loop on the B-H plane A plot of magnetic flux density B vertical versus magnetising field H horizontal. From the origin, an initial curve rises and saturates at positive B. As H decreases the curve traces above, reaching B_r at H equals 0. It then crosses zero at minus H_c, saturates at negative B, and returns symmetrically. The enclosed area is shaded lightly. H (A/m) B (T) initial magnetisation +B_r (retentivity) −B_r −H_c +H_c (coercivity) saturation +B_s −B_s area = energy lost / m³ per cycle
The hysteresis loop. Dashed: the initial magnetisation curve from the virgin state. Solid: the steady-state loop traced on subsequent cycles. Four labelled points: saturation $\pm B_s$, retentivity $\pm B_r$ (where the loop crosses the vertical axis), coercivity $\pm H_c$ (where the loop crosses the horizontal axis). The shaded area equals the energy dissipated per cubic metre per cycle.

Let me narrate what happens, point by point, as you cycle the current.

(a) From virgin state to saturation. As H grows from zero, B grows — but not linearly. At first the curve rises steeply as easy-to-flip domains align with H. Then it bends over as the remaining, harder-to-flip domains resist. Eventually every domain is aligned with H and M saturates at some maximum value M_s. Past this point, B = \mu_0(H + M_s) continues to creep up, but only linearly with H — the ferromagnetic contribution is spent. This upper value is saturation flux density B_s. For pure iron, B_s \approx 2.1 T.

(b) Reducing H back to zero. Now decrease the current. H comes down, but B does not follow the same path. The domains that were forced into alignment do not all relax back. When H = 0, there is still a positive magnetisation left — the retentivity B_r. The iron is now a permanent magnet, even with no external field. This is the phenomenon the Moradabad blacksmith exploits.

(c) Reversing the current. To get rid of that residual magnetisation, you have to push back. Reverse the current and grow H in the opposite direction. At some specific value, H = -H_c, the magnetisation collapses to zero. That value is the coercivity. The larger H_c, the harder it is to erase — the better the magnet is at being a magnet.

(d) Reverse saturation. Keep going past -H_c and the domains re-align in the new direction, reaching -B_s.

(e) Back up. Now increase H from -H_s back toward 0 and positive. The curve traces the lower branch — through -B_r (negative retentivity) and +H_c (needs positive coercivity to demagnetise).

After the first cycle, the system is on the steady-state loop and never again visits the initial virgin curve. Every subsequent traversal follows the same outer loop.

Energy dissipated per cycle

Here is the result that makes hysteresis quantitative rather than just descriptive. The energy dissipated as heat per unit volume per cycle equals the area enclosed by the B–H loop.

Step 1. Consider the magnetic energy balance. When the applied H increases by dH, work is done by the source driving the coil. The differential work per unit volume of the material is (from Maxwell's theory of electromagnetism):

dw = H\,dB

Why: the power delivered by the source per unit volume of the core equals H \cdot dB/dt. So the energy per unit volume over a small time interval is H \cdot dB. This is the rigorous magnetic analogue of the mechanical formula dW = F\,dx.

Step 2. Integrate around one full cycle of the loop.

w_{\text{cycle}} = \oint H\,dB

Why: the energy delivered by the source over one complete traversal is the closed-contour integral of H\,dB. For a reversible process (say a linear paramagnetic material), this integral is zero because B is a single-valued function of H and the path retraces. For hysteresis, the path encloses area and the integral is positive.

Step 3. Interpret geometrically. The closed integral \oint H\,dB equals the signed area enclosed by the loop in the B-vs-H plane. Because the loop is traversed in a specific direction (counter-clockwise in the sign convention used here), the area is positive and equals the energy delivered to the material but not stored — it must have gone to heat.

\boxed{\;w_{\text{cycle}} = \text{(area enclosed by B-H loop)}\;\text{(J/m}^3\text{)}\;}

Step 4. Multiply by the driving frequency f to get power loss per cubic metre:

P_{\text{hyst}} = f \cdot w_{\text{cycle}} \quad \text{(W/m}^3\text{)}

For a transformer running at f = 50\ \text{Hz} with a core made of silicon steel (loop area \approx 200\ \text{J/m}^3), this gives P_{\text{hyst}} \approx 10{,}000\ \text{W/m}^3 = 10\ \text{kW/m}^3. A transformer core of 1 cubic metre dissipates 10 kW to hysteresis alone — before you add eddy-current losses. This is why transformer cores run warm to the touch, and why large ones need oil-based cooling.

