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Electricity and Magnetism

Magnetic Field Intensity

Magnetic field intensity (often denoted by H or magnetic field strength) is a vector field quantity that represents the magnitude of the magnetic field produced by external currents without taking into account the magnetization of the medium. Its SI unit is ampere per meter (A/m) and its dimensional formula is [M0L-1T0A1].

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of magnetic field intensity, Biot-Savart Law, Ampere's Circuital Law, magnetic field due to straight conductor, circular coil, and long solenoid.

PGT extension

Relation between B, H, and M (B = μ0(H + M)), magnetic susceptibility, hysteresis loops, toroid calculations, and boundary conditions for magnetic fields across interfaces.

Prerequisites

Vector calculus (Line and surface integrals)
Concept of electric current and current density
Fundamental magnetic forces and Lorentz force

Definition and Physical Significance of H

Magnetic field intensity H is defined as the magnetic field produced by free or conduction currents alone.
It is independent of the magnetic properties of the medium through which it passes.
The total magnetic induction B inside a medium is the sum of the external field contribution (μ₀H) and the material's magnetization contribution (μ₀M).

Biot-Savart Law

Provides the differential magnetic field dB at a point due to a current-element Idl.
Mathematically expressed as d$B = (μ_{0}/4π) * (Idl × r̂)/r^{2}$.
The direction of dB is given by the right-hand thumb rule for cross product of Idl and position vector r.

Ampere's Circuital Law

States that the line integral of magnetic field intensity H around any closed loop is equal to the total free current enclosed by the loop.
Expressed mathematically as ∮ H · dl = I_{free, enclosed}.
Particularly useful for highly symmetric systems such as solenoids, toroids, and straight current-carrying cylinders.

Relation Between B, H, and Magnetization M

The constitutive relation in magnetism is given by $B = μ_{0}(H + M)$.
Magnetization M is defined as the magnetic dipole moment per unit volume of the material.
For linear isotropic media, $M = χ_{m}H$, leading to $B = μ_{0}(1 + χ_{m})H = μH$, where χ_m is magnetic susceptibility and μ is magnetic permeability.

Derivations

Magnetic Field Intensity Inside a Long Solenoid

Conditions: Infinitely long solenoid, tightly wound with n turns per unit length carrying current I.

Consider a rectangular Amperean loop of length l, with one side parallel to the axis inside the solenoid and the opposite side far outside where the field is zero.
Apply Ampere's Circuital Law: ∮ H · d$l = I_{enc}$.
The line integral evaluates only along the internal side of length l, since the perpendicular sides contribute zero and the external side experiences zero field.
Total enclosed free current I_enc = n · l · I.
Equate the line integral to the enclosed current: H × l = n · l · I.

Result: H = nI

Magnetic Field Intensity at the Center of a Circular Current-Carrying Coil

Conditions: Circular coil of radius R having N turns and carrying current I.

Consider an infinitesimally small current element Idl on the circular loop.
Using the Biot-Savart law, the angle between Idl and position vector r is 90 degrees (sin θ = 1).
The differential magnetic induction is d$B = (μ_{0}/4π) * (I dl / R^{2})$.
Integrate around the entire circumference: $B = ∫ dB = (μ_{0}I / 4πR^{2}) ∮ dl = (μ_{0}I / 4πR^{2})(2πRN)$.
Convert magnetic induction B to magnetic field intensity H by dividing by μ₀ (in vacuum, $H = B / μ_{0})$.

