Centrale Physique 2 PC 2017

Thème de l'épreuve Interférométrie atomique
Principaux outils utilisés thermodynamique, mécanique quantique, optique ondulatoire, propagation d'onde
Mots clefs tuyère, nombre de Mach, onde stationnaire, polarisabilité, diffraction, interféromètre de Mach-Zehnder, effet Sagnac
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Énoncé complet

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Rapport du jury

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Énoncé obtenu par reconnaissance optique des caractères


S*
9 ?2m`2b

*H+mHi`B+2b miQ`Bbû2b

kyRd

S?vbB[m2 k
AMi2`7û`QKûi`B2 iQKB[m2

G2b QM/2b /2 KiB`2 T2`K2ii2Mi /2 `ûHBb2` /2b 2tTû`B2M+2b /BMi2`7û`QKûi`B2 iQmi 
¨ 7Bi MHQ;m2b ¨ +2HH2b
`2M+QMi`û2b 2M QTiB[m2X PM bBMiû`2bb2 B+B ¨ mM BMi2`7û`QKi`2 /2 J+?@w2?M/2` ¨ 
QM/2b /2 KiB`2 U};m`2 RVX
lM i2H BMi2`7û`QKi`2  ûiû +QMbi`mBi T` /2b +?2`+?2m`b /m G#Q`iQB`2 *QHHBbBQMb- 
;`û;ib- 2i _û+iBpBiû
UG*_V ¨ hQmHQmb2X G bQm`+2 2bi mM D2i bmT2`bQMB[m2 +QHHBKiû /iQK2b /2 HBi?BmK 
d- H2b KB`QB`b 2i HK2b
bûT`i`B+2b bQMi `ûHBbûb ¨ HB/2 /QM/2b HmKBM2mb2b bm` H2b[m2HH2b H2b iQK2b bQMi 
/Bz`+iûb c 2M}M- bm` HmM2
/2b bQ`iB2b /2 HBMi2`7û`QKi`2- mM /ûi2+i2m` K2bm`2 HBMi2MbBiû /m 7Bb+2m iQKB[m2 
`ûbmHiMi /2 HBMi2`7û`2M+2
2Mi`2 H2b QM/2b /2 KiB`2 [mB b2 T`QT;2Mi /Mb H2b /2mt #`b /2 HBMi2`7û`QKi`2X
UJRV

UJkV

UJjV

aQm`+2

.ûi2+i2m`

6B;m`2 R a+?ûK /2 T`BM+BT2 /2 HBMi2`7û`QKi`2 /2 J+?@w2?M/2` ¨ QM/2b /2 KiB`2
UJRV- UJkV 2i UJjV bQMi /2b KB`QB`b THMb c  2i  bQMi H2b 72Mi2b /2 +QHHBKiBQM 
/m D2i bmT2`bQMB[m2 c 
2bi H 72Mi2 /2 /ûi2+iBQM c H /BbiM+2 2Mi`2 /2mt QM/2b biiBQMMB`2b 2bi    NN
G2b +QMbiMi2b T?vbB[m2b miBH2b m T`Q#HK2 BMbB [mmM 7Q`KmHB`2 bQMi /QMMûb 2M }M 
/ûMQM+ûX
G2b p2+i2m`b mMBiB`2b b2HQM - -  bQMi MQiûb  -  -  X

