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).