Why a soft magnet wastes less energy

Look at the loop. Its area depends on how wide it is (coercivity H_c) and how tall it is (B_s, which is similar for all iron-based materials at \sim 2 T). The width dominates. A material with tiny H_c has a pencil-thin loop; a material with large H_c has a fat one.

This single-number contrast — a factor of 10^3 in coercivity — is why the two material families exist and why a different alloy ships in the Maruti motor than in the Tarapur transformer.

Soft vs hard magnetic materials

The two families diverge by more than coercivity alone. Here is the full contrast table.

Property Soft (e.g. silicon steel, mu-metal, soft ferrites) Hard (e.g. Alnico, ferrite, NdFeB)
Coercivity H_c Low (1 to 100 A/m) Very high (10^4 to 10^6 A/m)
Retentivity B_r Low (but not zero) High (up to ~1.5 T for NdFeB)
Loop area Tiny Large
Magnetise by Weak external field Strong external field + curing
Demagnetise by Weak opposing field Strong opposing field or high heat
Application Transformers, motors' cores, inductors, magnetic shielding Permanent magnets in speakers, motors, sensors, MRI shim coils
Indian examples Silicon steel cores at Tarapur, Kudankulam, NTPC plants; mu-metal shielding at TIFR cyclotron Maruti Suzuki eV30 traction motor (NdFeB); Indian speakers and earbuds (ferrite or NdFeB); hard disk read heads
Comparison of soft and hard magnetic hysteresis loops Two B-H plots side by side. Left: a narrow tall loop representing a soft magnet. Right: a wide rectangular loop representing a hard magnet. Axes and key points are labelled. H B Soft (silicon steel) H B Hard (NdFeB)
Schematic comparison. Silicon steel (left) has a thin loop — tiny area, tiny energy loss per cycle. NdFeB (right) has a broad, near-rectangular loop — huge coercivity means the magnet resists demagnetisation under normal conditions.

Explore the loop — retentivity, coercivity, and saturation

The interactive below draws a simplified hysteresis loop whose coercivity H_c you can vary. Drag the control and watch how the shape, the retentivity, and the loop area all change together.

Interactive: varying coercivity reshapes the hysteresis loop An interactive B versus H plot. A draggable parameter hc represents the coercivity. As hc changes, two curves representing the upper and lower branches of the hysteresis loop change width together, and a readout shows the loop area. H (A/m, scaled) B (T) upper branch lower branch drag to change H_c
Each branch of the loop is modelled as $B = 2\tanh(H \pm H_c)$, a smooth approximation. As you drag the coercivity slider, the two branches separate, the retentivity at $H=0$ grows, and the enclosed area expands. The readout gives a rough loop-area proxy. Real hysteresis loops have sharper corners but identical qualitative behaviour.

Worked examples

Example 1: Transformer hysteresis loss at Tarapur

A laminated silicon-steel core in a 50 Hz transformer has a hysteresis-loop area of 190\ \text{J/m}^3 per cycle. The core volume is 0.15\ \text{m}^3. Find the power dissipated to hysteresis and the annual energy loss.

Thin hysteresis loop of silicon steel A narrow hysteresis loop with small enclosed area. Axes are B vertical and H horizontal. The loop is drawn in red with a lightly shaded interior. H B area = 190 J/m³
A skinny silicon-steel loop. Narrow loops are exactly what transformer designers want.

Step 1. Compute energy per cycle per cubic metre. Given as the loop area: w = 190\ \text{J/m}^3.

Step 2. Energy per cycle for the whole core:

W_{\text{core}} = w \cdot V = 190\ \text{J/m}^3 \times 0.15\ \text{m}^3 = 28.5\ \text{J/cycle}

Why: every cubic metre of core material loses w joules each cycle. Total volume times loss density equals total loss per cycle.

Step 3. Multiply by frequency:

P = f \cdot W_{\text{core}} = 50 \times 28.5 = 1425\ \text{W} \approx 1.43\ \text{kW}

Step 4. Annual energy (one year = 365 \times 24 = 8760 hours):

E_{\text{year}} = P \cdot t = 1425\ \text{W} \times 8760\ \text{h} = 12.48 \times 10^6\ \text{Wh} = 12.48\ \text{MWh}

Why: the transformer runs continuously. Power times time gives total energy. Multiply by the cost of electricity (\sim ₹5/kWh industrial rate) and you get around ₹62,000 per transformer per year — just for hysteresis, not including eddy currents or copper losses.

