Elementi di Fotometria

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& & & OP = xx o + yy 0 + zz 0

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­ x = r sin(θ ) cos(φ ) ° ® y = r sin(θ ) sin(φ ) ° z = r cos(θ ) ¯ r

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& & & & & & OP = xxo + yy 0 + zz 0 = r sin(θ ) cos(φ ) xo + r sin(θ ) sin(φ ) y 0 + r cos(θ ) z 0

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∂OP FKH UDSSUHVHQWD LO YHWWRUH WDQJHQWH DOOD θ − curva RVVLD DOOD ∂θ FXUYD FDUDWWHUL]]DWD GD θ r YDULDELOH H GD r = cos t φ = cos t ∂ θ OP =

•

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•

6L KD ∂ r OP

& & & = sin(θ ) cos(φ ) x0 + sin(θ ) sin(φ ) y 0 + cos(θ ) z 0

& & & = r cos(θ ) cos(φ ) x0 + r cos(θ ) sin(φ ) y 0 − r sin(θ ) z 0 & & & ∂ φ OP = − r sin(θ ) sin(φ ) x 0 + r sin(θ ) cos(φ ) y 0 − 0 z 0

∂ θ OP

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§sin(θ )cos(φ ) r cos(θ )cos(φ ) −r sin(θ )sin(φ )· & & & ¨ ¸ ∂r OP, ∂θ OP, ∂φ OP = (x o , y 0, z 0 ) sin(θ )sin(φ ) r cos(θ )sin(φ ) r sin(θ )cos(φ ) ¸ ¨ ¸ ¨ cos( θ ) −r sin( θ ) 0 ¹ ©

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/D PDWULFH SUHVHQWH QHOOD UHOD]LRQH q GHWWD PDWULFH MDFRELDQD HG q LQGLFDWD FRQ LO VLPEROR J

§ sin(θ ) cos(φ ) r cos(θ ) cos(φ ) − r sin(θ ) sin(φ ) · ¨ ¸ J = ¨ sin(θ ) sin(φ ) r cos(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸ ¨ cos(θ ) ¸ − r sin(θ ) 0 © ¹

9DOXWLDPR RUD LO GHWHUPLQDQWH GL J

sin(θ ) cos(φ ) r cos(θ ) cos(φ ) − r sin(θ ) sin(φ )

J = sin(θ ) sin(φ ) cos(θ ) − r sin(θ ) sin(φ )

r cos(θ ) sin(φ )

r sin(θ ) cos(φ ) =

− r sin(θ )

0

sin(θ ) sin(φ ) r cos(θ ) sin(φ ) cos(θ )

− r sin(θ )

− r sin(θ ) cos(φ )

sin(θ ) cos(φ ) r cos(θ ) cos(φ ) cos(θ )

− r sin(θ )

= − r sin(θ ) sin(φ )[−r sin(θ ) 2 sin(φ ) − r cos(θ ) 2 sin(φ )] − − r sin(θ ) cos(φ )[− r sin(θ ) 2 cos(φ ) − r cos(θ ) 2 cos(φ )]

J = r 2 sin(θ ) sin(φ ) 2 + r 2 sin(θ ) cos(φ ) 2 = r 2 sin(θ ) J = r 2 sin(θ )

=

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∂ r OP • ∂ r OP = & & & & = [sin(θ ) cos(φ ) x0 + sin(θ ) sin(φ ) y 0 + cos(θ ) z 0 ] • [sin(θ ) cos(φ ) x0 + & & sin(θ ) sin(φ ) y 0 + cos(θ ) z 0 ] = = sin(θ ) 2 cos(φ ) 2 + sin(θ ) 2 sin(φ ) 2 + cos(θ ) 2 = sin(θ ) 2 [cos(φ ) 2 + sin(φ ) 2 ] + cos(θ ) 2 = = sin(θ ) 2 + cos(θ ) 2 = 1 3HUWDQWR VL SXz FRQFOXGHUH