A *`+iû`BbiBQM /2 H bQm`+2 iQKB[m2
G T`Q/m+iBQM /mM2 pT2m` iQKB[m2 b2z2+im2 2M +?mzMi HûHûK2Mi +?BKB[m2 TQm` 
ii2BM/`2 mM2 T`2bbBQM
/2 pT2m` bim`Mi2 bm{bMi2X G2 HBi?BmK vMi mM2 T`2bbBQM /2 pT2m` bim`Mi2 7B#H2 mt 
i2KTû`im`2b
2tTû`BK2MiH2K2Mi ++2bbB#H2b- QM H2 pTQ`Bb2 /Mb mM ;x /`;QM bQmb ?mi2 T`2bbBQM- 
}M /Q#i2MB` mM2
bm`T`2bbBQM THmb BKTQ`iMi2 /Mb H2 7Qm`X G2 KûHM;2 ;x2mt BMbB Q#i2Mm +QMiB2Mi 
KDQ`BiB`2K2Mi /2 H`;QM
2i mM2 7B#H2 T`QTQ`iBQM /2 HBi?BmKX *2 ;x 2bi 2MbmBi2 ++ûHû`û ¨ i`p2`b mM2 
imv`2 /2 HQM;m2m` M2ii2K2Mi
bmTû`B2m`2 ¨ b b2+iBQM U};m`2 kV }M /2 7Q`K2` mM D2i iQKB[m2X PM bQm?Bi2 /Mb 
+2ii2 T`iB2 +`+iû`Bb2` H2
D2i iQKB[m2 BMbB +`ûûX
A(z )
Chambre
a vide

P0 , T 0
Four

z
0

6B;m`2 k a+?ûK /2 H imv`2X PM  `2T`ûb2Miû [m2H[m2b HB;M2b /2 Hû+QmH2K2Mi BMbB 
[m2 H bm`7+2  
T2`T2M/B+mHB`2 mt HB;M2b /2 Hû+QmH2K2MiX

kyRd@yk@kk R8,Ry,jy

S;2 Rfd

PM +QMbB/û`2` H2b ?vTQi?b2b bmBpMi2b ,
 /Mb H2 7Qm`- H2 ;x 2bi ¨ H T`2bbBQM  2i H i2KTû`im`2  c
 QM bmTTQb2 [m2 H2 HB#`2 T`+Qm`b KQv2M /Mb H2 7Qm` 2i /Mb H imv`2 2bi i`b 
BM7û`B2m` m /BKi`2 /2
H imv`2X *2ii2 ?vTQi?b2 T2`K2i /2 /û+`B`2 H2 ~mB/2 +QKK2 mM KBHB2m +QMiBMm 2i 
/miBHBb2` H2b HQBb /2 H
/vMKB[m2 /2b ~mB/2b c
 H2b /Bzû`2Mi2b [mMiBiûb T?vbB[m2b [mB /û+`Bp2Mi Hû+QmH2K2Mi UT`2bbBQM   - 
i2KTû`im`2   - Kbb2 pQHm@
KB[m2   - 2Mi?HTB2 KbbB[m2   - pBi2bb2   V QMi KK2 pH2m` 2M iQmi TQBMi /2 H 
bm`7+2   [mQ++mT2
H2 ~mB/2- T2`T2M/B+mHB`2K2Mi mt HB;M2b /2 Hû+QmH2K2Mi U};m`2 kVX .Mb H K2bm`2 Q 
H2 /BKi`2 /2 H
imv`2 p`B2 H2Mi2K2Mi p2+ - QM bbBKBH2 H bm`7+2   m THM    X *2ii2 TT`QtBKiBQM 
2Mi`BM2
[m2 H2b +QKTQbMi2b i`Mbp2`bH2b /2   bQMi T`iQmi Mû;HB;2#H2b /2pMi H +QKTQbMi2 
/2   b2HQM Ht2  c
 Hû+QmH2K2Mi 2bi bmTTQbû mMB/BK2MbBQMM2H H2 HQM; /2 Ht2 - biiBQMMB`2 2i T`7Bi c
 H2 ;x 2bi bmTTQbû T`7Bi- KQMQiQKB[m2- /2 Kbb2 KQHB`2     LHNPM 2i /2 +Q2{+B2Mi 
/B#iB[m2
   bmTTQbû BM/ûT2M/Mi /2 H i2KTû`im`2 c
 H pBi2bb2 KQv2MM2  /2b iQK2b /Mb H2 7Qm` 2bi Mû;HB;û2 /2pMi H pBi2bb2 /2b 
iQK2b /Mb H2 D2iX

PM MQi2`   - Q  2bi H +QMbiMi2 /2b ;x T`7BibX

PM /QMM2 H2b T`Ki`2b /2 H2tTû`B2M+2 bmBpMib ,
 /BKi`2 /2 +QH /2 H imv`2    ËN c
 i2KTû`im`2 /m ;x /Mb H2 7Qm`    , c
 T`2bbBQM /m ;x /Mb H2 7Qm`    NCBSX
AX 

