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Electricity and Magnetism
Magnetic Hysteresis
Hysteresis is the lagging of magnetic induction (B) behind the magnetizing field intensity (H) when a ferromagnetic material is subjected to a cyclic magnetizing process. This phenomenon results in energy dissipation per cycle equal to the area of the B-H loop.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of magnetic hysteresis, B-H curve, retention, coercivity, and core selection criteria for transformers and electromagnets.
PGT extension
Quantitative calculation of hysteresis energy loss per unit volume per cycle, comparison of soft and hard magnetic materials, and application in alternating current circuits.
Prerequisites
Magnetic field intensity (H) and magnetic induction (B)
Intensity of magnetization (I) and magnetic susceptibility
Domain theory of ferromagnetism
B-H Loop (Hysteresis Loop)
When a ferromagnetic specimen is taken through a complete cycle of magnetization, the curve of B versus H forms a closed loop known as the hysteresis loop.
The non-linear path reflects the irreversible domain wall motion and domain rotation under the influence of an external magnetic field.
The loop exhibits rotational symmetry about the origin for symmetric cyclic magnetization.
Remanence and Coercivity
Retentivity (or Remanence, B_r): The value of magnetic induction B left in the material when the magnetizing field H is reduced to zero from saturation.
Coercivity (H_c): The magnitude of the reverse magnetizing field required to reduce the residual magnetic induction B to zero.
Materials with high retentivity and high coercivity are termed hard magnetic materials, while those with low retentivity and low coercivity are soft magnetic materials.
Hysteresis Loss
Work done per unit volume in taking a ferromagnetic material through one complete cycle of magnetization is equal to the area enclosed by the B-H loop.
The energy lost per unit volume per cycle is given by the integral W = oint H dB.
This dissipated energy appears as thermal energy (heat) in magnetic cores of transformers, generators, and inductors.
Soft versus Hard Ferromagnetic Materials
Soft Magnetic Materials (e.g., Soft Iron, Permalloy): Narrow hysteresis loop, low coercivity, low hysteresis loss, high initial permeability, used in transformer cores and electromagnets.
Hard Magnetic Materials (e.g., Alnico, Neodymium-Iron-Boron): Wide hysteresis loop, high retentivity, high coercivity, used for making permanent magnets.
The choice of material depends strictly on minimizing core losses in AC machinery or maximizing permanent retention in transducers.
Derivations
Work Done per Unit Volume in Magnetization
Conditions: Quasi-static cyclic magnetization of a ferromagnetic material
Consider a solenoid of n turns per unit length carrying current I, producing a magnetic field intensity H = nI.
When current changes by dI, the induced electromotive force in the coil opposes the change, and the work done dW per unit volume against this back emf in time dt is given by dW = H dB.
Integrating over a complete cycle of magnetization to find the total work done (energy loss) per unit volume: W = oint dW = oint H dB.
Result: W = oint H dB, which represents the area of the B-H loop.
Steinmetz Empirical Formula for Hysteresis Power Loss
Conditions: Ferromagnetic core subjected to an alternating magnetic field of frequency f
Empirical observation shows that the area of the hysteresis loop depends on the maximum flux density B_{max} raised to a power alpha, known as the Steinmetz coefficient (typically between 1.6 and 2.0).
Energy loss per unit volume per cycle is expressed as $W_h = \eta B_{max}^{\alpha}$, where eta is the material-dependent Steinmetz constant.
If the alternating field operates at a frequency f, the power loss per unit volume P_h is the energy loss per cycle multiplied by the frequency.
