Electromagnetic Formula

MIDTERM FORMULA SHEET VECTOR IDENTITIES G G G G G G G G G A ⋅ ( B × C ) = C ⋅ ( A × B ) = B ⋅ (C × A) G G G G G G G G G A × ( B × C ) = B( A ⋅ C ) − C ( A ⋅ B) ∇(Φ + Ψ ) = ∇Φ + ∇Ψ G G G G ∇ ⋅ ( A + B) = ∇ ⋅ A + ∇ ⋅ B G G G G ∇ × ( A + B) = ∇ × A + ∇ × B ∇(ΦΨ ) = Φ∇Ψ + Ψ∇Φ  Φ  Ψ∇Φ − Φ∇Ψ ∇  = Ψ2 Ψ

(

∇Φ n = nΦ n−1∇Φ G G G ∇ ⋅ (ΦA) = A ⋅∇Φ + Φ∇ ⋅ A G G G G G G ∇ ⋅ ( A × B ) = B ⋅∇ × A − A ⋅∇ × B G G G ∇ × (ΦA) = ∇Φ × A + Φ∇ × A G G G G G G G G G G ∇ × ( A × B ) = A∇ ⋅ B − B∇ ⋅ A + ( B ⋅∇) A − ( A ⋅∇) B G G G G G G G G G G ∇( A ⋅ B ) = A × (∇ × B ) + B × (∇ × A) + ( A ⋅∇) B + ( B ⋅∇) A ∇ ⋅∇Φ = ∇ 2Φ G ∇ ⋅∇ × A = 0 ∇ × ∇Φ = 0 G G G ∇ ×∇ × A = ∇∇ ⋅ A − ∇ 2 A VECTOR INTEGRAL THEOREMS G G G ∫∫∫ (∇ ⋅ A)dv = w ∫∫ A ⋅ ds (Divergence theorem, Gauss identity) v

S[ v ]

G G ∫∫ (∇ × A) ⋅ ds = S

v∫

G G A ⋅ dl (Curl theorem 1, Stokes’ theorem)

C[ S ]

G G G G ˆ ∇ × A dv = ds × A ≡ n × A ( ) ( )ds (Curl theorem 2) ∫∫∫ w ∫∫ w ∫∫ v

S[ v ]

SOME INTEGRALS OFTEN MET IN EM PROBLEMS 1 ∫ x dx = ln | x | +C 1 x ∫ (a 2 ± x 2 )3 / 2 dx = ± a 2 a 2 ± x 2 + C x 1 ∫ (a 2 + x 2 )3 / 2 dx = − a 2 + x 2 + C

S[ v ]

)

x2 x 2 2 ∫ (a 2 + x 2 )3 / 2 dx = − a 2 + x 2 + ln x + a + x + C 1 1 x ∫ a 2 + x 2 dx = a arctan a + C 1  1 a−x  x  2a ln  a + x  = − a arctanh  a  ,| x |< a 1      ∫ x 2 − a 2 dx =  1  − arccoth  x  ,| x |> a    a a x 1 2 2 ∫ a 2 + x 2 dx = 2 ln a + x + C x 2 2 ∫ a 2 + x 2 dx = a + x + C 1 2 2 ∫ a 2 + x 2 dx = ln( x + a + x ) + C 1 1  a + a 2 + x2  +C ln  dx = − ∫ x a2 + x2  a  x   1 ∫ x sin(ax)dx = a 2 [sin(ax) − ax cos(ax)] 1 ∫ x cos(ax)dx = a 2 [cos(ax) + ax sin(ax)]

(

)

MIDTERM FORMULA SHEET  x ag − bf 1  = arctan ∫ (ax 2 + b) fx 2 + g b ag − bf  b fx 2 + g  dx

 , (ag > bf )  

π

∫ tan xdx = − ln | cos x | +C , x ≠ (2k + 1) 2 ∫ cot xdx = ln | sin x | +C , x ≠ 2kπ 1

 x

x = R sin θ cos φ y = R sin θ sin φ z = R cos θ

x π

∫ cos x dx = ln tan  2 + 4  + C SOME USEFUL DEFINITE INTGERALS 2π 0 , m ≠ n ∫ sin mx ⋅ sin nx dx = π , m = n ≠ 0 0 2π