Result: H = \frac{NI}{2R}

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Biot-Savart Law (Differential Form)	$$dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2}$$	dB = Magnetic field, I = Current, dl = Length element, r = Distance, θ = Angle between dl and r Tesla (T)	General applicability in vacuum/air
Ampere's Circuital Law	$$\oint H \cdot dl = I_{enc}$$	H = Magnetic field intensity, dl = Path element, $I_enc = Enclosed free current$ Ampere (A)	Steady currents and high symmetry paths
Magnetic Field Intensity of a Long Solenoid	$$H = nI$$	n = Turns per unit length, I = Current A/m	Long solenoid approximation (length >> radius)
Magnetic Field Intensity Inside a Toroid	$$H = \frac{NI}{2\pi r}$$	N = Total turns, I = Current, r = Radial distance from center A/m	Circular toroidal core of radius r
Constitutive Magnetic Relation	$$B = \mu_0(H + M)$$	B = Magnetic induction, H = Magnetic field intensity, M = Magnetization, \m$u_0 = Permeability of free space$ Tesla (T)	Valid for macroscopic magnetic media
Magnetic Susceptibility Relationship	$$M = \chi_m H$$	M = Magnetization, \ch$i_m = Magnetic susceptibility$, H = Magnetic field intensity Dimensionless (M and H both in A/m)	Linear isotropic magnetic materials
Magnetic Field Intensity due to Long Straight Wire	$$H = \frac{I}{2\pi r}$$	I = Current, r = Perpendicular distance from wire A/m	Infinitely long straight conductor
Relation between B and H in Vacuum	$$B = \mu_0 H$$	B = Magnetic induction, \m$u_0 = Permeability of free space$, H = Magnetic field intensity T and A/m	Vacuum or non-magnetic medium (M = 0)

Graphs and experiments

Hysteresis Loop (B vs H Curve)

Draw: A closed loop showing non-linear relationship between B and H during cyclic magnetization and demagnetization, marking retentivity and coercivity.

Interpret: Area of the hysteresis loop represents thermal energy loss per unit volume per magnetization cycle in ferromagnetic materials.

Exam use: Frequently tested for identifying soft and hard ferromagnetic materials based on loop area, retentivity, and coercivity.

Magnetic Field Intensity H vs Distance r for a Long Straight Conductor

Draw: A rectangular hyperbola curve showing H decreasing inversely with distance r (H ∝ 1/r).

Interpret: Demonstrates the rapid fall-off of magnetic field intensity away from a linear current source.

Exam use: Conceptual questions comparing field intensity at different radial distances from a cable.

Magnetization M vs Magnetic Field Intensity H

Draw: Linear plots for paramagnetic (positive constant slope), diamagnetic (negative constant slope), and asymptotic/saturation curves for ferromagnetic materials.

Interpret: The slope of M versus H yields the magnetic susceptibility χ_m of the material.

Exam use: Numerical and theoretical questions regarding susceptibility calculations.

Solved numericals

TGT core

A long solenoid of length 2.0 m has 2000 turns and carries a current of 1.5 A. Calculate the magnetic field intensity H inside the solenoid.

Identify given parameters: Total turns N = 2000, Length L = 2.0 m, Current I = 1.5 A.
Calculate turns per unit length: $n = N / L = 2000 / 2.0 = 1000 turns/m$.
Use formula for magnetic field intensity of a solenoid: H = nI.
Substitute values: $H = 1000 m^{-1} × 1.5 A = 1500 A/m$.

Answer: 1500 A/m

PGT extension

A magnetic material is placed in an external magnetic field intensity H of 1000 A/m. If the resulting magnetization M is 3000 A/m, find the magnetic induction B and the magnetic susceptibility of the material.

Given: $H = 1000 A/m$, $M = 3000 A/m$, μ₀ = 4π × 10^-7 T·m/A.
Calculate magnetic susceptibility χ_m using $M = χ_{m}H: χ_{m} = M / H = 3000 / 1000 = 3$.
Calculate magnetic induction B using $B = μ_{0}(H + M): B = (4π × 10^{-7}) × (1000 + 3000)$.
Simplify expression: $B = (4π × 10^{-7}) × 4000 = 1.6π × 10^{-3} T ≈ 5.027 × 10^{-3} T$.

Answer: χ_m = 3, $B = 5.03 × 10^{-3} T$

TGT core

A circular coil of 50 turns and radius 0.1 m carries a current of 2.0 A. What is the magnetic field intensity at the center of the coil?

Identify given parameters: N = 50, R = 0.1 m, I = 2.0 A.
Use the formula for magnetic field intensity at the center of a circular coil: $H = (NI) / (2R)$.
Substitute the values: $H = (50 × 2.0) / (2 × 0.1)$.
Calculate result: $H = 100 / 0.2 = 500 A/m$.

Answer: 500 A/m

Assumptions and limitations

Ampere's Circuital Law calculations assume high degree of symmetry (such as infinite length or closed toroidal geometry) to easily extract H from the line integral.
Biot-Savart Law assumes steady (time-independent) currents; for time-varying currents, displacement current contributions must be included via Maxwell-Ampere law.