Result: $P_h = \eta f B_{max}^{1.6}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Hysteresis Energy Loss per Cycle
$$W = oint H dB$$
H = Magnetic field intensity, B = Magnetic induction Joule per cubic meter (J/m^3)
One complete symmetric magnetic cycle
Steinmetz Hysteresis Power Loss
$$P_h = \eta f B_{max}^{1.6}$$
$P_h = Power loss per unit volume$, eta = Steinmetz constant, f = Frequency, $B_{max} = Maximum flux density$ W/m^3
Standard ferromagnetic AC core application
Magnetic Induction Relation
$$B = \mu_0 (H + M)$$
B = Magnetic induction, m$u_0 = Permeability of free space$, H = Magnetic field intensity, M = Magnetization Tesla (T)
General macroscopic magnetic medium
Magnetic Susceptibility
$$\chi_m = \frac{M}{H}$$
ch$i_m = Magnetic susceptibility$, M = Magnetization, H = Magnetic field intensity Dimensionless
Linear isotropic media (approximate for ferromagnets at small H)
Relative Permeability
$$\mu_r = 1 + \chi_m$$
m$u_r = Relative permeability$, ch$i_m = Magnetic susceptibility$ Dimensionless
Non-conducting magnetic media
Total Core Loss
$$P_{total} = P_h + P_e$$
$P_{total} = Total core loss$, $P_h = Hysteresis loss$, $P_e = Eddy current loss$ W
Alternating magnetic excitation
Coercive Field Energy Equivalence
$$U_c = \int_{0}^{H_c} B dH$$
$U_c = Energy density associated with coercive threshold$ J/m^3
Demagnetization quadrant
Total Energy Loss for Core Volume V
$$E = V oint H dB$$
E = Total energy lost per cycle, V = Volume of the core Joule (J)
Uniform magnetic field across core volume V
Graphs and experiments
Standard B-H Hysteresis Loop
Draw: A closed, centrally symmetric loop plotting B (y-axis) against H (x-axis), indicating saturation point, retentivity on the B-axis, and coercivity on the H-axis.
Interpret: Shows the non-linear multi-valued relation between B and H, illustrating residual magnetism and energy dissipation per cycle.
Exam use: Direct identification of retentivity, coercivity, and calculation of hysteresis energy loss from loop area.
Comparison of Soft Iron and Steel Hysteresis Loops
Draw: Two loops on the same axes: Soft iron with a tall, narrow loop (high B_r, very low H_c); Steel with a shorter, wider loop (moderate B_r, high H_c).
Interpret: Soft iron loses magnetization easily and has minimal area (low loss), whereas steel retains magnetization strongly (high coercivity).
Exam use: Conceptual questions matching core material properties with appropriate engineering applications (transformers vs permanent magnets).
Draw: A toroidal specimen wound with primary and secondary coils connected to a variable DC source, reversing switch, and a ballistic galvanometer.
Interpret: Used experimentally to map the B-H curve point-by-point by measuring induced charge flow upon field reversal.
Exam use: VIVA and conceptual questions regarding experimental determination of magnetic induction B.
Solved numericals
TGT core
The area of a B-H hysteresis loop for a given ferromagnetic material is equivalent to 250 J/m^3. If the core volume is 0.02 m^3, find the total thermal energy lost per minute when the alternating magnetic field operates at a frequency of 50 Hz.
Energy loss per unit volume per cycle (W) = 250 J/m^3.
Volume of the core (V) = 0.02 m^3.
Energy loss per cycle for the entire cor$e = W * V = 250 * 0.02 = 5 Joules$.
Frequency f = 50 Hz means 50 cycles per second. Number of cycles per minut$e = 50 * 60 = 3000 cycles$.
Total energy loss per minut$e = Energy loss per cycle * Cycles per minute = 5 J * 3000 = 15000 Joules$.
Answer: 15000 J (or 15 kJ)
PGT extension
In a transformer core, the hysteresis power loss is 400 W at a frequency of 50 Hz. Keeping the maximum flux density constant, what will be the hysteresis power loss when the frequency is increased to 60 Hz?
Hysteresis power loss is given by Steinmetz relation: $P_h = \eta f B_{max}^{1.6}$.
