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Physics LibreTexts

14.4: Basic Equations for Electromagnetics and Applications

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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

Wireless Communications and Radar

\mathrm{G}(\theta, \phi)=\mathrm{P}_{\mathrm{r}} /\left(\mathrm{P}_{\mathrm{R}} / 4 \pi \mathrm{r}^{2}\right) \nonumber

\mathrm{P_{R}=\int_{4 \pi} P_{r}(\theta, \phi, r) r^{2} \sin \theta \ d \theta d \phi} \nonumber

\mathrm P_{\mathrm{rec}}=\mathrm{P}_{\mathrm{r}}(\theta, \phi) \mathrm{A}_{\mathrm{e}}(\theta, \phi) \nonumber

\mathrm{A}_{\mathrm{e}}(\theta, \phi)=\mathrm{G}(\theta, \phi) \lambda^{2} / 4 \pi \nonumber

\mathrm{R}_{\mathrm{r}}=\mathrm{P}_{\mathrm{R}} \big/\left\langle\mathrm{i}^{2}(\mathrm{t})\right\rangle \nonumber

\mathrm{E}_{\mathrm{ff}}(\theta \cong 0)=\left(\mathrm{je}^{\mathrm{jkr}} / \lambda \mathrm{r}\right) \int_{\mathrm{A}} \mathrm{E}_{\mathrm{t}}(\mathrm{x}, \mathrm{y}) \mathrm{e}^{\mathrm{jk}_{\mathrm{x}}{\mathrm{x}+\mathrm{jk}}_{ \mathrm{y}}{\mathrm{y}}} \mathrm{dxdy} \nonumber

\mathrm P_{\mathrm{rec}}=\mathrm{P}_{\mathrm{R}}\left(\mathrm{G} \lambda / 4 \pi \mathrm{r}^{2}\right)^{2} \sigma_{\mathrm{s}} / 4 \pi \nonumber

\underline{\mathrm{\vec E}}=\sum_{\mathrm{i}} \mathrm{a}_{\mathrm{i}} \overrightarrow{\mathrm{\underline E}}_{\mathrm{i}} \mathrm{e}^{-\mathrm{jk} \mathrm{r}_{1}}=(\text { element factor })(\text { array } \mathrm{f}) \nonumber

\mathrm{E}_{\text {bit }} \geq \sim 4 \times 10^{-20}[\mathrm{J}] \nonumber

\underline{\mathrm Z}_{12}=\underline{\mathrm Z}_{21} \text { if reciprocity } \nonumber

\left(\mathrm{d}^{2} / \mathrm{d} \mathrm z^{2}+\omega^{2} \mathrm{LC}\right) \underline{\mathrm{V}}(\mathrm{z})=0 \nonumber

\mathrm{\underline V}(\mathrm{z})=\underline{\mathrm{V}}_{+}\mathrm{e}^{-\mathrm{jk} \mathrm{z}}+\underline{\mathrm{V}}_{-} \mathrm{e}^{+\mathrm{jk} \mathrm{z}} \nonumber

\mathrm{\underline{I}(z)=Y_{o}\left[\underline{V}_{+}e^{-j k z}-\underline{V}_{-} e^{+j k z}\right]} \nonumber

\mathrm{k}=2 \pi / \lambda=\omega / \mathrm{c}=\omega(\mu \varepsilon)^{0.5} \nonumber

\mathrm{\underline{Z}(z)=\underline{V}(z) / \underline{I}(z)=Z_{o} \underline{Z}_{n}(z)} \nonumber

\mathrm{\underline{Z}_{n}(z)=[1+\underline{\Gamma}(z)]/[1-\underline{\Gamma}(z)]=R_{n}+j X_{n}} \nonumber

\underline{\Gamma}(\mathrm{z})=\left(\underline{\mathrm{V}}_{-}/\underline{\mathrm{V}}_{+}\right) \mathrm{e}^{2 \mathrm{jkz}}=\left[\underline{\mathrm{Z}}_{\mathrm{n}}(\mathrm{z})-1\right]/\left[\mathrm{\underline Z}_{\mathrm{n}}(\mathrm{z})+1\right] \nonumber