Common misconceptions

Common error: Magnetic field intensity H depends directly on the magnetic properties (permeability) of the surrounding medium.

Correct: Magnetic field intensity H depends solely on free/conduction currents and geometrical configuration, independent of the medium.

The medium's property (permeability μ or susceptibility χ_m) affects the magnetic induction B and magnetization M, but by definition, H is the auxiliary field generated purely by external free currents.

Common error: The SI unit of magnetic field intensity H is Tesla.

Correct: The SI unit of H is Ampere per meter (A/m), whereas Tesla (T) is the unit of magnetic induction B.

Tesla measures magnetic flux density or induction, while H represents current per unit length driving the field.

Quick revision

Magnetic field intensity H has units of A/m and is produced exclusively by free currents.
Fundamental formula for a long solenoid is H = nI.
Constitutive relation connecting fields is $B = μ_{0}(H + M)$.
Magnetic susceptibility is defined by the relation $M = χ_{m}H$.

Must remember

Magnetic field intensity $H$ is defined exclusively by free or conduction currents and is independent of the medium's magnetic properties.
The constitutive relation connecting magnetic induction $B$, field intensity $H$, and magnetization $M$ is given by $B = \mu_0(H + M)$.
For linear isotropic media, the magnetization is proportional to $H$, expressed as $M = \chi_m H$, leading to $B = \mu H$.
The SI unit of magnetic field intensity $H$ is $\text{A/m}$, while the unit of magnetic induction $B$ is $\text{Tesla (T)}$.
Ampere's Circuital Law states that $\oint H \cdot dl = I_{\text{free, enclosed}}$, making it ideal for highly symmetric geometries like solenoids and toroids.
The Biot-Savart Law provides the differential magnetic field contribution $dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2}$.

Glossary

Magnetic Field Intensity ($H$)An auxiliary vector field representing the strength of a magnetic field produced by free currents, measured in $\text{A/m}$.

Magnetic Induction ($B$)The total magnetic flux density inside a medium, incorporating both external free currents and internal material response, measured in Teslas.

Magnetization ($M$)The net magnetic dipole moment per unit volume induced inside a magnetic material, measured in $\text{A/m}$.

Magnetic Susceptibility ($\chi_m$)A dimensionless proportionality constant indicating the degree of magnetization of a material in response to an applied magnetic field.

Permeability ($\mu$)The property of a medium that describes its ability to support a magnetic field, equal to $\mu_0(1 + \chi_m)$.

Comparison between Magnetic Field Intensity ($H$) and Magnetic Induction ($B$)

Feature	Magnetic Field Intensity ($H$)	Magnetic Induction ($B$)
Primary Cause	Generated solely by free/conduction currents	Generated by both free currents and bound/material currents
Medium Dependence	Independent of the surrounding magnetic medium	Dependent on the magnetic permeability of the medium
SI Unit	$\text{A/m}$ (Ampere per meter)	$\text{T}$ (Tesla) or $\text{Wb/m}^2$
Dimensional Formula	$[L^{-1} I]$	$[M T^{-2} I^{-1}]$

Practice Quiz

Choose an option to reveal the explanation.

1. What is the SI unit of magnetic field intensity $H$?

Explanation: Magnetic field intensity $H$ represents current per unit length driving a magnetic field, hence its SI unit is $\text{A/m}$. Tesla is the unit of magnetic induction $B$.

2. Which of the following best describes magnetic field intensity $H$ in a material medium?

Explanation: By definition, $H$ is produced by free or conduction currents alone and is completely independent of the magnetic properties (such as permeability or susceptibility) of the medium.

3. The constitutive relation connecting $B$, $H$, and $M$ in a magnetic medium is given by:

Explanation: The macroscopic magnetic induction inside a medium is given by $B = \mu_0(H + M)$, where $\mu_0 H$ is the contribution from external free currents and $\mu_0 M$ is from material magnetization.

4. For a linear isotropic magnetic medium, the relationship between magnetization $M$ and magnetic field intensity $H$ is expressed as:

Explanation: For linear isotropic media, magnetization is directly proportional to the magnetic field intensity through magnetic susceptibility $\chi_m$, i.e., $M = \chi_m H$.

