Optique géométrique : lentilles (PCSI)

L1L2f0
1f0 2= 2f0 1
L1L2=f0
2L2
AB L14f0
1
O1A=−4f0
1AB L1
→A1B1
L2
→A0B0
1
O2A0−1 O2A1=1 f0
2
1
O1A1−1 O1A=1 f0
1⇒O1A1=f0
1(−4f0 1)
f0
1−4f0 1=4
3f0
1O2A1=O2O1+O1A1=4
3f0
1−2f0 1=
−2
3f0
1=−1
3f0
2O2A0=−f0
2
2
L1L2f0
1= 5 cm
f0
2= 20 cm ∆ = F0 1F2= 17 cm
L2AB L1
AB A0B0
L1
α0AB
α0
αα
AB L1 →A1B1
L2
→A0B0
A1=F2O1A1=f0
1+ ∆ 1
O1A1−1 O1A=1 f0
1⇒O1A=−f0
1(∆+f0 1)
∆
L1=A1B1
AB =O1A1 O1A=−∆ f0
1
tanα0=A1B1
O2A1=A1B1 AB
AB
O2A1=−∆ f0
1
AB
−f0
2
A pp '25cm tanα =AB
pp

f0
l1x1α1
l2x2
α2
x1x2x0
1x0 2
x1α1
x0
1
α1x0
2
F
x0
2α2x0 1
α2α1
x1
x2α2x0
1
x2α1
x1
Mx2
f0α2=Mx1 f0α1
α1, α2, x2x0
2x1x0 1
x1−x0
1
l1+f0=−α1x0
1=x1+α1(l1+f0)
x0
2
f0=α1x0
2=α1f0x0
1−x0 2
f0=−α2α2=−x1 f0−
α1l1
f0
x0
1−x2
l2+f0=−α2x2=−x1l2 f0+α1f0−l1l2 f0M=
−l2
f01−l1l2 f02
−1−l1
f0!

f0
r0δϕ
r0f0
M
δφ =4GM
c2r0r0G
r0
D r0
α= 2 10−6
D= 109al 1al '9.1015m
δϕ =tanδϕ =r0
f0⇒f0=c2r2
0
4GM
f0
D D =c2r2
0
4GM ⇔r0=±2√GMD cr0
r0
D=2 cqGM D
M=D
Gαc 22'1039kg

f0L
d1
f0
D
¯
OA =x
•D x1x2
¯
OA0¯ OA D
•D x1x2
f0
{lentille +miroir}AB →
A1B1→A2B2→A0B0
•¯ OA ¯ OA0¯ OA1¯ OA2
•d2{lentille +miroir}
f0