\mathrm{\underline{Z}(z)=Z_{0}\left(\underline{Z}_{L}-j Z_{0} \tan k z\right) /\left(\underline{Z}_{0}-j Z_{L} \tan k z\right)} \nonumber

\mathrm{VSWR}=\left|\underline{\mathrm{V}}_{\max }\right| /\left|\underline{\mathrm{V}}_{\min }\right|=\mathrm{R}_{\max } \nonumber

\theta_{\mathrm{r}}=\theta_{\mathrm{i}} \nonumber

\sin \theta_{\mathrm{t}} / \sin \theta_{\mathrm{i}}=\mathrm{k}_{\mathrm{i}} / \mathrm{k}_{\mathrm{t}}=\mathrm{n}_{\mathrm{i}} / \mathrm{n}_{\mathrm{t}} \nonumber

\theta_{\mathrm{c}}=\sin ^{-1}\left(\mathrm{n}_{\mathrm{t}} / \mathrm{n}_{\mathrm{i}}\right) \nonumber

\theta>\theta_{\mathrm{c}} \Rightarrow \overrightarrow{\mathrm{\underline E}}_{\mathrm{t}}=\overrightarrow{\mathrm{\underline E}}_{\mathrm{i}} \mathrm{\underline{T}e}^{+\alpha x-\mathrm{j} \mathrm{k}_{\mathrm{z}^{\mathrm{z}}}} \nonumber

\underline{\mathrm{\vec k}}=\overrightarrow{\mathrm{k}}^{\prime}-\mathrm{j} \overrightarrow{\mathrm{k}}^{\prime \prime} \nonumber

\underline{\Gamma}=\underline{\mathrm{T}}-1 \nonumber

\operatorname{At} \omega_{0}, \ \left\langle\mathrm{w}_{\mathrm{e}}\right\rangle=\left\langle\mathrm{w}_{\mathrm{m}}\right\rangle \nonumber

\left\langle\mathrm{w}_{\mathrm{e}}\right\rangle=\int_{\mathrm{V}}\left(\varepsilon|\overrightarrow{\mathrm{\underline E}}|^{2} \big/ 4\right) \mathrm{d} \mathrm{v} \nonumber

\left\langle\mathrm{w}_{\mathrm{m}}\right\rangle=\int_{\mathrm{V}}\left(\mu|\overrightarrow{\mathrm{\underline H}}|^{2} \big/ 4\right) \mathrm{d} \mathrm{v} \nonumber

\mathrm{Q}_{\mathrm{n}}=\omega_{\mathrm{n}} \mathrm{W}_{\mathrm{Tn}} / \mathrm{P}_{\mathrm{n}}=\omega_{\mathrm{n}} / 2 \alpha_{\mathrm{n}} \nonumber

\mathrm{f}_{\mathrm{mnp}}=(\mathrm{c} / 2)\left([\mathrm{m} / \mathrm{a}]^{2}+[\mathrm{n} / \mathrm{b}]^{2}+[\mathrm{p} / \mathrm{d}]^{2}\right)^{0.5} \nonumber

\mathrm{S}_{\mathrm{n}}=\mathrm{j} \omega_{\mathrm{n}}-\alpha_{\mathrm{n}} \nonumber

Optical Communications

\mathrm{E}=\mathrm{hf}, \text { photons or phonons } \nonumber

\mathrm{hf} / \mathrm{c}=\text { momentum }\left[\mathrm{kg} \ \mathrm{ms}^{-1}\right] \nonumber

\mathrm{dn}_{2} / \mathrm{dt}=-\left[\mathrm{An}_{2}+\mathrm{B}\left(\mathrm{n}_{2}-\mathrm{n}_{1}\right)\right] \nonumber

Acoustics

\mathrm{P}=\mathrm{P}_{\mathrm{o}}+\mathrm{p}, \ \overrightarrow{\mathrm{U}}=\overrightarrow{\mathrm{U}}_{\mathrm{o}}+\mathrm{u} \ \ \left(\overrightarrow{\mathrm{U}}_{\mathrm{o}}=0 \text { here }\right) \nonumber