Since B_{max} and the material constant eta remain constant, P_h is directly proportional to frequency f (P_h ∝ f).
Therefore, P_{h2} / $P_{h1} = f_2 / f_1$.
Substitute the given values: P_{h2} / $400 = 60 / 50$.
$P_{h2} = 400 * (6/5) = 400 * 1.2 = 480 W$.
Answer: 480 W
PGT extension
A sample of iron weighing 7.8 kg is subjected to an alternating magnetic field of frequency 50 Hz. If the density of iron is 7800 kg/m^3 and the hysteresis loop area corresponds to 300 J/m^3, calculate the annual hysteresis energy loss assuming continuous operation.
Volume of the iron sample $V = Mass / Density = 7.8 kg / 7800 kg/m^3 = 0.001 m^3$.
Energy loss per cycle for the sampl$e = Loop Area * Volume = 300 J/m^3 * 0.001 m^3 = 0.3 J/cycle$.
Power loss $P_h = Energy loss per cycle * Frequency = 0.3 J * 50 s^{-1} = 15 W (Joules per second)$.
Total time in one yea$r = 365 days * 24 hours/day * 3600 seconds/hour = 3.1536 * 10^7 seconds$.
Total annual energy los$s = Power * Time = 15 W * 3.1536 * 10^7 s = 4.73 * 10^8 Joules$.
Answer: 4.73 * 10^8 J
Assumptions and limitations
Assumption: Quasi-static conditions are maintained during the tracing of the B-H loop, ignoring dynamic eddy current contributions at high frequencies.
Limitation: Steinmetz empirical power law (alpha approx 1.6) fails at extremely high magnetic induction values approaching magnetic saturation.
Common misconceptions
Common error: Hysteresis loss depends only on the maximum magnetic field H.
Correct: Hysteresis loss depends on the area of the B-H loop, which is determined by both the maximum field intensity and the magnetic properties (retentivity and coercivity) of the specific material.
Different materials subjected to the same maximum H will exhibit drastically different loop areas and energy losses.
Common error: Soft iron retains permanent magnetism permanently.
Correct: Soft iron has low retentivity and very low coercivity, making it lose its magnetization easily once the external field is removed.
High coercivity and high retentivity (hard magnetic materials) are required for permanent magnets.
Quick revision
Hysteresis is the lagging of B behind H in a cyclic magnetization process.
Area of the B-H loop equals thermal energy dissipated per unit volume per cycle.
Retentivity is remaining B at H = 0; Coercivity is reverse H needed to make B = 0.
Soft magnetic materials have narrow loops and low losses, ideal for AC transformer cores.
Hard magnetic materials have wide loops and high coercivity, ideal for permanent magnets.
Steinmetz hysteresis power loss formula: $P_h = \eta f B_{max}^{1.6}$.
Must remember
The area enclosed by the B-H loop represents the energy loss per unit volume per cycle ($W = \oint H \, dB$).
Retentivity ($B_r$) is the magnetic induction left when the magnetizing field $H$ is reduced to zero.
Coercivity ($H_c$) is the reverse magnetic field required to reduce residual induction to zero.
Soft magnetic materials have a narrow hysteresis loop, low coercivity, and low hysteresis loss, making them ideal for transformer cores.
Hard magnetic materials have a wide hysteresis loop, high retentivity, and high coercivity, making them ideal for permanent magnets.
Steinmetz hysteresis power loss is given by $P_h = \eta f B_{max}^{1.6}$.
Relative permeability is related to magnetic susceptibility by $\mu_r = 1 + \chi_m$.
Glossary
HysteresisThe lagging of magnetic induction ($B$) behind the magnetizing field ($H$) during cyclic magnetization.
Retentivity (Remanence, $B_r$)The residual magnetic induction left in a ferromagnetic material when the external field $H$ becomes zero.
Coercivity ($H_c$)The magnitude of the reverse magnetic field needed to bring residual magnetic induction down to zero.