5. What is the magnetic field intensity $H$ inside an infinitely long straight wire carrying current $I$ at a perpendicular distance $r$?

Explanation: Using Ampere's circuital law $\oint H \cdot dl = I$, we get $H(2\pi r) = I$, which gives $H = \frac{I}{2\pi r}$.

6. Ampere's Circuital Law is mathematically expressed as:

Explanation: Ampere's circuital law states that the line integral of magnetic field intensity $H$ around any closed loop equals the total free current enclosed by that loop: $\oint H \cdot dl = I_{\text{enc}}$.

7. The differential form of the Biot-Savart law in free space for a current element $I dl$ at distance $r$ is:

Explanation: The Biot-Savart law gives the differential magnetic field $dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2}$.

8. What is the magnetic field intensity $H$ inside a long solenoid having $n$ turns per unit length carrying a current $I$?

Explanation: For a long solenoid, application of Ampere's law gives $H = nI$, independent of the core medium's permeability.

9. What is the magnetic field intensity $H$ at a radial distance $r$ inside a toroid having $N$ total turns and carrying current $I$?

Explanation: For a toroid of radius $r$, the magnetic field intensity is $H = \frac{NI}{2\pi r}$, derived by integrating $H$ over a circular path of radius $r$ enclosing $NI$ current.

10. In a vacuum or non-magnetic medium where magnetization $M = 0$, what is the relation between magnetic induction $B$ and magnetic field intensity $H$?

Explanation: Since $B = \mu_0(H + M)$, setting $M = 0$ for vacuum yields $B = \mu_0 H$.

11. If the magnetic susceptibility $\chi_m$ of a medium is $99$, what is its relative permeability $\mu_r$?

Explanation: Relative permeability is given by $\mu_r = 1 + \chi_m$. Therefore, $\mu_r = 1 + 99 = 100$.

12. Which law is particularly useful for calculating magnetic fields in highly symmetric systems like solenoids and toroids?

Explanation: Ampere's Circuital Law simplifies calculations significantly for highly symmetric systems where symmetry allows easy evaluation of the line integral of $H$.

13. What does the direction of the differential magnetic field $dB$ in the Biot-Savart law correspond to?

Explanation: The Biot-Savart law contains the term $Idl \times \hat{r}$, meaning the direction of $dB$ follows the right-hand thumb rule for the cross product of $Idl$ and $\hat{r}$.

14. If a magnetic material has a negative magnetic susceptibility $\chi_m$, the material is classified as:

Explanation: Diamagnetic materials have a small negative magnetic susceptibility ($\chi_m < 0$), causing them to weakly oppose applied magnetic fields.

15. Which physical quantity has the same dimensions and SI unit as magnetic field intensity $H$?

Explanation: Both magnetic field intensity $H$ and magnetization $M$ represent current per unit length and are measured in $\text{A/m}$ with dimensions $[L^{-1} I]$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A long solenoid of length $1.0\text{ m}$ has $1000$ turns and carries a current of $2.0\text{ A}$. What is the magnetic field intensity $H$ inside the solenoid?

Explanation: Turns per unit length $n = \frac{N}{L} = \frac{1000}{1\text{ m}} = 1000\text{ turns/m}$. The magnetic field intensity is $H = nI = 1000 \times 2.0 = 2000\text{ A/m}$ (independent of core permeability).

2. The magnetic susceptibility of a paramagnetic medium is $3 \times 10^{-4}$. If the magnetic field intensity is $10^4\text{ A/m}$, what is the magnetization $M$ of the medium?

Explanation: Using $M = \chi_m H$, we get $M = (3 \times 10^{-4}) \times (10^4\text{ A/m}) = 3\text{ A/m}$?

3. An infinitely long straight wire carries a current of $10\text{ A}$. What is the magnitude of magnetic field intensity $H$ at a distance of $0.05\text{ m}$ from the wire?

Explanation: Using $H = \frac{I}{2\pi r}$, we get $H = \frac{10}{2\pi(0.05)} = \frac{10}{0.1\pi} = \frac{100}{\pi}\text{ A/m}$.

