Chapter 3 - Magnetostatics & Magnetic Boundary Conditions

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📄 Section: 03 Chapter Map - Magnetostatics & Magnetic Materials

03 Chapter Map - Magnetostatics & Magnetic Materials

Chapter 3 Overview & Map of Content (MOC): Fundamental postulates of magnetostatics, Lorentz force equation, Biot-Savart law applications (finite straight wire, circular loop, coaxial cable), Ampere’s circuital law, vector magnetic potential A, vector Poisson equation, magnetization M, bound currents (Jms, Jm), magnetostatic boundary conditions (B1n=B2n, H1t-H2t=Js), magnetic refraction law (tanθ1/tanθ2=μ1/μ2), magnetic energy density, and Hall effect.

📚 Study Notes Index

• 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation
• 3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable)
• 3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A)
• 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect
• 00 Chapter 3 Active-Recall Diagnostic Quiz

🔗 Related Resources

• Course Teaching Plan: ECE 2105 - Electromagnetic Fields and Waves
• Master Course Portal: 00 ECE 2105 Complete Course Master Map

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📄 Section: 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation

3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation

Related Concepts: 2.01 Fundamental Postulates of Electrostatics & Gauss’s Law Applications | 3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable) | 3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A)

3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation

Overview: Magnetostatics is the study of steady (time-invariant) magnetic fields produced by constant direct currents (DC). The field behavior is anchored by two fundamental postulates: Conservation of Magnetic Flux (∇ · B = 0) and Ampere’s Curl Postulate (∇ × H = J).

1. Fundamental Postulates of Magnetostatics in Free Space [PYQ: 2015, 2016, 2018, 2021]

Postulate Name	Differential (Point) Form	Integral Form	Physical Significance
Divergence Postulate	∇ · B = 0	∮S B · ds = 0	No Isolated Magnetic Monopoles: Magnetic flux lines always form continuous closed loops.
Curl Postulate	∇ × H = J	∮C H · dl = Ienclosed	Ampere’s Law: Circulation of magnetic field equals total enclosed free current.

2. The Lorentz Force Equation [PYQ: 2018, 2021]

When a test charge q moves with velocity u in a region containing coexisting electric field E and magnetic field B, the total force experienced is the vector sum:

F = Fe + Fm = q(E + u × B) [Newtons]

2.1 Key Physical Features of Magnetic Force (Fm = qu × B)

1. Perpendicularity: Magnetic force Fm is always perpendicular to both charge velocity u and magnetic field B.

2. Zero Work Done: Because Fm · dl = q(u × B) · (u dt) = 0, a static magnetic field does no work on a moving charge; it alters particle direction but cannot change kinetic energy or speed.

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📄 Section: 3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable)

title: “3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable)”

aliases:

• Biot-Savart Law
• Finite Straight Wire Derivation
• Circular Loop Field Derivation
• Biot-Savart vs Ampere Law

tags:

• ece2105
• electromagnetics
• magnetostatics
• biot-savart
• study-order/3.02

type: uni-note

course: “ECE 2105 - Electromagnetic Fields and Waves”

chapter_map: “03 Chapter Map - Magnetostatics & Magnetic Materials”

term: 2-1

teacher: Mashuk Sir / Sanglap Sir

status: enhanced

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Related Concepts: 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation | 3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A) | 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect

3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable)

Overview

The Biot-Savart Law calculates the magnetic flux density dB produced at a point in space by a differential current element I dl’.

1. Statement & Vector Formula of Biot-Savart Law [PYQ: 2015, 2018, 2023, 2025]

dB = (u0 I / 4pi) * (dl’ x aR / R^2) ⇒ B = (u0 I / 4pi) * Integral (dl’ x aR / R^2)

1.1 Biot-Savart Law vs. Ampere’s Circuital Law Comparison [PYQ: 2015, 2018, 2023, 2025]

Feature	Biot-Savart Law	Ampere’s Circuital Law
Mathematical Method	Direct vector integration over current distribution.	Line integral along closed path C (Integral H . dl = I).
Geometry Requirement	General. Works for asymmetric wires and arbitrary loops.	Practical only for highly symmetric systems.
Electrostatic Analogy	Analogous to Coulomb’s Law.	Analogous to Gauss’s Law.

2. Master Derivation 1: Finite Straight Wire of Length 2L [PYQ: Heavily Tested: 2015, 2016, 2021, 2023, 2025]

Let a straight wire of length 2L lie along the z-axis from z = -L to z = +L, carrying current I. Find B at observation point P(r, 0, 0) at distance r on the x-axis.