∂ r OP • ∂ r OP = 1 ∂ θ OP • ∂ θ OP = & & & & = [r cos(θ ) cos(φ ) x0 + r cos(θ ) sin(φ ) y 0 − r sin(θ ) z 0 ] • [r cos(θ ) cos(φ ) x0 & & + r cos(θ ) sin(φ ) y 0 − r sin(θ ) z 0 ] = r 2 cos(θ ) 2 cos(φ ) 2 + r 2 cos(θ ) 2 sin(φ ) 2 + r sin(θ ) 2 = ‰

3HUWDQWR VL SXz FRQFOXGHUH

∂ θ OP • ∂ θ OP = r 2 ‰

∂ φ OP • ∂ φ OP = & & & & = [−r sin(θ ) sin(φ ) x0 + r sin(θ ) cos(φ ) y 0 ] • [− r sin(θ ) sin(φ ) x0 + r sin(θ ) cos(φ ) y 0 ] = r 2 sin(θ ) 2 sin(φ ) 2 + r 2 sin(θ ) 2 cos(φ ) 2 = r 2 sin(θ ) 2 [sin(φ ) 2 + cos(φ ) 2 ] = r 2 sin(θ ) 2 3HUWDQWR VL SXz FRQFOXGHUH

∂ φ OP • ∂ φ OP = r 2 sin(θ ) 2 ‰

∂ r OP • ∂ θ OP =

(OHPHQWL GL )RWRPHWULD &DSLWROR

& & & = [sin(θ ) cos(φ ) x0 + sin(θ ) sin(φ ) y 0 + cos(θ ) z 0 ] •

& & & • [r cos(θ ) cos(φ ) x0 + r cos(θ ) sin(φ ) y 0 − r sin(θ ) z 0 ] =

= [ r cos(θ ) sin(θ ) cos(φ ) 2 + r cos(θ ) sin(θ ) sin(φ ) 2 − r sin(θ ) cos(θ ) = = r cos(θ ) sin(θ )[cos(φ ) 2 + sin(φ ) 2 ] − r sin(θ ) cos(θ ) = r cos(θ ) sin(θ ) − r sin(θ ) cos(θ ) = 0

3HUWDQWR VL SXz FRQFOXGHUH

∂ r OP • ∂ θ OP = 0 ‰

∂ r OP • ∂ φ OP = & & & = [sin(θ ) cos(φ ) x0 + sin(θ ) sin(φ ) y 0 + cos(θ ) z 0 ] •

& & • [− r sin(θ ) sin(φ ) x0 + r sin(θ ) cos(φ ) y 0 ] =

= [−r sin(θ ) 2 sin(φ ) cos(φ ) + r sin(θ ) 2 cos(φ ) sin(θ ) sin(φ ) = 0

3HUWDQWR VL SXz FRQFOXGHUH

∂ r OP • ∂ φ OP = 0 ‰

∂ θ OP • ∂ φ OP = & & & = [r cos(θ ) cos(φ ) x0 + r cos(θ ) sin(φ ) y 0 − r sin(θ ) z 0 ] •

& & • [− r sin(θ ) sin(φ ) x0 + r sin(θ ) cos(φ ) y 0 ] =

& = [−r 2 sin(θ ) sin(φ ) cos(θ ) cos(φ ) x0 + r 2 cos(θ ) sin(φ )r sin(θ ) cos(φ )] = 0

3HUWDQWR VL SXz FRQFOXGHUH

∂ θ OP • ∂ φ OP = 0 'DOOH HODERUD]LRQL SUHFHGHQWL VL GHGXFH FKH LO WHQVRUH PHWULFR VL ULFRUGD FKH LO WHQVRUH PHWULFR LQ XQR VSD]LR YHWWRULDOH D WUH GLPHQVLRQL q FRVWLWXLWR GD QRYH FRPSRQHQWL g ij GDWH GDO SURGRWWR VFDODUH GHO YHWWRUH GL EDVH i − esimo SHU LO YHWWRUH GL EDVH j − esimo H SXz HVVHUH UDSSUHVHQWDWR LQ IRUPD PDWULFLDOH WUDPLWH XQD PDWULFH LQGLFDWD FRQ LO VLPEROR G q GDWR GD

§ ∂ r OP • ∂ r OP · §1 0 0 0 ¨ ¸ ¨ ¨ ¸ = ¨0 r 2 G = ∂ θ OP • ∂ θ OP 0 0 ¨ ¸ ¨ ∂ φ OP • ∂ φ OP ¸ ¨© 0 0 0 0 © ¹