_ûHBbiBQM /mM D2i bmT2`bQMB[m2

AXXRV Zm2 T2mi@QM /B`2 /m /û#Bi KbbB[m2  /2 +2i û+QmH2K2Mi \ *QKK2Mi 
b2tT`BK2@i@BH 2M 7QM+iBQM /2
  -   2i   X
AXXkV ú+`B`2 H `2HiBQM `2HBMi   -   2i   X 1tT`BK2` H p`BiBQM E /2 H2Mi?HTB2 
KbbB[m2 /m ;x
2M 7QM+iBQM /2 -  2i /2 H p`BiBQM E /2 b i2KTû`im`2X
AXXjV G2b i`Mb72`ib i?2`KB[m2b 2Mi`2 H2 ;x 2i H2b T`QBb /2 H imv`2 bQMi 
Mû;HB;2#H2b- /2 bQ`i2 [m2 H2 ;x
bû+QmH2 b2HQM mM2 i`Mb7Q`KiBQM /B#iB[m2 2i `ûp2`bB#H2X 1M /û/mB`2 mM2 `2HiBQM 
2Mi`2   2i   BMbB
[mmM2 `2HiBQM 2Mi`2 E-   2i EX
AXX9V G2b 7Q`+2b /2 T2bMi2m` ûiMi Mû;HB;û2b /2pMi H2b 7Q`+2b T`2bbMi2b 2t2`+û2b 
bm` H2 ~mB/2- `2HB2` H
pBi2bb2   ¨ H2Mi?HTB2 KbbB[m2   ¨ H TQbBiBQM  2i ¨ H2Mi?HTB2 KbbB[m2  /Mb H2 
7Qm`X
AXX8V PM `TT2HH2 [m2 H +ûHû`Biû /m bQM  /Mb mM ;x 2bi `2HBû2 ¨ b Kbb2 pQHmKB[m2 
 2i b +QKT`2bbB#BHBiû

 T`   
X
Bb2Mi`QTB[m2  

.QMM2` H2tT`2bbBQM /2 H +ûHû`Biû /m bQM /Mb H imv`2   ¨ H#b+Bbb2  2M 7QM+iBQM 
/2 -  2i   X
.Mb H bmBi2- QM MQi2`   H2 `TTQ`i /2 H pBi2bb2 /2b iQK2b /m D2i   bm` H 
+ûHû`Biû /m bQM ¨ H#b+Bbb2
- TT2Hû H2 MQK#`2 /2 J+?

AXXeV § T`iB` /2 H2Mb2K#H2 /2b `2HiBQMb T`û+û/2Mi2b- ûi#HB` H `2HiBQM bmBpMi2- 
+QMMm2 bQmb H2 MQK /2
`2HiBQM /2 >m;QMBQi ,
E

E       

AXXdV Gû+QmH2K2Mi 2bi TT2Hû bm#bQMB[m2- Qm bmT2`bQMB[m2- b2HQM [m2    - Qm    X 
.û/mB`2 /2 H
`2HiBQM /2 >m;QMBQi [m2- TQm` Tbb2` /mM û+QmH2K2Mi bm#bQMB[m2 ¨ mM û+QmH2K2Mi 
bmT2`bQMB[m2- H b2+iBQM /2
H imv`2 /QBi Mû+2bbB`2K2Mi +QKTQ`i2` mM KBMBKmKX Zm2 pmi H2 MQK#`2 /2 J+? ¨ +2i 
2M/`QBi \
AX" 