Result: P \approx 1.43\ \text{kW} of continuous hysteresis loss per transformer, about 12.5 MWh per year.

What this shows: a 5\% reduction in loop area across the transformers of a single substation saves tens of thousands of rupees per year. This is why metallurgists chase grain-oriented silicon steels — every millijoule of loop area matters at grid scale.

Example 2: Demagnetising a carbon-steel chisel

A carbon-steel chisel has been magnetised to remanent flux density B_r = 1.1\ \text{T}. Its coercivity is H_c = 3200\ \text{A/m}. You wrap a single-layer solenoid of n = 2000 turns per metre around it and drive a current through the winding to demagnetise it. What minimum current must the solenoid carry?

Solenoid wound around a chisel to demagnetise it A horizontal iron chisel with a coil of wire wound uniformly around it. Arrows indicate the direction of current flow, and a label marks the external H field needed to drive B to zero. carbon-steel chisel (remanent B_r = 1.1 T) H = nI (reverse direction, magnitude = H_c)
A solenoid wound around the chisel produces an axial $H$ field inside it. To erase the remanent magnetisation, drive $H$ up to the coercivity in the opposite direction.

Step 1. The condition to erase the magnetisation is |H| \ge H_c in the reverse direction.

H_{\min} = H_c = 3200\ \text{A/m}

Step 2. The solenoid produces H = nI:

I = \frac{H_c}{n} = \frac{3200\ \text{A/m}}{2000\ \text{turns/m}} = 1.6\ \text{A}

Why: for a long solenoid, H inside depends only on the turns per unit length and the current — not on the core material or its permeability. This is the defining convenience of H over B when you design coil-based demagnetisers.

Step 3. Sanity-check the direction. The current must flow in the opposite sense to whatever direction originally magnetised the chisel. In practice you don't know that direction, so a bulk eraser uses an AC current whose amplitude is slowly reduced — the chisel is cycled through ever-smaller hysteresis loops until B settles at zero.

Result: a minimum demagnetising current of about 1.6\ \text{A} through a 2000-turn-per-metre coil, applied in the reverse direction.

What this shows: coercivity is not a mysterious material parameter — it translates directly into a number of amperes you must drive through your coil. This is exactly the calculation behind commercial "degaussing" units used at TV-repair shops (erasing old CRT monitors) and at airport security (demagnetising hard drives).

Example 3: Energy cost of a tape recorder's record head

A magnetic recording tape advancing at 4.76\ \text{cm/s} past a record head deposits a new magnetisation pattern whose energy density (integrated over the record-head loop) is 450\ \text{J/m}^3. The tape is 12.7\ \text{mm} wide and 12\ \mu\text{m} thick. How much power does the record head dissipate, ignoring electrical losses?

Step 1. Compute the rate at which tape volume passes the head:

\dot{V} = v \cdot w \cdot h = (0.0476\ \text{m/s})(0.0127\ \text{m})(12 \times 10^{-6}\ \text{m})
\dot{V} = 7.25 \times 10^{-9}\ \text{m}^3/\text{s}

Why: each second, a slab of tape v metres long and cross-section w \times h passes through the field of the record head and gets cycled once.

Step 2. Power dissipated:

P = w_{\text{cycle}} \cdot \dot{V} = 450 \cdot 7.25 \times 10^{-9}\ \text{J/s} = 3.26 \times 10^{-6}\ \text{W}
P \approx 3.3\ \mu\text{W}

Step 3. Interpret.

Why: a cassette tape dissipates only microwatts in the tape itself — tiny, which is why the tape does not heat up noticeably. Modern hard-disk storage is similar but with orders-of-magnitude smaller volumes and orders-of-magnitude higher coercivity (per bit you want it hard to erase). The same hysteresis-area calculation governs both.

Result: about 3\ \mu\text{W} dissipated in the tape material per record head.

What this shows: hysteresis is not only about big transformers. Every bit written to a hard disk, every frame of audio laid down on a tape, costs the loop-area energy — integrated over the volume of material cycled. The same formula spans eight orders of magnitude of physical scale.