0

· ¸ 0 ¸ 2 2¸ r sin(θ ) ¹

Ë EHQH RVVHUYDUH FKH L YHWWRUL GHOOD EDVH QDWXUDOH QRQ VRQR D PRGXOR XQLWDULR RVVLD QRQ VRQR GHL YHUVRUL 6H YRJOLDPR RWWHQHUH L YHUVRUL VHFRQGR OR VFKHPD ULSRUWDWR LQ )LJXUD q VXIILFLHQWH GLYLGHUH FLDVFXQ YHWWRUH GL EDVH SHU LO SURSULR PRGXOR RWWHQHQGR TXLQGL •

& r0 = ∂ r OP

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&

1 r

•

θ 0 = ∂ θ OP

•

φ0 =

&

1 ∂ φ OP r sin(θ )

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GXH SRVVLELOLWj H ULFRUGDQGR OH UHOD]LRQL DEELDPR

& & & & V = v x x0 + v y y 0 + v z z 0 = v r ∂ r OP + vθ ∂ θ OP + vφ ∂ φ OP

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& & & & V = wr r0 + wθ θ 0 + wφ φ 0

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& & & & & & & V • ∂ r OP = v r = (v x x0 + v y y 0 + v z z 0 ) • [sin(θ ) cos(φ ) x0 + sin(θ ) sin(φ ) y 0 + cos(θ ) z 0 ] = = v x sin(θ ) cos(φ ) + v y sin(θ ) sin(φ ) + v z cos(θ )

v r = v x sin(θ ) cos(φ ) + v y sin(θ ) sin(φ ) + v z cos(θ ) & & & & & & & • V • ∂ θ OP = vθ = (v x x 0 + v y y 0 + v z z 0 ) • [r cos(θ ) cos(φ ) x 0 + r cos(θ ) sin(φ ) y 0 − r sin(θ ) z 0 ] =

•

= v x r cos(θ ) cos(φ ) + v y r cos(θ ) sin(φ ) − v z r sin(θ ) vθ = v x r cos(θ ) cos(φ ) + v y r cos(θ ) sin(φ ) − v z r sin(θ ) & & & & & & & V • ∂ φ OP = vθ = (v x x0 + v y y 0 + v z z 0 ) • [−r sin(θ ) sin(φ ) x0 + r sin(θ ) cos(φ ) y 0 − 0 z 0 ] = = −v x r sin(θ ) sin(φ ) + v y r sin(θ ) cos(φ )

(OHPHQWL GL )RWRPHWULD &DSLWROR

vφ = −v x r sin(θ ) sin(φ ) + v y r sin(θ ) cos(φ ) 3HUWDQWR VL SRUUH

& V = v r ∂ r OP + vθ ∂ θ OP + vφ ∂ φ OP =

= [v x sin(θ ) cos(φ ) + v y sin(θ ) sin(φ ) + v z cos(θ )]∂ r OP + [v x r cos(θ ) cos(φ ) + v y r cos(θ ) sin(φ ) − v z r sin(θ )]∂ θ OP

+ [−v x r sin(θ ) sin(φ ) + v y r sin(θ ) cos(φ )]∂ φ OP /D SHUPHWWH GL GHWHUPLQDUH O¡HVSUHVVLRQH GL XQ YHWWRUH LQ XQ ULIHULPHQWR VIHULFR LQ IXQ]LRQH GHOOH FRPSRQHQWL GHOOR VWHVVR LQ XQ ULIHULPHQWR FDUWHVLDQR 6H DSSOLFKLDPR OD DL WUH YHWWRUL GL EDVH FDUWHVLDQL VL KD & • SHU x o v x = 1, v y = v z = 0 GD FXL VHJXH

& xo = sin(θ ) cos(φ )∂ r OP + r cos(θ ) cos(φ )∂ θ OP − r sin(θ ) sin(φ )∂ φ OP

•

SHU

& y o v x = v z = 0, v y = 1 GD FXL VHJXH

•

&

& y o = sin(θ ) sin(φ )∂ r OP + r cos(θ ) sin(φ )∂ θ OP + r sin(θ ) cos(φ )∂ φ OP

