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Physics LibreTexts

14.4: Basic Equations for Electromagnetics and Applications

( \newcommand{\kernel}{\mathrm{null}\,}\)



Fundamentals

f=q(E+v×μoH)[N]

×E=B/t

cEds=ddtABda

×H=J+D/t

cHds=AJda+ddtADda

D=ρADda=Vρdv

B=0ABda=0

J=ρ/t

E= electric field (Vm1)

H= magnetic field (Am1)

D= electric displacement (Cm2)

B= magnetic flux density (T) Tesla (T)= Weber m2=10,000 gauss 

ρ= charge density (Cm3)

J= current density (Am2)

σ= conductivity (Siemens m1)

Js= surface current density (Am1)

ρs= surface charge density (Cm2)ε0=8.85×1012 Fm1

μo=4π×107 Hm1

c=(εoμo)0.53×108 ms1

e=1.60×1019 C

Ey(z,t)=E+(zct)+E(z+ct)=Re{E_y(z)ejωt}

Hx(z,t)=η1o[E+(zct)E(z+ct)] [ or (ωtkz) or (tz/c)]

A(E×H)da+(d/dt)V(ε|E|2/2+μ|H|2/2)dv=VEJ dv (Poynting Theorem) 

Media and Boundaries

D=εoE+P

D=ρf, τ=ε/σ

εoE=ρf+ρp

P=ρp, J=σE

B=μH=μo(H+M)

ε=εo(1ω2p/ω2)

ωp=(Ne2/mεo)0.5

εeff=ε(1jσ/ωε)

A set of boundary condition equations for electromagnetic fields at a surface, with a shaded column emphasizing specific terms and a diagram showing a normal vector above a surface labeled 1 and 2.

Electromagnetic Quasistatics

2Φ=0

KCL:iIi(t)=0 at node 

KVL:iVi(t)=0 around loop 

C=Q/V=Aε/d[F]

L=Λ/I

i(t)=Cdv(t)/dt

v(t)=Ldi(t)/dt=dΛ/dt

Cparallel =C1+C2

Cseries =(C11+C12)1

we=Cv2(t)/2;wm=Li2(t)/2

Lsolenoid =N2μA/W

τ=RC,τ=L/R

Λ=ABdˉa (per turn) 

f=q(E+v×μoH)[N]

fz=dwT/dz

F=I×μoH[Nm1]

Ee=v×μoH  inside wire 

P=ωT=WTdVolume /dt[W]

Max f/A=B2/2μo,D2/2εo[Nm2]

vi =dwTdt+fdzdt

Electromagnetic Waves

(2με2/t2)E=0 [ Wave Eqn.]

(2+k2)E_=0,E_=E_0ejkr

k=ω(με)0.5=ω/c=2π/λ

k2x+k2y+k2z=k2o=ω2με

vp=ω/k, vg=(k/ω)1

θr=θi

sinθt/sinθi=ki/kt=ni/nt

θc=sin1(nt/ni)

θ>θcE_t=E_iT_e+αxjkzz

k_=kjk

Γ_=T_1

T_TE=2/(1+[ηocosθt/ηtcosθi])

T_TM=2/(1+[ηtcosθt/ηicosθi])

θB=tan1(εt/εi)0.5 for TM

Pd|J_S|2/2σδ [Wm2]

E=ϕA/t,B=×A

Φ_(r)=V(ρ_(r)ejk|rr|/4πεo|rr|)dv

A_(r)=Vμo(J_(r)ejk|rr|/4π|rr|)dv

E_ff=ˆϑ(jηokI_d/4πr)ejkrsinθ

2Φ_+ω2μ0ε0Φ_=ρ/ε0

2A_+ω2μoεoA_=μoJ_

Forces, Motors, and Generators

f=q(E+v×μoH)[N]

fz=dwT/dz

F=I×μoH[Nm1]

Ee=v×μoH  inside wire 

P=ωT=WTdVolume /dt[W]

Max f/A=B2/2μo, D2/2εo[Nm2]

vi=dwTdt+fdzdt

f=ma=d(mv)/dt

x=xo+vot+at2/2

P=fv [W]=Tω

wk=mv2/2

T=I dω/dt

I=imir2i

Circuits

KCL:iIi(t)=0 at node 

KVL:iVi(t)=0 around loop 

C=Q/V=Aε/d[F]

L=Λ/I

i(t)=C dv(t)/dt

v(t)=L di(t)/dt=dΛ/dt

Cparallel =C1+C2

Cseries =(C11+C12)1

we=Cv2(t)/2; wm=Li2(t)/2

Lsolenoid =N2μA/W

τ=RC,τ=L/R

Λ=ABda (per turn) 

Z_ series=R+jωL+1/jωC

Y_par =G+jωC+1/jωL

Q=ωowT/Pdiss=ωo/Δω

ωo=(LC)0.5

v2(t)/R=kT

Limits to Computation Speed

dv(z)/dz=Ldi(z)/dt

di(z)/dz=Cdv(z)/dt

d2v/dz2=LC d2v/dt2

v(z,t)=f+(tz/c)+f(t+z/c)=g+(zct)+g(zct)

i(t,z)=Yo[f+(tz/c)f(t+z/c)]

c=(LC)0.5=1/με

Zo=Y1o=(L/C)0.5

ΓL=f/f+=(RLZo)/(RL+Zo)

v(z,t)=g+(zct)+g(z+ct)