 Z-axis

+L (Wire Top)

|

| R = sqrt(r^2 + z’^2)

| /

dz’ ----+(z’) -------⇒ P(r, 0, 0) [Observation Point at distance r]

| \

| \

-L (Wire Bottom)

2.1 Step-by-Step Integration

1. Current Element: dl’ = az dz’. Distance vector R = ar r - az z’.
2. Cross Product: dl’ x R = (az dz’) x (ar r - az z’) = aphi r dz’.
3. Integral Setup: B = aphi (u0 I r / 4pi) * Integral from -L to +L of dz’ / (r^2 + z’^2)^(3/2)
4. Evaluate Integral (Trigonometric Substitution z’ = r tan theta): Integral = 2L / (r^2 * sqrt(r^2 + L^2))
5. Final Result:

B = aphi (u0 I L) / (2pi r sqrt(r^2 + L^2))

Infinite Wire Limiting Case (L → infinity) [PYQ: 2015, 2021, 2025]

When L >> r, L / sqrt(r^2 + L^2) → 1: B = aphi (u0 I / 2pi r)

3. Master Derivation 2: Circular Current Loop along Axis [PYQ: 2016, 2018, 2020, 2022]

For a circular current loop of radius b in the xy-plane carrying current I, the magnetic flux density at height z along the z-axis is:

B = az (u0 I b^2) / (2 (b^2 + z^2)^(3/2))

• At Center (z=0): B = az (u0 I / 2b)

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📄 Section: 3.03 Ampere_s Circuital Law & Vector Magnetic Potential (A)

3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A)

Related Concepts: 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation | 3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable) | 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect

3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A)

Overview

Ampere’s Circuital Law provides a rapid algebraic method to calculate magnetic fields for symmetric current distributions. For general non-symmetric fields, Vector Magnetic Potential A serves as the magnetic analogue to electric potential V.

1. Vector Magnetic Potential (A) Definition & Identity Proof [PYQ: 2016, 2018, 2019, 2021]

Since ∇ ⋅ B = 0, and the divergence of any vector curl is identically zero (∇ ⋅ (∇ × A) ≡ 0), we define A as:

B = ∇ × A [Wb/m² or Tesla]

2. Vector Poisson’s Equation (∇² A = -μ₀ J) [PYQ: 2019, 2024]

Substitute B = μ₀ H and B = ∇ × A into Ampere’s postulate ∇ × H = J:

∇ × (∇ × A) = μ₀ J

Using vector identity ∇ × (∇ × A) = ∇ (∇ ⋅ A) - ∇² A, and choosing the Coulomb Gauge (∇ ⋅ A = 0):

∇² A = -μ₀ J (Vector Poisson’s Equation)

2.1 Integral Expressions for A

• Volume Current (J): A = (μ₀ / 4π) ∫_V (J / R) dv’
• Line Current (I): A = (μ₀ I / 4π) ∫_L (dl’ / R)

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📄 Section: 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect

Related Concepts: 3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A) | 2.04 Electrostatic Boundary Conditions, Refraction Law & Poisson-Laplace Equations | 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation

3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect

> [!abstract] Overview > In magnetic media, atomic electron orbits create microscopic magnetic dipoles defining the Magnetization Vector $\vec{M}$**. Field transitions across magnetic boundaries are governed by Magnetostatic Boundary Conditions and the Hall Effect*.*

1. Magnetization Vector $\vec{M}$ & Bound Currents [PYQ: 2018, 2021]

The Magnetization Vector $\vec{M}$ is defined as the magnetic dipole moment per unit volume: $\vec{M}=\lim \_\Delta v\to 0\frac{\sum \vec{m}\_i}{\Delta v}[\text{A/m}]$

1.1 Equivalent Bound Current Densities [PYQ: 2018, 2021]

• Surface Bound Current Density: $\vec{J}\_ms=\vec{M}\times \hat{a}\_n[\text{A/m}]$
• Volume Bound Current Density: $\vec{J}\_m=\nabla \times \vec{M}[{\text{A/m}}^2]$

2. Magnetostatic Boundary Conditions & Refraction Law [PYQ: 2016, 2019, 2022, 2025]

 Medium 1 (mu_1) B1n ====⇒ B2n (Normal B is continuous)

-----------------------+---------------- Surface Current Js

Medium 2 (mu_2) H1t -----⇒ H2t (H1t - H2t = Js)



2.1 Normal Boundary Condition ($B\_1n=B\_2n$) [PYQ: 2016, 2019, 2022, 2025]

Applying $\oint \_S\vec{B}\cdot d\vec{s}=0$ over a pillbox:

$B\_1n=B\_2n⟹\mu \_1H\_1n=\mu \_2H\_2n$

Physical Origin: Non-existence of Magnetic Monopoles

This boundary condition is derived from Gauss’s law for magnetism, which states that the divergence of the magnetic flux density is zero ($\nabla \cdot \vec{B}=0$). Physically, this represents the non-existence of isolated magnetic monopoles (magnetic charge), meaning magnetic flux lines always form closed loops and must enter and leave any closed boundary in equal amounts. Consequently, the normal component of $\vec{B}$ is always continuous across any interface.