SHU z o v x

= v y = 0, v z = 1 GD FXL VHJXH

& z o = cos(θ )∂ r OP − r sin(θ )∂ θ OP ,Q IRUPD PDWULFLDOH OH SUHFHGHQWL UHOD]LRQL IRUQLVFRQR OD VHJXHQWH HTXD]LRQH

§ sin(θ )cos(φ ) sin(θ )sin(φ ) cos(θ ) ¡ & & & ¨ ¸ (x o , y 0 , z 0 ) = ∂r OP, ∂θ OP, âˆ‚Ď† OP r cos(θ )cos(φ ) r cos(θ )sin(φ ) −r sin(θ ) ¨ ¸ ¨ ¸ θ )sin( φ ) r sin( θ )cos( φ ) 0 −r sin( Š š

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J

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sin(θ ) sin(φ ) cos(θ ) · § sin(θ ) cos(φ ) ¨ ¸ = ¨ r cos(θ ) cos(φ ) r cos(θ ) sin(φ ) − r sin(θ ) ¸ ¨ − r sin(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸ 0 © ¹

,QROWUH OD SXz HVVHUH HVSUHVVD LQ IRUPD PDWULFLDOH FRPH VHJXH

§ v r · § sin(θ ) cos(φ ) sin(θ ) sin(φ ) cos(θ ) ·§ v x · ¨ ¸ ¨ ¸¨ ¸ ¨ vθ ¸ = ¨ r cos(θ ) cos(φ ) r cos(θ ) sin(φ ) − r sin(θ ) ¸¨ v y ¸ ¨ v ¸ ¨ − r sin(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸¨ v ¸ 0 ¹© z ¹ © φ¹ © 5LFRUGDQGR OD VWUXWWXUD GL J

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§ vr · § vx · ¨ ¸ ¨ ¸ −1 ¨ vθ ¸ = J ¨ v y ¸ ¨v ¸ ¨v ¸ © z¹ © φ¹

§ vx · ¨ ¸ ¨vy ¸ = ¨v ¸ © z¹ RVVLD LQ WHUPLQL HVSOLFLWL

§ vr · ¨ ¸ J ¨ vθ ¸ ¨v ¸ © φ¹

§ v x · § sin(θ ) cos(φ ) r cos(θ ) cos(φ ) − r sin(θ ) sin(φ ) ·§ v r · ¨ ¸ ¨ ¸¨ ¸ ¨ v y ¸ = ¨ sin(θ ) sin(φ ) r cos(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸¨ vθ ¸ ¨ v ¸ ¨ cos(θ ) ¸¨ v ¸ − r sin(θ ) 0 © z¹ © ¹© φ ¹ /D LQYHUWH GXQTXH OD H TXLQGL OD H SHUPHWWH LQ VRVWDQ]D GL HVSULPHUH OH FRPSRQHQWL FDUWHVLDQH GL XQ YHWWRUH LQ IXQ]LRQH GL TXHOOH VIHULFKH /H H OH VL FKLDPDQR OHJJL GL WUDVIRUPD]LRQH GHOOH FRPSRQHQWL GL XQ YHWWRUH SDVVDQGR GD XQ VLVWHPD GL ULIHULPHQWR DG XQ DOWUR H VLFFRPH FRPH YHWWRUH VL SXz HVVHUH SUHVR LO YHWWRUH SRVL]LRQH OH FXL FRPSRQHQWL LQ FRRUGLQDWH FDUWHVLDQH VRQR OH FRRUGLQDWH GL SXQWR HG LQ FRRUGLQDWH VIHULFKH q LO UDJJLR YHWWRUH RVVLD

& & & OP = xxo + yy 0 + zz 0 = r∂ r OP

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& & V • V = (v r ∂ r OP + vθ ∂ θ OP + vφ ∂ φ OP ) • (v r ∂ r OP + vθ ∂ θ OP + vφ ∂ φ OP ) & & 2 2 2 2 2 2 V • V = (v r ) + r (vθ ) ∂ + r sin(θ ) (vφ )

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g 22 = r 2

•

g 33 = r 2 sin(θ ) 2

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=

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(OHPHQWL GL )RWRPHWULD &DSLWROR