VTh=2f+(t), RTh=Zo

Power Transmission

(d2/dz2+ω2LC)V_(z)=0

V_(z)=V_+ejkz+V_e+jkz

I_(z)=Yo[V_+ejkzV_e+jkz]

k=2π/λ=ω/c=ω(με)0.5

Z_(z)=V_(z)/I_(z)=ZoZ_n(z)

Z_n(z)=[1+Γ_(z)]/[1Γ_(z)]=Rn+jXn

Γ_(z)=(V_V_+)e2 jkz=[Z_n(z)1]/[Z_n(z)+1]

Z_(z)=Z0(Z_LjZ0tankz)/(Z_0jZLtankz)

VSWR=|V_max|/|V_min|=Rmax

Wireless Communications and Radar

G(θ,ϕ)=Pr/(PR/4πr2)

PR=4πPr(θ,ϕ,r)r2sinθ dθdϕ

Prec=Pr(θ,ϕ)Ae(θ,ϕ)

Ae(θ,ϕ)=G(θ,ϕ)λ2/4π

Rr=PR/i2(t)

Eff(θ0)=(jejkr/λr)AEt(x,y)ejkxx+jkyydxdy

Prec=PR(Gλ/4πr2)2σs/4π

E_=iaiE_iejkr1=( element factor )( array f)

Ebit ≥∼4×1020[J]

Z_12=Z_21 if reciprocity 

(d2/dz2+ω2LC)V_(z)=0

V_(z)=V_+ejkz+V_e+jkz

I_(z)=Yo[V_+ejkzV_e+jkz]

k=2π/λ=ω/c=ω(με)0.5

Z_(z)=V_(z)/I_(z)=ZoZ_n(z)

Z_n(z)=[1+Γ_(z)]/[1Γ_(z)]=Rn+jXn

Γ_(z)=(V_/V_+)e2jkz=[Z_n(z)1]/[Z_n(z)+1]

Z_(z)=Z0(Z_LjZ0tankz)/(Z_0jZLtankz)

VSWR=|V_max|/|V_min|=Rmax

θr=θi

sinθt/sinθi=ki/kt=ni/nt

θc=sin1(nt/ni)

θ>θcE_t=E_iT_e+αxjkzz

k_=kjk

Γ_=T_1

Atω0, we=wm

we=V(ε|E_|2/4)dv

wm=V(μ|H_|2/4)dv

Qn=ωnWTn/Pn=ωn/2αn

fmnp=(c/2)([m/a]2+[n/b]2+[p/d]2)0.5

Sn=jωnαn

Optical Communications

E=hf, photons or phonons 

hf/c= momentum [kg ms1]

dn2/dt=[An2+B(n2n1)]

Acoustics

P=Po+p, U=Uo+u  (Uo=0 here )

p=ρou/t

u=(1/γPo)p/t

(2k22/t2)p=0

k2=ω2/c2s=ω2ρo/γPo

cs=vp=vg=(γPo/ρo)0.5 or (K/ρo)0.5

ηs=p/u=ρocs=(ρoγPo)0.5 gases 

ηs=(ρoK)0.5 solids, liquids 

p,u continuous at boundaries 

p_=p_+ejkz+p_e+jkz

u_z=η1s(p_+ejkzp_e+jkz)

Aupda+(d/dt)V(ρo|u|2/2+p2/2γPo)dV

Mathematical Identities

sin2θ+cos2θ=1

cosα+cosβ=2cos[(α+β)/2]cos[(αβ)/2]

H_(f)=+h(t)ejωtdt

ex=1+x+x2/2!+x3/3!+

sinα=(ejαejα)/2j

cosα=(ejα+ejα)/2

Vector Algebra

=ˆx/x+ˆy/y+ˆz/z

ˉAˉB=AxBx+AyBy+AzBz

2ϕ=(2/x2+2/y2+2/z2)ϕ

(×ˉA)=0

×(×ˉA)=(ˉA)2ˉA

Gauss and Stokes’ Theorems

\oiint_{\mathrm{V}}(\nabla \bullet \overrightarrow{\mathrm{G}}) \mathrm{d} \mathrm{v}=\oiint_{\mathrm{A}} \overrightarrow{\mathrm{G}} \bullet \mathrm{d} \overrightarrow{\mathrm{a}} \nonumber

\oiint_{\mathrm{A}}(\nabla \times \overrightarrow{\mathrm{G}}) \bullet \mathrm{d} \overrightarrow{\mathrm{a}}=\oint_{\mathrm{c}} \overrightarrow{\mathrm{G}} \bullet \mathrm{d} \overrightarrow{\mathrm{s}} \nonumber

Complex Numbers and Phasors

\mathrm{v}(\mathrm{t})=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{\underline{V}e}^{\mathrm{j} \omega \mathrm{t}}\right\} \text { where } \underline{\mathrm{V}}=|\mathrm{V}| \mathrm{e}^{\mathrm{j} \phi} \nonumber

\mathrm{e}^{\mathrm{j} \omega \mathrm{t}}=\cos \omega \mathrm{t}+\mathrm{j} \sin \omega \mathrm{t} \nonumber

Spherical Trigonometry

\int_{4 \pi} \mathrm{r}^{2} \sin \theta \ \mathrm{d} \theta \mathrm{d} \phi=4 \pi \nonumber


This page titled 14.4: Basic Equations for Electromagnetics and Applications is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David H. Staelin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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