4. In a magnetic material, the magnetic induction $B$ is $1.2\text{ T}$ and the magnetic field intensity $H$ is $10^3\text{ A/m}$. Given $\mu_0 = 4\pi \times 10^{-7}\text{ T m/A}$, what is the magnetization $M$ approximately equal to?

Explanation: From $B = \mu_0(H + M)$, we have $H + M = \frac{B}{\mu_0} = \frac{1.2}{4\pi \times 10^{-7}} \approx 9.549 \times 10^5\text{ A/m}$. Since $H = 10^3\text{ A/m}$ is negligible compared to this term, $M \approx 9.55 \times 10^5\text{ A/m}$.

5. A toroid has $500$ turns and carries a current of $1.5\text{ A}$. If the mean radius of the toroid is $0.2\text{ m}$, what is the magnetic field intensity $H$ inside the core?

Explanation: Using $H = \frac{NI}{2\pi r}$, we substitute $N = 500$, $I = 1.5\text{ A}$, and $r = 0.2\text{ m}$: $H = \frac{500 \times 1.5}{2\pi(0.2)} = \frac{750}{0.4\pi} = \frac{3750}{\pi}\text{ A/m}$.

6. If the relative permeability of a ferromagnetic substance is $2000$, its magnetic susceptibility $\chi_m$ is:

Explanation: Since $\mu_r = 1 + \chi_m$, magnetic susceptibility is $\chi_m = \mu_r - 1 = 2000 - 1 = 1999$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which statement correctly distinguishes magnetic field intensity $H$ from magnetic induction $B$?

Explanation: By definition, $H$ is the auxiliary field generated purely by external conduction currents, while $B$ incorporates the material's response via magnetization $M$.

2. Calculate the magnetic field intensity $H$ at a point $0.1\text{ m}$ away from a long straight conductor carrying $5\text{ A}$ current.

Explanation: Using $H = \frac{I}{2\pi r} = \frac{5}{2\pi(0.1)} = \frac{5}{0.2\pi} = \frac{25}{\pi}\text{ A/m}$.

3. If the magnetic susceptibility of a medium is $-0.5$, what type of magnetic material is it and what is its relative permeability?

Explanation: A negative susceptibility ($\chi_m = -0.5$) characterizes diamagnetic materials. The relative permeability is $\mu_r = 1 + \chi_m = 1 + (-0.5) = 0.5$.

4. What happens to the magnetic field intensity $H$ inside a toroidal coil if the core material is changed from air to soft iron?

Explanation: Magnetic field intensity $H$ depends only on free current and geometry ($H = \frac{NI}{2\pi r}$), so changing the core medium does not alter $H$ (though magnetic induction $B$ would increase greatly).

5. In a linear magnetic medium, if $H$ is doubled, what happens to the magnetization $M$?

Explanation: Since $M = \chi_m H$ for a linear medium, doubling $H$ results in a proportional doubling of $M$.

6. The line integral of $H$ around a closed loop is $40\text{ A}$. What is the total free current enclosed by the loop?

Explanation: By Ampere's Circuital Law, $\oint H \cdot dl = I_{\text{enc}}$. Thus, if the line integral is $40\text{ A}$, the enclosed free current is $40\text{ A}$.

7. What is the dimensional formula for magnetic field intensity $H$?

Explanation: Since $H$ is measured in $\text{A/m}$ (Current / Length), its dimensional formula is $[L^{-1} I]$.

8. A solenoid of length $0.5\text{ m}$ has $400$ turns and carries a current of $3\text{ A}$. Find the magnetic field intensity $H$ inside.

Explanation: Turns per unit length $n = \frac{400}{0.5} = 800\text{ turns/m}$. Then $H = nI = 800 \times 3 = 2400\text{ A/m}$.

9. Which of the following relations correctly links magnetic induction $B$, magnetic field intensity $H$, and magnetic permeability $\mu$ in a homogeneous isotropic medium?

Explanation: Since $B = \mu_0(H + M) = \mu_0(1 + \chi_m)H = \mu H$, the relation is simply $B = \mu H$.

10. Why is Ampere's Circuital Law generally more convenient than the Biot-Savart Law for calculating fields of highly symmetric configurations?

Explanation: High symmetry allows $H$ to be pulled out of the line integral $\oint H \cdot dl$, reducing vector integration to a simple algebraic multiplication.