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10 −9 F −7 H μ 0 = 4π 10 36π m m

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& & j = 0 H ρ = 0

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(

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•

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&

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, x 2 , x 3 )

4

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∂E i ∂x j

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& E = E i [Φ ( x 1 , x 2 , x 3 , x 4 )]ei

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j

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i

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−∞

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[m ] DOORUD RFFRUUH XWLOL]]DUH O·HTXLYDOHQ]D HVSUHVVD GDOOD 2

I=

10,76

π

I a ≅ 3,4262 I a

9HGLDPR XQ HVHPSLR QXPHULFR

•

α = (1,3)−1

•

S1 = 3,35 m 2 S a ≅ 36,05 ft 2

•

L1 = 23,37

π cd La = 23,37 ≅ 6,82 fL 2 10,76 m

'D TXHVWL GDWL GL LQSXW VHJXH

•

I = I1 =

•

Ia =

23,37 ⋅ 3,35 cd ≅ 60,20 cd 1,3

6,82 ⋅ 36,05 fL ⋅ ft 2 ≅ 189,13 fL ⋅ ft 2 1,3

( VL YHULILFD FKH

I a 189,13 ≅ ≅ 3,14 60,20 I BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

(OHPHQWL GL )RWRPHWULD 6LPEROL XWLOL]]DWL

6,0%2/, 87/,==$7,

& V

9(7725(

* & V • W

352'2772 6&$/$5( 75$ , 9(7725, V ( W

* & V × W

352'2772 9(7725,$/( 75$ , 9(7725, V ( W

' ∇ •V

',9(5*(1=$ '(/ 9(7725( V

' ∇ ×V

52725( '(/ 9(7725( V

∂x f

'(5,9$7$ 3$5=,$/( '(//$ )81=,21(

&

&

&

&

&

&

9$5,$%,/( x 266,$

f 5,63(772 $//$

∂f ∂x

∂2 f ∂x∂y

∂ xy f

'(5,9$7$ 0,67$ 6(&21'$ '(//$ )81=,21(

i = 1..n

,1',&( i &+( 38Ó $6680(5( 7877, , 9$/25, ,17(5, '$ 1 $ n

λ

/81*+(==$ '·21'$

T

3(5,2'2 ', 81·21'$ 6,1862,'$/(

ν

)5(48(1=$ ', 81·21'$ 6,1862,'$/(

> sr @

f 266,$

67(5$',$17,

Re(α )

3$57( 5($/( ', 81$ 48$17,7­ &203/(66$ α

³³ ...dS

,17(*5$/( (67(62 $' 81$ 683(5),&,( S &+,86$

s

(OHPHQWL GL )RWRPHWULD 6LPEROL XWLOL]]DWL

...dτ ³³³ τ

,17(*5$/( (67(62 $' 81 92/80( 683(5),&,( &+,86$ S

T

7(1625( '(/ 6(&21'2 25',1(

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τ

/,0,7$72 '$//$

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•

3HU O·RWWLFD LO WHVWR GL *ULJRUM 6 /DQGVEHUJ µ2WWLFDµ LQ GXH YROXPL HGLWR LQ ,WDOLD GDOOH HGL]LRQL 0LU

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7DEHOOD 9DORUL GL LOOXPLQDPHQWR

7DEHOOD &RQYHUVLRQL PHWULFKH OLQHDUL

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,1',&( '(//( ),*85(

)LJXUD &RRUGLQDWH 6IHULFKH )LJXUD %DVH QDWXUDOH H YHUVRUL )LJXUD (OHPHQWR GL VXSHUILFLH )LJXUD ² $QJROR VROLGR LQILQLWHVLPR )LJXUD ² VLPLOLWXGLQH WUD FRQL )LJXUD ² $QJROR VROLGR ULIHULWR D VXSHUILFLH REOLTXD )LJXUD &DPSR JHQHUDWR GDOOD IRU]D SHVR )LJXUD &DPSR HOHWWULFR )LJXUD /LQHH GL IOXVVR )LJXUD /LQHH G IRU]D QHO FDVR GL XQ GLSROR HOHWWULFR )LJXUD 7XER GL IOXVVR )LJXUD )OXLGR DWWUDYHUVR XQ FLOLQGUR LQILQLWHVLPR )LJXUD 9HWWRUH GL 3R\QWLQJ QHO FDVR GL RQGH SLDQH )LJXUD &XUYH GL YLVLELOLWj )LJXUD )OXVVR /XPLQRVR )LJXUD 'LPLQX]LRQH GHO IOXVVR GDOOD GLVWDQ]D GDOOD VRUJHQWH )LJXUD /XPLQDQ]D GL XQD VRUJHQWH HVWHVD