1biBKiBQM /2b ;`M/2m`b i?2`KQ/vMKB[m2b ¨ HBbbm2 /2 H2tTMbBQM

AX"XRV 1tT`BK2` H i2KTû`im`2   /m ;x 2M 7QM+iBQM /2  -  2i /m MQK#`2 /2 J+?   X 
*QKK2Mi H
i2KTû`im`2 ûpQHm2@i@2HH2 /Mb H /B`2+iBQM  /m D2i \
AX"XkV 1tT`BK2` H /2MbBiû iQKB[m2 /m D2i   2M 7QM+iBQM /2  -  2i /m MQK#`2 /2 
J+?   X SQm`[mQB
T2mi@QM +QMbB/û`2` [m¨ HBbbm2 /2 H T?b2 /2tTMbBQM- HQ`b[m2  2bi bm{bKK2Mi ;`M/- 
H i2KTû`im`2 2i H
pBi2bb2 /2b iQK2b /m D2i MûpQHm2Mi THmbX G2 MQK#`2 /2 J+? ii2BMi ¨ HBbbm2 /2 H 
T?b2 /2tTMbBQM b2` MQiû
T` H bmBi2  X
AX"XjV 1tT`BK2` /2 KK2 H pBi2bb2 /m D2i   2M 7QM+iBQM /2  - -  2i /m MQK#`2 /2 
J+?   X 1M
bmTTQbMi [m2  2bi ;`M/ /2pMi mM2 [mMiBiû [m2 HQM T`û+Bb2`- KQMi`2` [m2 H 
pBi2bb2 ii2BMi2 2M bQ`iB2 /2
H imv`2  M2 /ûT2M/ Tb /2  2i /QMM2` bQM 2tT`2bbBQM 2M 7QM+iBQM /2  -  2i X 
GûpHm2` MmKû`B[m2K2MiX
kyRd@yk@kk R8,Ry,jy

S;2 kfd

AX"X9V PM /QMM2 bm` H };m`2 j H2b `ûbmHiib /2 K2bm`2b /m MQK#`2 /2 J+? i2`KBMH  
Q#i2Mmb T` mM2
2tTû`B2M+2 bBKBHB`2- KBb TQm` /2b T`Ki`2b 2tTû`BK2Mimt /Bzû`2MibX 1M /û/mB`2 
mM2 2biBKiBQM /2 H pH2m`
/2  TQm` H2b T`Ki`2b /2 H2tTû`B2M+2 ûim/Bû2- TmBb /2 H pH2m` /2 H i2KTû`im`2  
ii2BMi2 ¨ HBbbm2
/2 H2tTMbBQMX G?vTQi?b2 7Bi2 ¨ H [m2biBQM T`û+û/2Mi2 2bi@2HH2 pû`B}û2 \
jy

ky

Ry

6B;m`2 j

UN  1B  , V

LQK#`2 /2 J+? ii2BMi ¨ HBbbm2 /2 H2tTMbBQM 2M 7QM+iBQM /2b T`Ki`2b /2 H bQm`+2

AA .Bz`+iBQM /m D2i iQKB[m2 T` mM2 QM/2 biiBQMMB`2
§ HB/2 /mM Hb2` /2 HQM;m2m` /QM/2    ON- QM +`û2 mM2 QM/2 THM2 2i T`Q;`2bbBp2- 
TQH`Bbû2 `2+iBHB;M2@
K2Mi b2HQM  - /2 p2+i2m` /QM/2   2i /2 TmHbiBQM  - [mB b2 T`QT;2 /Mb H 
/B`2+iBQM  X G2 +?KT ûH2+i`B[m2
/2 +2ii2 QM/2 bû+`Bi ,
           DPT     