Common confusions

If you came here to understand what the hysteresis loop is and how to use its area to compute energy loss, you have what you need. What follows is the microscopic domain-wall picture, the Stoner–Wohlfarth model, and the link to ferroelectric hysteresis.

Magnetic domains and the origin of hysteresis

Inside a ferromagnetic crystal, atomic moments align not all at once but in regions called domains — typically 1\ \mu\text{m} to 100\ \mu\text{m} across — within which all atomic moments point the same way. Different domains point in different directions, so a bulk piece of unmagnetised iron has essentially zero net M despite every individual atom being a tiny magnet.

When you apply H, three things happen in sequence:

  1. Domain wall motion. Domains aligned with H grow at the expense of misaligned ones. The boundaries (domain walls) move through the crystal. This is reversible for weak H and gives the initial nearly-linear part of the virgin curve.

  2. Irreversible wall displacement. Stronger H pushes walls past crystal defects — dislocations, grain boundaries, non-magnetic inclusions — that pin them. Once a wall pops past a pinning centre, it cannot easily return: when H is removed, the wall stays put, leaving residual M. This is the microscopic origin of retentivity.

  3. Domain rotation. At the highest fields, whole domains whose internal moments are still not aligned with H rotate into alignment. This gives the final approach to saturation.

The coercivity is essentially the external field needed to unpin domain walls and rotate the magnetisation back. In a soft magnet you want few defects (pure, well-annealed material) so walls move freely — low H_c. In a hard magnet you want many strong pinning centres (fine precipitates, grain-boundary phases) — high H_c.

The Stoner–Wohlfarth model — rectangular loops from anisotropy

For single-domain particles (fine enough that no wall fits inside them), domain-wall physics is replaced by coherent rotation of a single moment. The Stoner–Wohlfarth model captures this by minimising a free energy

\mathcal{F}(\theta) = K_u \sin^2(\theta - \theta_e) - \mu_0 H M_s \cos\theta

where \theta is the angle of the moment with respect to an applied field, \theta_e is the easy axis, K_u is the uniaxial anisotropy constant (a material-dependent energy density that prefers alignment along a crystallographic axis), and M_s is the saturation magnetisation.

Setting \partial\mathcal{F}/\partial\theta = 0 gives stable equilibria. The moment hops discontinuously between two stable states when the field crosses a threshold:

H_K = \frac{2K_u}{\mu_0 M_s}

called the anisotropy field. For H applied along the easy axis, the loop is perfectly rectangular with coercivity H_c = H_K.

Real polycrystalline magnets average over grain orientations, rounding the corners, but the principle is the same: high anisotropy makes high coercivity. NdFeB has K_u \approx 5 \times 10^6\ \text{J/m}^3 — about 50 times larger than iron's 10^5\ \text{J/m}^3 — which is exactly why NdFeB makes the strongest commercial permanent magnets despite not having the largest saturation M_s.

Connection to ferroelectric hysteresis

The exact same loop shape appears in ferroelectric materials — barium titanate, PZT, the stuff of piezoelectric sensors — with the electric polarisation P playing the role of M and the electric field E playing the role of H. The energy dissipated per cycle is again the loop area \oint E\,dP.

This is not a coincidence. Both phenomena arise from a double-well free energy landscape with two equivalent minima (domains of opposite polarisation) separated by a barrier that an external field can tilt. The mathematics is identical; only the physical labels change. It is a lovely example of universality in condensed-matter physics.

Why eddy currents are the other half of the transformer loss story

Hysteresis is one of two major loss mechanisms in a transformer core. The other is eddy currents — induced circulating currents in the bulk of the conducting core, driven by the changing flux (Faraday's law; see faraday-s-law and lenz-s-law). Eddy losses scale as f^2 (quadratic in frequency) while hysteresis losses scale as f (linear).

At 50 Hz in a well-designed laminated core, hysteresis and eddy losses are comparable — each is a few watts per kilogram of core. The engineering solution:

  • Eddy currents → laminate the core (thin insulated sheets), which breaks the conduction loops.
  • Hysteresis → choose an alloy with the thinnest possible loop (grain-oriented silicon steel, amorphous metallic glass).

Together, a modern 100 kVA distribution transformer achieves > 98\% efficiency. That last 2% is mostly hysteresis plus eddy currents, and the chase for 99% is still the front line of power-grid materials research.

Where this leads next