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(OHPHQWL GL )RWRPHWULD ,QGLFH GHOOH HTXD]LRQL

,1',&( '(//( (48$=,21,

­ x = r sin(θ ) cos(φ ) ° ÂŽ y = r sin(θ ) sin(φ ) ° z = r cos(θ ) ÂŻ ∂ r OP

& & & = sin(θ ) cos(φ ) x0 + sin(θ ) sin(φ ) y 0 + cos(θ ) z 0

§ sin(θ ) cos(φ ) r cos(θ ) cos(φ ) − r sin(θ ) sin(φ ) ¡ ¸ & & & ¨ ∂ r OP, ∂ θ OP, ∂ φ OP = ( xo , y 0 , z 0 ) ¨ sin(θ ) sin(φ ) r cos(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸ ¸ ¨ cos(θ ) − r sin(θ ) 0 š Š

(

)

§ ∂ r OP • ∂ r OP ¡ §1 0 0 0 ¨ ¸ ¨ ¨ ¸ = ¨0 r 2 0 0 ∂ θ OP • ∂ θ OP G = ¨ ¸ ¨ 0 0 ∂ φ OP • ∂ φ OP ¸ ¨Š 0 0 Š š

(w

0

¡ ¸ 0 ¸ 2 2¸ r sin(θ ) š

= v r , wθ = rvθ , wφ = r sin(θ )vφ )

r

= [v x sin(θ ) cos(φ ) + v y sin(θ ) sin(φ ) + v z cos(θ )]∂ r OP + [v x r cos(θ ) cos(φ ) + v y r cos(θ ) sin(φ ) − v z r sin(θ )]∂ θ OP

+ [−v x r sin(θ ) sin(φ ) + v y r sin(θ ) cos(φ )]∂ φ OP

sin(θ ) sin(φ ) cos(θ ) ¡ § sin(θ ) cos(φ ) ¨ ¸ & & & ( x o , y 0 , z 0 ) = ∂ r OP, ∂ θ OP, ∂ φ OP ¨ r cos(θ ) cos(φ ) r cos(θ ) sin(φ ) − r sin(θ ) ¸ ¨ − r sin(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸ 0 Š š

(

)

§ v r ¡ § sin(θ ) cos(φ ) sin(θ ) sin(φ ) cos(θ ) ¡§ v x ¡ ¨ ¸ ¨ ¸¨ ¸ ¨ vθ ¸ = ¨ r cos(θ ) cos(φ ) r cos(θ ) sin(φ ) − r sin(θ ) ¸¨ v y ¸ ¨ v ¸ ¨ − r sin(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸¨ v ¸ 0 šŠ z š Š φš Š § v x ¡ § sin(θ ) cos(φ ) r cos(θ ) cos(φ ) − r sin(θ ) sin(φ ) ¡§ v r ¡ ¨ ¸ ¨ ¸¨ ¸ ¨ v y ¸ = ¨ sin(θ ) sin(φ ) r cos(θ ) sin(φ ) r sin(θ ) cos(φ ) ¸¨ vθ ¸ ¨ v ¸ ¨ cos(θ ) ¸¨ v ¸ − r sin(θ ) 0 Š zš Š šŠ φ š & & 2 2 2 2 2 2 V • V = (v r ) + r (vθ ) ∂ + r sin(θ ) (vφ ) ds