\nabla \mathrm{p}=-\rho_{\mathrm{o}} \partial \overrightarrow{\mathrm{u}} / \partial \mathrm{t} \nonumber

\nabla \bullet \overrightarrow{\mathrm{u}}=-\left(1 / \gamma \mathrm{P}_{\mathrm{o}}\right) \partial \mathrm{p} / \partial \mathrm{t} \nonumber

\left(\nabla^{2}-\mathrm{k}^{2} \partial^{2} / \partial \mathrm{t}^{2}\right) \mathrm{p}=0 \nonumber

\mathrm{k}^{2}=\omega^{2} / \mathrm{c}_{\mathrm{s}}^{2}=\omega^{2} \rho_{\mathrm{o}} / \gamma \mathrm{P}_{\mathrm{o}} \nonumber

\mathrm{c}_{\mathrm{s}}=\mathrm{v}_{\mathrm{p}}=\mathrm{v}_{\mathrm{g}}=\left(\gamma \mathrm{P}_{\mathrm{o}} / \rho_{\mathrm{o}}\right)^{0.5} \text { or }\left(\mathrm{K} / \rho_{\mathrm{o}}\right)^{0.5} \nonumber

\eta_{\mathrm{s}}=\mathrm{p} / \mathrm{u}=\rho_{\mathrm{o}} \mathrm{c}_{\mathrm{s}}=\left(\rho_{\mathrm{o}} \gamma \mathrm{P}_{\mathrm{o}}\right)^{0.5} \text { gases } \nonumber

\eta_{\mathrm{s}}=\left(\rho_{\mathrm{o}} \mathrm{K}\right)^{0.5} \text { solids, liquids } \nonumber

\mathrm{p}, \overrightarrow{\mathrm{u}}_{\perp} \text { continuous at boundaries } \nonumber

\mathrm{\underline p}=\mathrm{\underline p}_{+} \mathrm{e}^{-\mathrm{jkz}}+\mathrm{\underline p}_{-} \mathrm{e}^{+\mathrm{jkz}} \nonumber

\mathrm{\underline{ u}_{z}=\eta_{s}^{-1}\left(\underline{p}_{+} e^{-j k z}-\underline{p}_{-}e^{+j k z}\right)} \nonumber

\mathrm{\int_{A} \overrightarrow{u} p \bullet d \overrightarrow{a}+(d / d t) \int_{V}\left(\rho_{o}|\overrightarrow{u}|^{2} / 2+p^{2} / 2 \gamma P_{o}\right) d V} \nonumber

Mathematical Identities

\sin ^{2} \theta+\cos ^{2} \theta=1 \nonumber

\cos \alpha+\cos \beta=2 \cos [(\alpha+\beta) / 2] \cos [(\alpha-\beta) / 2] \nonumber

\mathrm{\underline{H}(f)=\int_{-\infty}^{+\infty} h(t) e^{-j \omega t} d t} \nonumber

\mathrm{e^{x}=1+x+x^{2} / 2 !+x^{3} / 3 !+\ldots} \nonumber

\mathrm{\sin \alpha=\left(e^{j \alpha}-e^{-j \alpha}\right) / 2 j} \nonumber

\mathrm{\cos \alpha=\left(e^{j \alpha}+e^{-j \alpha}\right) / 2} \nonumber

Vector Algebra

\nabla=\mathrm{\hat{x} \partial / \partial x+\hat{y} \partial / \partial y+\hat{z} \partial / \partial z } \nonumber

\mathrm{\bar{A} \bullet \bar{B}=A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z}} \nonumber

\mathrm{\nabla^{2} \phi=\left(\partial^{2} / \partial x^{2}+\partial^{2} / \partial y^{2}+\partial^{2} / \partial z^{2}\right) \phi} \nonumber

\nabla \bullet (\nabla \times \bar{\mathrm A})=0 \nonumber

\nabla \times(\nabla \times \mathrm{ \bar{A})=\nabla(\nabla \bullet \bar{A})-\nabla^{2} \bar{A}} \nonumber

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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