MIT OpenCourseWare http://ocw.mit.edu 6.013/ESD.013J Electromagnetics and Applications, Fall 2005
Please use the following citation format: Markus Zahn, Erich Ippen, and David Staelin, 6.013/ESD.013J Electromagnetics and Applications, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons AttributionNoncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
6.013 Formula Sheet # 1 Cartesian Coordinates (x, y, z) ηf ηf ηf ix + iy + iz ηx ηy ηz
ηAx ηAy ηAz
�·A = + + ηx ηy ηz „ « „ « „ « ηAz ηAy ηAx ηAz ηAy ηAx − − − + iy + iz � × A = ix ηy ηz ηz ηx ηx ηy �f =
Cartesian x y z ix
= = = =
Cylindrical r cos � r sin � z cos �ir − sin �i�
= = = =
iy
=
sin �ir + cos �i�
=
iz
=
iz
=
Cylindrical r � z ir i� iz
= = = = = =
Cartesian p x2 + y 2 y tan−1 x z cos �ix + sin �iy − sin �ix + cos �iy iz
= = = = = =
Spherical r sin σ � r cos σ sin σir + cos σiφ i� cos σir − sin σiφ
Spherical r σ
= =
= =
Cylindrical � r2 + z 2 cos−1 � z
= =
� sin σir + cos σiz
=
cos σir − sin σiz
η2f η2f η2f � f = + + 2 2 ηx ηy ηz 2
Spherical r sin σ cos � r sin σ sin � r cos σ sin σ cos �ir + cos σ cos �iφ − sin �i� sin σ sin �ir + cos σ sin �iφ + cos �i� cos σir − sin σiφ
2
Cylindrical Coordinates (r, �, z) ηf 1 ηf ηf ir + i� + iz ηr r η� ηz
1 η 1 ηA� ηAz
�·A = (rAr ) + + r ηr r η� ηz
" ` # ´ „ « „ « ηA� 1 ηAz ηAr ηAz 1 η rA� ηAr − − + i� + iz � × A = ir − r η� ηz ηz ηr r ηr η� „ « 2f 2 f
1 η η η ηf 1 r + 2 + �2 f = r ηr ηr r η�2 ηz 2
�f =
Spherical Coordinates (r, σ, �) ηf 1 ηf 1 ηf ir + iφ + i� ηr r ησ r sin σ η� 1 η (sin σAφ ) 1 η ` 2 ´ 1 ηA� �·A = 2 r Ar + + r ηr r sin σ ησ r sin σ η� # " ` ´ η sin σA� 1 ηAφ
− � × A = ir r sin σ ησ η�
" ` ´# » – η rA� 1 1 ηAr 1 η (rAφ ) ηAr − − + iφ + i� r sin σ η� ηr r ηr ησ „ « „ « 1 η ηf 1 1 η ηf η 2 f
r2 + 2 sin σ + 2 �2 f = 2 r ηr ηr r sin σ ησ ησ r sin2 σ η�2
�f =
pCartesian x2 + y 2 + z 2 z cos−1 �
x2 +y 2 +z 2
cot−1 x y sin σ cos �ix + sin σ sin �iy + cos σiz iφ = cos σ cos �ix + cos σ sin �iy − sin σiz i� = − sin �ix + cos �iy Geometric relations between coordinates and unit and spherical coordinate systems. � ir
1
= =
r 2 +z 2
= i� vectors for Cartesian, cylindrical,
6.013 Formula Sheet # 1 Vector Identities
Maxwell’s Equations Integral Differential Boundary Conditions Faraday’s Law R H � � � d � × E = − σB n × (E2 − E1 ) = 0 L E · dl = − dt S B · dS σt HAmpere’s Law R with Maxwell’s R Displacement Current Correction d H · dl = S Jf · dS + dt D · dS � × H = Jf + σD n × (H2 − H1 ) = Kf L S σd HGauss’s Law R � · D = �f n · (D2 − D1 ) = �f H S D · dS = V �f dV B · dS = 0 �·B= 0 n · (B2 − B1 ) = 0 S Conservation of Charge H R σ� σ� d � · Jf + σtf = 0 n · (J2 − J1 ) + σtf = 0 S Jf · dS + dt V �f dV = 0 Usual Linear Constitutive Laws D = πE B = µH � Jf = �(E + v × B) = �E [Ohm’s law for moving media with velocity v]
(A × B) · C = A · (B × C) = (C × A) · B A × (B × C) = B(A · C) − C(A · B)
� · (� × A) = 0 � × (�f ) = 0
�(f g) = f �g + g�f
�(A · B) = (A · �)B + (B · �)A + A × (� × B) + B × (� × A) � · (f A) = f � · A + (A · �)f
� · (A × B) = B · (� × A) − A · (� × B)
� × (A × B) = A(� · B) − B(� · A) + (B · �)A − (A · �)B
� × (f A) = �f × A + f � × A
1 �(A · A) 2 � × (� × A) = �(� · A) − �2 A (� × A) × A = (A · �)A −
Physical Constants Constant Symbol Value Speed of light in vacuum c 2.9979 × 108 � 3 × 108 Elementary electron charge e 1.602 × 10−19 Electron rest mass me 9.11 × 10−31 Electron charge to mass ratio me 1.76 × 1011 e Proton rest mass mp 1.67 × 10−27 Boltzmann constant k 1.38 × 10−23 Gravitation constant G 6.67 × 10−11 Acceleration of gravity g 9.807 −9 Permittivity of free space π0 8.854 × 10−12 � 10 36ρ −7 Permeability of free space µ0 4φ × 10 Planck’s constant h 6.6256 × 10−34 q µ0 Impedance of free space ρ0 = 376.73 � 120φ π
Integral Theorems Line Integral of a Gradient Z b �f · dl = f (b) − f (a) a
Divergence Theorem: Z I � · AdV = A · dS V
S
Corollaries Z I �f dV = f dS S ZV I � × AdV = − A × dS V
0
Avogadro’s number
S
Stokes’ Theorem: I Z A · dl = (� × A) · dS L
S
Corollary I Z f dl = − �f × dS L
S
2
N
6.023 × 1023
units m/sec coul kg coul/kg kg joule/� K nt-m2 /(kg)2 m/(sec)2 farad/m henry/m joule-sec ohms atoms/mole