2

= (dr ) + r 2 (dθ ) ∂ + r 2 sin(θ ) 2 (dφ ) 2

2

2

dS

= r 2 sin(θ ) dθ dφ

dV

= r 2 sin(θ ) dr dθ dφ

(OHPHQWL GL )RWRPHWULD ,QGLFH GHOOH HTXD]LRQL

&

∇ • V

&

∇ • V

=

1 1 ∂ r r 2 vr + ∂ θ (sin(θ )vθ ) + ∂ φ (vφ ) 2 sin(θ ) r

=

1 1 1 ∂ r r 2 wr + ∂ θ (sin(θ ) wθ ) + ∂ φ (wφ ) 2 r sin(θ ) r sin(θ ) r

(

)

(

)

dΩ

=

dS r2

dS

=

dS1 dS 2 = 2 (r1 ) (r2 ) 2

dΩ

=

dS = sin(θ ) dθ dφ r2

Ω

= ³³ S

dΩ

=

φ + Δφ θ + Δθ dS =³ dφ ³ sin(θ ) dθ = Δφ [cos(θ ) − cos(θ + Δθ )] 2 φ θ r

dS ′ dS = cos(ψ ) 2 2 r r

­1 °σ ° °1 ® °σ °1 ° ¯σ

dx 1 = v x ( x, y , z ) dt ρ dy 1 = v y ( x, y, z ) dt ρ dz 1 = v z ( x, y , z ) dt ρ & & Ψ = ³³ V • n dS S

&

dΨ cos(ψ )dS & dΨ = (∂ x v x + ∂ y v y + ∂ z v z ) dzdxdy = ∇ • Vdτ & & & Ψ = ³³ V • n dS = ³³³ ∇ • Vdτ V

=

τ

S

Ψ

=

Ω2

&

& &

³ V (Ω) dΩ = ³³V • n sin(θ )dθdφ

Ω1

&

S

V ( Ω ) =

dΨ dΩ n

QTOT

= ¦ ni q i i =1

(OHPHQWL GL )RWRPHWULD ,QGLFH GHOOH HTXD]LRQL

Ψ

& & = ³³ V • n dS = ³³ (v x n x + v y n y + v z n z ) dxdy S

S

& & 2 2 2 Ψ = ³³ V • n dS = ³³ v r n r + r [vθ nθ + sin(θ ) vφ nφ ] r dθ dφ

{

S

S

S

S

}

& & 2 Ψ = ³³ V • n dS = ³³ ( wr n r + wθ nθ + wφ nφ ) r dθ dφ Ω2 & & & & & = ³³ V • n dS = ³³ V cos(ψ )dS = ³³ V dS ′ = ³ V r 2 dΩ

Ψ

S

S

Ω1

S

& & ∂ & & E • t ds = − B • ndS ³l ∂t ³³S

³ H • t ds = ³³

& &

S

l

& & & & ∂ j • ndS + ³³ D • ndS ∂t S

& & ∂B ∇ × E = − ∂t & & & ∂D ∇ × H = j + ∂t & ∂2 & ∇ E = εμ 2 E ∂t 2

­ 2 ρ ∂2 εμ ∇ Π = Π=− ° 2 ε0 ∂t ° ® & 2 °∇ 2 A& = εμ ∂ A& = − j °¯ ε0 ∂t 2 & & & & E = E x [Φ ( x, y , z , t )] x 0 + E y [Φ ( x, y , z , t )] y 0 + E z [Φ ( x, y , z , t )] z 0 ∂

2

j

E i = ∂ 2 Φ E i ⋅ (∂ j Φ ) = ∂ 2 Φ E i ⋅ (k j ) 2

2

ª 3 (k h )2 º» ¦ « 3 & ª 2º 2 i 2 » ∂ 2 4 E i ei ∇ E = «¦ (k h ) » ∂ Φ E ei = « h =1 2 « ω » ¬ h =1 ¼ «¬ »¼ ª 3 2 º « ¦ (k h ) » » = v 2 = εμ « h =1 2 « ω » «¬ »¼ & & ' & ' & ' & & & E = E x [ K • r − ωt )]x 0 + E y [ K • r − ωt ] y 0 + E z [ K • r − ωt ] z 0