Steinmetz Constant ($\eta$)A material-dependent empirical constant used in calculating hysteresis power loss.
Magnetic Susceptibility ($\chi_m$)The ratio of magnetization $M$ to the magnetic field intensity $H$.
Soft vs Hard Magnetic Materials
Property
Soft Magnetic Materials
Hard Magnetic Materials
Hysteresis Loop
Narrow loop
Wide loop
Coercivity ($H_c$)
Low
High
Retentivity ($B_r$)
Low to Moderate
High
Hysteresis Loss
Low
High
Applications
Transformer cores, electromagnets
Permanent magnets
Practice Quiz
Choose an option to reveal the explanation.
1. What does the area of a closed B-H hysteresis loop represent?
Explanation: The work done per unit volume in taking a ferromagnetic material through one complete cycle of magnetization is given by $W = \oint H \, dB$, which equals the area of the B-H loop.
2. Which of the following materials is most suitable for making transformer cores?
Explanation: Soft iron has a narrow hysteresis loop, low coercivity, and low hysteresis loss, making it ideal for alternating current machinery like transformer cores.
3. Retentivity is defined as the value of magnetic induction $B$ when the magnetizing field $H$ is:
Explanation: Retentivity (or remanence, $B_r$) is the magnetic induction retained in the material when the external magnetizing field $H$ is reduced to zero from saturation.
4. Coercivity ($H_c$) is the magnitude of the reverse magnetizing field required to:
Explanation: Coercivity measures the resistance of a magnetic material to demagnetization, specifically the reverse field needed to nullify residual induction $B$.
5. According to Steinmetz empirical law, hysteresis power loss $P_h$ varies with maximum flux density $B_{max}$ as:
Explanation: Steinmetz hysteresis power loss equation is given by $P_h = \eta f B_{max}^{1.6}$, showing a 1.6 power dependence on maximum flux density.
6. Permanent magnets are constructed using materials characterized by:
Explanation: Hard magnetic materials used for permanent magnets require high retentivity so they retain strong magnetism and high coercivity so they resist accidental demagnetization.
7. The relation between magnetic induction $B$, magnetic field intensity $H$, and magnetization $M$ is given by:
Explanation: The general macroscopic magnetic relation is $B = \mu_0(H + M)$, accounting for both external field contribution and material magnetization.
8. Magnetic susceptibility $\chi_m$ is expressed in terms of magnetization $M$ and field intensity $H$ as:
Explanation: Magnetic susceptibility is defined as the dimensionless ratio $\chi_m = \frac{M}{H}$ for linear isotropic media.
9. What happens to the energy dissipated due to hysteresis in a transformer core?
Explanation: The energy lost per unit volume per cycle during hysteresis appears as thermal energy (heat) in magnetic cores.
10. Relative permeability $\mu_r$ and magnetic susceptibility $\chi_m$ are related by:
Explanation: For non-conducting magnetic media, the relative permeability is derived as $\mu_r = 1 + \chi_m$.
11. Which material is classified as a hard magnetic material?
Explanation: Alnico is a classic hard magnetic alloy used for permanent magnets, whereas soft iron, permalloy, and silicon steel are soft magnetic materials.
12. If the volume of a magnetic core is $V$, the total energy lost per cycle is given by:
Explanation: Since $\oint H \, dB$ represents energy loss per unit volume, multiplying by core volume $V$ gives the total energy lost per cycle: $E = V \oint H \, dB$.
13. Why do soft magnetic materials have narrow hysteresis loops?
Explanation: Soft magnetic materials exhibit low coercivity and highly reversible/easy domain wall motion, resulting in very narrow hysteresis loops.
14. Which of the following statements regarding hysteresis is correct?
Explanation: The non-linear path of the B-H curve reflects irreversible domain wall motion and rotation under an external magnetic field, and the loop exhibits rotational symmetry about the origin.