2.2 Tangential Boundary Condition ($H\_1t-H\_2t=J\_s$) [PYQ: 2016, 2019, 2022, 2025]

Applying $\oint \_C\vec{H}\cdot d\vec{l}=I\_\text{enc}$ over a rectangular loop:

$\hat{a}\_n2\times (\vec{H}\_1-\vec{H}\_2)=\vec{J}\_s⟹H\_1t-H\_2t=J\_s$

(If boundary carries no free surface current $J\_s=0$, $H\_1t=H\_2t$).

2.3 Refraction Law for Magnetic Field Lines [PYQ: 2022, 2025]

$\frac{\tan \theta \_1}{\tan \theta \_2}=\frac{\mu \_1}{\mu \_2}$

3. The Hall Effect Mechanics & Voltage Formula [PYQ: 2018, 2021, 2024]

> [!abstract] Hall Effect Principle [PYQ: 2018, 2021, 2024] > When a current $I$ flows through a conducting strip of width $w$ placed in a transverse magnetic field $B$, magnetic force $\vec{F}\_m=q(\vec{u}\times \vec{B})$ deflects charge carriers to one side, establishing a transverse electric field $E\_H$ and Hall Voltage $V\_H$**:

$V\_H=E\_H\cdot w=\frac{BI}{nqw}$

Where $n$ is carrier density, $q$ is carrier charge, and $w$ is strip width. Used to measure magnetic fields and identify semiconductor carrier type (N-type vs. P-type).

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📄 Section: 00 Chapter 3 Active-Recall Diagnostic Quiz

00 Chapter 3 Active-Recall Diagnostic Quiz (Magnetostatics)

Overview: Test your conceptual understanding and mathematical recall of magnetostatic postulates, Biot-Savart law applications, Ampere’s law, vector magnetic potential A, bound current densities, magnetic boundary conditions, and the Hall effect.

Question 1: Magnetostatic Fundamental Postulates

State the differential and integral forms of the two fundamental postulates of magnetostatics in free space.

Solution:

1. Divergence Postulate (No Monopoles): ∇ ⋅ B = 0 ⟺ ∮ B ⋅ ds = 0
2. Curl Postulate (Ampere’s Law): ∇ × H = J ⟺ ∮ H ⋅ dl = I_enclosed

Question 2: Lorentz Force Equation

Write the Lorentz force equation for a charge q moving with velocity u in coexisting electric and magnetic fields.

Solution:

F = q(E + u × B)

Question 3: Biot-Savart Law vs. Ampere’s Law

When is it mathematically preferred to use Ampere’s Circuital Law rather than the Biot-Savart Law to calculate B?

Solution:

Ampere’s Law (∮ H ⋅ dl = I_enc) is preferred when the current distribution exhibits high spatial coordinate symmetry (infinite lines, cylinders, solenoids, toroids), allowing H to be factored out of the integral without explicit vector integrations.

Question 4: Field at Center of Circular Loop

Write the expression for magnetic flux density B at the center of a circular current loop of radius b carrying current I.

Solution:

B = a_z (μ₀ I / 2b)

Question 5: Vector Magnetic Potential Definition

Define Vector Magnetic Potential A and show why ∇ ⋅ B = 0 is automatically satisfied.

Solution:

Vector Magnetic Potential A is defined such that B = ∇ × A. Since the divergence of any vector curl is identically zero (∇ ⋅ (∇ × A) ≡ 0), taking the divergence yields ∇ ⋅ B = 0 automatically.

Question 6: Vector Poisson’s Equation

Write Vector Poisson’s equation for vector magnetic potential A in the Coulomb gauge (∇ ⋅ A = 0).

Solution:

∇² A = -μ₀ J

Question 7: Bound Current Densities

Define surface bound current density J_ms and volume bound current density J_m in terms of the magnetization vector M.

Solution:

• Surface Bound Current Density: J_ms = M × a_n
• Volume Bound Current Density: J_m = ∇ × M

Question 8: Normal Component Boundary Condition

State the boundary condition for the normal component of magnetic flux density B across an interface separating medium 1 (μ₁) and medium 2 (μ₂).

Solution:

B_1n = B_2n ⟹ μ₁ H_1n = μ₂ H_2n

The normal component of B is always continuous across any boundary.

Question 9: Tangential Component Boundary Condition

State the boundary condition for the tangential component of magnetic field intensity H across an interface carrying surface current density J_s.

Solution:

a_n2 × (H₁ - H₂) = J_s ⟹ H_1t - H_2t = J_s

If no free surface current exists (J_s = 0), H_1t = H_2t (tangential H is continuous).

Question 10: Hall Effect Voltage Formula

Write the formula for the Hall voltage V_H generated across a conductor strip of width w, carrier density n, carrying current I in magnetic field B.

Solution:

V_H = BI / nqw