AAX  PM/2 biiBQMMB`2
GQM/2 Hb2` b2 `û~û+?Bi 2M BM+B/2M+2 MQ`KH2 bm` mM KB`QB` THM bbBKBHû ¨ mM 
KB`QB` KûiHHB[m2 T`7Bi2K2Mi
+QM/m+i2m`- bBimû /Mb H2 THM    X
       TJO      X
JQMi`2` [m2 H2 +?KT ûH2+i`B[m2 /2 HQM/2 biiBQMMB`2 `ûbmHiMi2 bû+`Bi 
S`û+Bb2` H2tT`2bbBQM /2   2M 7QM+iBQM /2  -  - 2i  X
AAX"  LQiBQM /2 TQi2MiB2H HmKBM2mt
PM bBMiû`2bb2 /#Q`/ ¨ HBMi2`+iBQM /2 +2ii2 QM/2 biiBQMMB`2 p2+ H2b iQK2b /2 
HBi?BmKX GQ`b[mBHb TûMi`2Mi
 H2b iQK2b b2 TQH`Bb2MiX GQ`b[m2 H TmHbiBQM  2bi i`b ûHQB;Mû2 /2b
/Mb H xQM2 Q `;M2 H2 +?KT `ûbQMM+2b iQKB[m2b- H2b /BTH2b BM/mBib Qb+BHH2Mi ¨ H 
TmHbiBQM  /m Hb2` 2t+Bii2m` 2i 2M T?b2 p2+ HmB ,
 Q  2bi H TQH`Bb#BHBiû /2 HiQK2 /2 HBi?BmKX .Mb H2b
H2 /BTH2 iQKB[m2 BM/mBi bû+`Bi HQ`b     
Q       $N 2bi mM2
+QM/BiBQMb /2 H2tTû`B2M+2 UHb2` ?Q`b /2 `ûbQMM+2V- QM    
 
+QMbiMi2 +`+iû`BbMi H2 /BTH2 /2 HiQK2 /2 HBi?BmK 2i Q      2bi Hû+`i 2Mi`2 H 
TmHbiBQM Hb2` 
2i H TmHbiBQM  /2 H i`MbBiBQM iQKB[m2 H THmb T`Q+?2X G2 +?KT HmKBM2mt BMi2`;Bi 
p2+ H2 /BTH2 iQKB[m2
 
 [mBH BM/mBi pB mM2 ûM2`;B2 TQi2MiB2HH2      TT2Hû2 TQi2MiB2H HmKBM2mtX

AAX"XRV JQMi`2` [m2 H KQv2MM2 i2KTQ`2HH2 /2 +2 TQi2MiB2H HmKBM2mt bû+`Bi bQmb H 
7Q`K2
        DPT    

AAX"XkV 1tT`BK2`  2i  2M 7QM+iBQM /2b T`Ki`2b /m T`Q#HK2X
AAX*  ú[miBQM /2 T`QT;iBQM /2 HQM/2 /2 KiB`2
lM D2i bmT2`bQMB[m2 /2 HBi?BmK b2 T`QT;2 /Mb H2 THM  ¨ mM2 pBi2bb2    NT 2i 
i`p2`b2 HQM/2
HmKBM2mb2 biiBQMMB`2 ;ûMû`û2 T` mM KB`QB` UJV U};m`2 9VX lM iQK2 /2 HBi?BmK 2bi 
bbBKBHû ¨ mM T[m2i /QM/2b
/2 KiB`2 /û+`Bi T` b 7QM+iBQM /QM/2     /QMi H2 KQ/mH2 m +``û /QMM2 H /2MbBiû 
/2 T`Q
T`ûb2M+2 /2 HiQK2 /Mb H2bT+2 ¨ i`QBb /BK2MbBQMbX PM bBMiû`2bb2 B+B 
2t+HmbBp2K2Mi m T`Q+2bbmb ûHbiB[m2 /2
/BzmbBQM /2b iQK2b T` HQM/2 HmKBM2mb2 biiBQMMB`2- /2 bQ`i2 [m2 HûM2`;B2 /2 
HiQK2 T2mi i`2 +QMbB/û`û2
+QKK2 }tû2 2i û;H2 ¨ bQM ûM2`;B2 +BMûiB[m2 BM+B/2Mi2  X G T`iB2 bTiBH2    /2 b 
7QM+iBQM /QM/2 Q#ûBi
HQ`b ¨ Hû[miBQM /2 a+?`/BM;2`

kyRd@yk@kk R8,Ry,jy

S;2 jfd

¨ HBMiû`B2m` /2 HQM/2 biiBQMMB`2 U    V
¨ H2tiû`B2m` /2 HQM/2 biiBQMMB`2 U   Qm   V

