Magnetic Flow

Φ=∫B⋅en dS=∮A⋅dl\Phi = \int\bm B\cdot\bm e_n\:dS = \oint\bm A\cdot dl

Linked Flow

Λ=NΦ\Lambda = N\Phi

Self-Inductance and Mutual Inductance

Λ1=L1I1+MI2Λ2=L2I2+MI1\begin{gathered} \Lambda_1 = L_1I_1 + MI_2 \\ \Lambda_2 = L_2I_2 + MI_1 \end{gathered}

Magnetic Field Strength

Amperes Law:

∮H⋅dℓ=∫J⋅en dS=Iinside\oint\bm H\cdot d\ell = \int \bm J \cdot\bm e_n \: dS = I_{\textup{inside}}

Connection between magnetization

M,B\bm M, \bm B

and

H\bm H

:

{B=μ0(H+M)(holds generally)B=μrμ0H\begin{cases} \bm B = \mu_0(\bm H + \bm M) \quad \textup{(holds generally)}\\ \bm B = \mu_r\mu_0\bm H \end{cases}

Equivalent Current Density

Jm=∇×Mvolume current density\bm J_m = \bm\nabla\times\bm M \qquad \textup{volume current density}
Jm=∇×Msurface current density\bm J_m = \bm\nabla\times\bm M \qquad \textup{surface current density}

Boundary Conditions

{en2×(H1−H2)=JsBn  Continuous\begin{cases} \bm e_{n2}\times(\bm H_1 - H_2) = \bm J_s \\ \bm B_n \; \textup{Continuous} \end{cases}

Scalar Potential

From magnetic dipole

m\bm m

:

Vm=14πm⋅eRR2V_m = \frac 1{4\pi}\frac{\bm m\cdot\bm e_R}{R^2}

Magnetic Pole Density

{ρm=−∇⋅Mvolume pole densityρm,s=en1⋅(M1−M2)surface pole density\begin{cases} \rho_m = -\bm\nabla\cdot\bm M &\textup{volume pole density} \\ \rho_{m, s} = e_{n1}\cdot(\bm M_1 - \bm M_2) &\textup{surface pole density} \end{cases}

Magnetic Force Law

dFm=Idl×Bd\bm F_m = Id\bm l\times\bm B

Magetic moment for current loop

m=∫Ien dS\bm m = \int I\bm e_n\:dS

Torque on Magnetic Moment

Tm=m×B\bm T_m = \bm m\times\bm B

Maxwell's Voltage

∣T∣=12B⋅HB is a bisector to en and T|\bm T| = \frac 12\bm B\cdot\bm H \qquad \bm B \textup{ is a bisector to } \bm e_n \textup{ and } \bm T

Magnetic Energy

Wm=12∫J⋅A dv=12∫B⋅H dv=12∑i∑jLijIiIjW_m = \frac 12\int\bm J\cdot\bm A\:dv = \frac 12\int\bm B\cdot\bm H\: dv = \frac 12\sum_i\sum_jL_{ij}I_iI_j

Two coils:

Wm=12L1I12+12L2I22+MI1I2W_m = \frac 12L_1I_1^2 + \frac 12L_2I_2^2 + MI_1I_2

Reluctance

R=1μrμ0SR = \frac 1{\mu_r\mu_0S}