(OHPHQWL GL )RWRPHWULD ,QGLFH GHOOH HTXD]LRQL

&

& & & = E 0 e −i ( K •r −ωt ) & & − (α& • r& ) −i ( β& •r& −ωt ) E = E 0 e e & & & & & − (α& •r& ) & & [cos( β • r − ω t ) = Re[ E ] = E 0 Re[e − j ( K •r −ωt ) ] E 0 e & & & & & & & ( β • r0 − ω t ) = ( β • r − ω t ) β • ( r − r0 ) & & & & & & & (α • r0 ) = (α • r ) α • (r − r0 )

E

=

v

ω β cos(ψ )

& E ( x, y , z , t ) = &

1

+∞+∞ +∞

+∞

(2π )4 −³∞−³∞−³∞³−∞

E ( k x , k y , k z , ω )e

& i ( xk + yk + zk +ωt ) E (k x , k y , k z , ω )e x y z dxdydzdω

i ( xk x + yk y + zk z +ωt )

& & & = E (k x , k y , k z , ω )e i ( K •r +ωt )

&

' & = E × H & & & & & & ∂D & ∂B = 0 ∇ • P − j • E − E • −H• ∂t ∂t P

&& & & ∂ §1 & 2· §1 & 2· P n dS = j • E d τ + ε E d τ + ¨ ¸ ¨ μ H ¸ dτ ³³S ³³³ ³³³ ³³³ ¹ ¹ τ τ ©2 τ ∂t © 2 & & I λ = v (λ ) P (λ )

&

780 nm & & 1 jλt ( ) ( ) v λ P λ e d λ = v(λ ) P(λ )e jλt dλ ³−∞ ³ 2π 380 nm & & Φ = ³³ I • n dS

I

=

1 2π

∞

S

Φ

Ω2 & & & & & = ³³ I • n dS = ³³ I • n sin(θ )dθdφ = ³ I (Ω) dΩ S

Ω1

S

& dΦ I (Ω ) = dΩ & dΦ I (Ω) = dΩ

φ2 θ2 & & Φ = ³³ I • n dS = ³ dφ ³ I (θ , φ ) sin(θ ) dθ S

φ1

θ1

& & I L = L ′ & I

(OHPHQWL GL )RWRPHWULD ,QGLFH GHOOH HTXD]LRQL

L( x, y , z ,Ďˆ )

=

dÎŚ I = cos(Ďˆ ) dS dΊ cos(Ďˆ ) dS

Ď€ 2 dÎŚ = 2Ď€ Âł L cos(Ďˆ ) sin(Ďˆ ) dĎˆ dS 0

Ď€ 2

= 2Ď€L Âł cos(Ďˆ ) sin(Ďˆ ) dĎˆ = Ď€L

Lu

0

E

=

1 sb

dÎŚ dΊ I â‹… = I â‹… cos(Ďˆ ) = cos(Ďˆ ) dS dS r 2 =

1 phot

cd cd cd = − 4 2 = 10 4 2 = 10 4 nt 2 cm 10 m m =

lm lm lm = − 4 2 = 10 4 2 = 10 4 lux 2 cm 10 m m

1 cd cd ≅ 0,318 2 2 π m m

1 asb

=

1 L

1 cd 10 4 cd = = 10 4 asb 2 2 π cm π m

=

1mmL

= 10 −3 L = 10mL = 10asb =

10 cd π m2

1

cd 1 cd 1 cd 10 4 cd cd ≅ = = ≅ 10,76 2 2 2 −4 2 2 930,27 cm 930,27 10 m 930,27 m ft m

1

cd 1 cd cd ≅ ≅ 0,1548 2 2 2 6,4609 cm inch cm

lm 10 4 lm lm ≅ 10,76 2 1 2 ≅ 2 930,27 m ft m 1

lm 1 lm lm ≅ ≅ 0,1548 2 2 2 6,4609 cm inch cm

1 fL =

1 cd π ft 2

1 fL

=

§ 10 4 cd ¡ 1 cd 10,76 cd −4 ¨¨ ¸ = 10,76 10 −4 L = 1,076mL ≅ = 10 10 , 75 2 ¸ Ď€ ft 2 Ď€ m2 Ď€ m Š š

1 fL

=

1 cd 10,76 cd cd ≅ ≅ 3,4262 2 2 2 π ft π m m

(OHPHQWL GL )RWRPHWULD ,QGLFH GHOOH HTXD]LRQL

I = αLS

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