0

∫ cos mx ⋅ cos nx dx =  π ,

0 2π

∫

, m≠n m=n≠0

sin mx ⋅ cos nx dx = 0

0

π

0

, m≠n m=n≠0

0

, m≠n m=n≠0

∫ sin mx ⋅ sin nx dx = π / 2, 0

π

∫ cos mx ⋅ cos nx dx = π / 2, 0

, 0  ∫ sin mx ⋅ cos nx dx =  2m ,  m 2 − n 2 0 π π (a − b cos x)  , ∫ (a 2 + b2 − 2ab cos x) dx =  a  0, 0 π

x = r cos φ y = r sin φ z=z

r = x2 + y 2  y φ = arctan    x z=z

Rectangular ↔ Spherical

∫ sin x dx = ln tan  2  + C 1

COORDINATE TRANSFORMATIONS Rectangular ↔ Cylindrical

m + n = even number m + n = odd number a>b>0 b>a>0

R = x2 + y 2 + z 2

(

θ = arccos z / x 2 + y 2 + z 2 φ = arctan( y / x)

)

Cylindrical ↔ Spherical

r = R sin θ φ =φ z = R cos θ

R = r2 + z2 φ =φ θ = arccos z / r 2 + z 2

(

)

VECTOR TRANSFORMATIONS Rectangular Components ↔ Cylindrical Components ax = ar cos φ − aφ sin φ ar = ax cos φ + a y sin φ a y = ar sin φ + aφ cos φ aφ = − ax sin φ + a y cos φ a z = az az = az

Note: φ is the position angle of the point at which the vector exists. Rectangular Components ↔ Spherical Components ax = aR sin θ cos φ + aθ cosθ cos φ − aφ sin φ a y = aR sin θ sin φ + aθ cos θ sin φ + aφ cos φ az = aR cos θ − aθ sin θ aR = ax sin θ cos φ + a y sin θ sin φ + az cos θ aθ = ax cos θ cos φ + a y cos θ sin φ − a z cos θ aφ = − ax sin φ + a y cos φ

Note: φ and θ are the position angles of the point at which the vector exists.

MIDTERM FORMULA SHEET Cylindrical Components↔ Spherical Components ar = aR sin θ + aθ cos θ aR = ar sin θ + az cos θ aφ = aφ aθ = ar cos θ − az sin θ aφ = aφ a z = aR cos θ − aθ sin θ

Note: θ is the position angle of the point at which the vector exists. DERIVATIVES OF ELEMENTARY FUNCTIONS (const.)′ = 0 1 (arctan x)′ = ( x)′ = 1 1 + x2 1 ( x k )′ = kx k −1 (arc cot x)′ = − 1 + x2 (e x )′ = e x (sinh x)′ = cosh x (a x )′ = a x ln a (cosh x)′ = sinh x 1 1 (ln x)′ = (tanh x)′ = = 1 − tanh 2 x x cosh 2 x 1 (log a x)′ = , a ≠ 1, x > 0 1 (coth x)′ = − = 1 − coth 2 x x ln a 2 sinh x (sin x)′ = cos x 1 (cos x)′ = − sin x (arcsinh x)′ = 1 + x2 1 (tan x)′ = , x ≠ (2k + 1)π 1 cos 2 x (arccosh x)′ = ± , x >1 x2 − 1 1 (cot x)′ = − 2 , x ≠ kπ 1 sin x (arctanh x)′ = ,| x |< 1 1 1 − x2 (arcsin x)′ = ,| x |< 1 1 1 − x2 (arccoth x)′ = ,| x |> 1 2 1 − x 1 (arccos x)′ = − ,| x |< 1 1 − x2 DIFFERENTIAL OPERATORS Rectangular Coordinates ∂Φ ∂Φ ∂Φ ∇Φ = xˆ + yˆ + zˆ ∂x ∂y ∂z

G ∂F ∂Fy ∂Fz ∇⋅F = x + + ∂x ∂y ∂z G  ∂F ∂Fy   ∂Fx ∂Fz   ∂Fy ∂Fx  ∇ × F = xˆ  z − − −  + yˆ    + zˆ  ∂z  ∂x   ∂x ∂y   ∂z  ∂y ∂ 2Φ ∂ 2Φ ∂ 2Φ ∇ ⋅ (∇Φ ) ≡ ∇ 2 Φ ≡ ∆Φ = 2 + 2 + 2 ∂x ∂y ∂z G 2 2 2 2 ∇ F = xˆ∇ Fx + yˆ∇ Fy + zˆ∇ Fz Cylindrical Coordinates ∂Φ 1 ∂Φ ∂Φ ϕˆ + ∇Φ = rˆ + zˆ r ∂ϕ ∂r ∂z G 1 ∂ 1 ∂Fϕ ∂Fz (rFr ) + ∇⋅F = + r ∂r r ∂ϕ ∂z G  1 ∂Fz ∂Fφ  ˆ  ∂Fr ∂Fz   1 ∂ (rFφ ) 1 ∂Fr  ∇ × F = rˆ  − − −  +φ    + zˆ  ∂z  ∂r   r ∂r r ∂φ   ∂z  r ∂φ 1 ∂  ∂Φ  1 ∂ 2 Φ ∂ 2 Φ ∇ ⋅ (∇Φ ) ≡ ∇ 2 Φ ≡ ∆Φ = + r + r ∂r  ∂r  r 2 ∂φ 2 ∂z 2 G  ∂ 2 A 1 ∂Ar Ar 1 ∂ 2 Ar 2 ∂Aφ ∂ 2 Ar  ∇ 2 A = rˆ  2r + − + − + 2 + r ∂r r 2 r 2 ∂φ 2 r 2 ∂φ ∂z   ∂r  ∂ 2 Aφ 1 ∂Aφ Aφ 1 ∂ 2 Aφ 2 ∂Ar ∂ 2 Aφ  ˆ φ 2 + − 2 + 2 + 2 + + 2 2   ∂r r r φ ∂ ∂ r r r z φ ∂ ∂   2 2 2  ∂ A 1 ∂Az 1 ∂ Az ∂ Az  zˆ  2z + + 2 + 2  2 r r ∂ dr r φ ∂ ∂z   Spherical Coordinates ∂Φ ˆ 1 ∂Φ ˆ 1 ∂Φ θ+ ϕˆ R+ ∇Φ = R ∂θ R sin θ ∂ϕ ∂R G 1 ∂ 1 1 ∂Fϕ ∂ ( R 2 FR ) + ( Fθ sin θ ) + ∇⋅F = 2 R sin θ ∂θ R sin θ ∂ϕ R ∂R