15. In alternating current machinery, total core losses typically consist of:
Explanation: Total core loss under alternating excitation is given by $P_{total} = P_h + P_e$, combining hysteresis and eddy current losses.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The area of a B-H loop measured for a ferromagnetic substance measures:
Explanation: The closed loop integral $\oint H \, dB$ represents the mechanical work or energy lost as heat per unit volume per magnetization cycle, corresponding to the loop area.
2. For a permanent magnet, the material should have:
Explanation: Permanent magnets require high retentivity to keep strong residual magnetism and high coercivity so they are not easily demagnetized by stray fields.
3. The Steinmetz formula for power loss due to hysteresis is proportional to frequency $f$ and $B_{max}$ raised to the power of:
Explanation: Steinmetz empirical law states $P_h = \eta f B_{max}^{1.6}$, where the power exponent of $B_{max}$ is 1.6.
4. Which of the following is chosen as core material for electromagnets?
Explanation: Soft iron has high permeability and low retentivity/coercivity, allowing quick magnetization and demagnetization suitable for electromagnets.
5. Coercivity of a magnetic material is $4 \text{ A/m}$. What does this value signify?
Explanation: Coercivity specifies the reverse magnetic field intensity required to bring the residual magnetic induction $B$ down to zero.
6. If the relative permeability of a medium is given by $\mu_r$, its magnetic susceptibility $\chi_m$ is:
Explanation: Since $\mu_r = 1 + \chi_m$, it follows that magnetic susceptibility is $\chi_m = \mu_r - 1$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the physical significance of retentivity in a magnetic material?
Explanation: Retentivity ($B_r$) measures how much magnetic induction remains in the ferromagnetic substance after the external field $H$ is brought to zero.
2. If a core of volume $0.05 \text{ m}^3$ is subjected to a cycle where the B-H loop area corresponds to $200 \text{ J/m}^3$, what is the total energy loss per cycle?
Explanation: Total energy loss $E = V \times \text{loop area} = 0.05 \text{ m}^3 \times 200 \text{ J/m}^3 = 10 \text{ J}$.
3. Why are transformer cores laminated and made of soft magnetic materials?
Explanation: Soft magnetic materials reduce hysteresis loss (narrow B-H loop), while lamination reduces eddy current losses in AC transformers.
4. Which material property is crucial for a magnetic shield or sensor core requiring high initial permeability?
Explanation: High initial permeability and low coercivity are hallmark traits of soft magnetic materials, making them ideal for sensitive magnetic flux channeling.
5. In the B-H curve, what characterizes the saturation point?
Explanation: At saturation, domain alignment reaches its maximum limit, and increasing $H$ further yields negligible increase in $B$.
6. Calculate the Steinmetz hysteresis power loss per unit volume if $\eta = 10^{-4}$, frequency $f = 50 \text{ Hz}$, and $B_{max} = 1 \text{ T}$ ($1^{1.6} = 1$).
7. Which of the following correctly pairs a material with its magnetic classification?
Explanation: Neodymium-Iron-Boron is a modern high-strength hard magnetic material used for permanent magnets. The other options have inverted classifications.
8. What does the demagnetization quadrant of a hysteresis loop represent?
Explanation: The segment of the loop from remanence ($B_r$) to coercivity ($H_c$) is the demagnetization quadrant where reverse field removes residual induction.
9. If magnetic susceptibility $\chi_m$ of a ferromagnetic medium is very large, its relative permeability $\mu_r$ will be approximately equal to:
Explanation: Since $\mu_r = 1 + \chi_m$ and $\chi_m \gg 1$ for ferromagnets, $\mu_r \approx \chi_m$.
10. A student claims that hysteresis loss is independent of the frequency of the alternating magnetic field. Is this correct?
Explanation: While energy loss per cycle ($\oint H \, dB$) is independent of frequency, power loss is $P_h = \text{(energy per cycle)} \times f$, making total hysteresis power loss directly proportional to frequency.