Q       LH 2bi H Kbb2 /2 HiQK2 /2 HBi?BmK d 2i  bQM ûM2`;B2 iQiH2- û;H2 ¨ bQM 
ûM2`;B2 +BMûiB[m2
BM+B/2Mi2X
AAX*XRV _TT2H2` H2tT`2bbBQM /2 H HQM;m2m` /QM/2 /2 /2 "`Q;HB2  /2 HiQK2 
BM+B/2Mi 2i HûpHm2` MmKû@
`B[m2K2Mi /Mb H2 +b T`ûb2MiX
AAX*XkV PM bbBKBH2 H2 T[m2i /QM/2 iQKB[m2 BM+B/2Mi ¨ mM2 QM/2 THM2 /2 p2+i2m` 
/QM/2   X _2HB2`   ¨ H
[mMiBiû /2 KQmp2K2Mi /2 HiQK2   2i 2M /û/mB`2 H2tT`2bbBQM /2 bQM ûM2`;B2  X 
GûpHm2` MmKû`B[m2K2MiX
PM bmTTQb2` /Mb H bmBi2 [m2 H T`Q7QM/2m`  /m TQi2MiB2H HmKBM2mt 2bi #2m+QmT 
THmb 7B#H2 [m2 HûM2`;B2
 /2 HiQK2X
AAX*XjV ú+`B`2 Hû[miBQM /2 T`QT;iBQM- /Mb mM KBHB2m /BM/B+2   - /mM2 QM/2 
HmKBM2mb2      /2
p2+i2m` /QM/2 /Mb H2 pB/2  2i /2 TmHbiBQM X

AAX*X9V 1M /û/mB`2 [m2 H T`QT;iBQM /2 HQM/2 /2 KiB`2 ¨ HBMiû`B2m` /2 HQM/2 
biiBQMMB`2 2bi `û;B2 T` mM2
û[miBQM bBKBHB`2 ¨ +2HH2 /mM2 QM/2 HmKBM2mb2 /Mb mM KBHB2m /BM/B+2   KQ/mHû 
Tû`BQ/B[m2K2MiX Zm2H 2bi
H2 Tb  /2 +2 `ûb2m /BM/B+2 \ GûpHm2` MmKû`B[m2K2MiX .QMM2` H2tT`2bbBQM /2 
HKTHBim/2  /2 KQ/mHiBQM
/2 HBM/B+2 2M 7QM+iBQM /2  2i  X
AAX*X8V PM û+`Bi Hû[miBQM /2 T`QT;iBQM /2 HQM/2 /2 KiB`2 ¨ HBMiû`B2m` /2 HQM/2 
biiBQMMB`2 bQmb H
7Q`K2 bmBpMi2

1tT`BK2`  2M 7QM+iBQM /2  2i  X

     DPT    

UAAXRV

UJV

6B;m`2 9 _ûb2m HmKBM2mt /2 Tb  2i /ûTBbb2m` 

AAX.  _ûbQHmiBQM /2 Hû[miBQM /2 T`QT;iBQM
G2 7Bb+2m HmKBM2mt [mB +`û2 HQM/2 biiBQMMB`2 2bi HBKBiû i`Mbp2`bH2K2Mi T` mM 
/BT?`;K2 /2 `vQM
   NNX G2 `ûb2m HmKBM2mt bm` H2[m2H pQMi b2 /Bz`+i2` H2b QM/2b iQKB[m2b  /QM+ 
mM2 ûTBbb2m`   
U};m`2 9VX .2mt 72Mi2b i`b }M2b- /2 H`;2m`   ËN- bQMi TH+û2b H2 HQM; /m D2i 
bmT2`bQMB[m2- /2 KMB`2 ¨
bûH2+iBQMM2` i`b T`û+BbûK2Mi HM;H2 /BM+B/2M+2  /2b iQK2b bm` H2 `ûb2m HmKBM2mt 
UHQ`B2MiiBQM /2 HM;H2
 2bi T`û+Bbû2 bm` H };m`2 9VX .Mb +2b +QM/BiBQMb- QM KQMi`2 [m2 b2mH2b /2mt 
QM/2b THM2b b2 T`QT;2Mi
bBKmHiMûK2Mi /Mb H2 `ûb2m HmKBM2mt , HQM/2 BM+B/2Mi2    /KTHBim/2   2i /2 
p2+i2m` /QM/2   - 2i
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