MIDTERM FORMULA SHEET G ∇ × A = Rˆ

 ∂ ∂Aθ  ∂θ ( Aϕ sin θ ) − ∂ϕ   1  1 ∂AR ∂ θˆ  ( RAϕ )  + − R  sin θ ∂ϕ ∂R 

ϕˆ

1 R sin θ

G ˆ + θˆRdθ + ϕˆ R sin θ dϕ ; dl = RdR G ˆ 2 ds = RR sin θ dθ dϕ + θˆR sin θ dRdϕ + ϕˆ RdRdθ ;

 + 

dv = R 2 sin θ dRdθ dϕ

1 ∂ ∂A  ( RAθ ) − R   R  ∂R ∂θ 

1 ∂  2 ∂Φ  1 ∂  ∂Φ  1 R + 2  sin θ + 2 2 2 ∂R  R sin θ ∂θ  ∂θ  R sin θ R ∂R  2 2 G  ∂ AR 2 ∂AR 2 1 ∂ AR cot θ ∂AR ∇ 2 A = Rˆ  + − 2 AR + 2 + 2 + 2 R ∂R R R ∂θ 2 R ∂θ  ∂R ∂Aϕ ∂ 2 AR 2 ∂Aθ 2 cot θ 1 2 − − − A θ R 2 sin 2 θ ∂ϕ 2 R 2 ∂θ R2 R 2 sin θ ∂ϕ

∇ 2Φ =

∂ 2Φ ∂ϕ 2

 + 

 ∂ 2 Aθ 2 ∂Aθ Aθ 1 ∂ 2 Aθ cot θ ∂Aθ ˆ − + + 2 + θ 2 + R ∂R R 2 sin 2 θ R 2 ∂θ 2 R ∂θ  ∂R ∂ 2 Aθ 1 2 ∂AR 2 cot θ ∂Aϕ  + − + R 2 sin 2 θ ∂ϕ 2 R 2 ∂θ R 2 sin θ ∂ϕ  2  ∂ 2 Aϕ 2 ∂Aϕ Aϕ 1 ∂ Aϕ cot θ ∂Aϕ + − + + 2 + ϕˆ   ∂R 2 R ∂R R 2 sin 2 θ R 2 ∂θ 2 ∂ θ R 

∂ 2 Aϕ ∂AR 2 cot θ ∂Aθ  1 2 + +  R 2 sin 2 θ ∂ϕ 2 R 2 sin θ ∂ϕ R 2 sin θ ∂ϕ 

DIFFERENTIAL ELEMENTS Cartesian coordinates G G ˆ + ydy ˆ + zdz ˆ ˆ ˆ ; ds = xdydz ˆ dl = xdx + ydxdz + zdxdy ; dv = dxdydz Cylindrical coordinates G G ˆ + ϕˆ rdϕ + zdz ˆ ϕ dz + ϕˆ drdz + zrdrd ˆ ; ds = rrd ˆ dl = rdr ϕ ; dv = rdrdϕ dz Spherical coordinates

ELECTROMAGNETIC EQUATIONS Coaxial line 2πε µ b µ , F/m; L1 = 0 ln   + 0 , H/m C1 = ln(b / a) 2π  a  8π Twin-lead line 2  πε µ  h h C1 = F/m; L1 = ln +   − 1  H/m 2 π r h   r h     ln +   − 1 r  r  