Electromagnétisme : mouvement de particules chargées (PCSI)

n
−→
f=−m
τ−→
v
−→
E=Ex−→ ux+Ey−→ uy
−→
B=B0−→ uz
Ex
jy
(d2vx
dt2=−e mEx−vx τ
d2vy
dt2=−e mEy−vy τ(d2jx
dt2=ne2 mEx−jx τ
d2jy
dt2=ne2 mEy−jy τ(jx=σ0Ex
jy=σ0Ey
(d2vx
dt2=−e mEx−vx τ−e mB0vy
d2vy
dt2=−e mEy−vy τ+e mB0vx(σ0Ex=jx+eτ
mB0jy
−σ0Ey=−jy+eτ
mB0jx
1eτ
mB0
eτ
mB0−1
=
−1 + e2τ2
m2B2
0
jx=−1
1+ e2τ2
m2B2
0
σ0Exeτ
mB0
−σ0Ey−1
=σ0
1+ e2τ2
m2B2
0
Ex−σ0eτ
mB0
1+ e2τ2
m2B2
0
Eyjy=−1
1+ e2τ2
m2B2
0
1σ0Ex
eτ
mB0−σ0Ey
=
σ0
1+ e2τ2
m2B2
0
Ey+σ0eτ
mB0
1+ e2τ2
m2B2
0
Ex
q−→ E(−→ r)
m q
−→
E=−−−→ gradV
V(−→ r) = V0z2−x2+y2
2
−→
B=
0
0
B0

¨x
¨y
¨z
=
ω2
2ωc0
−ωcω2
20
0 0 −ω2
x
y
z
ω ωc
ξ=x+iy ξ
ωωcξ
x=−∞ x= 0 −→ B=B0−→ Uz
x= 0 x=L−→ E=E(t)−→ Ux
x=L x = +∞−→ B=B0−→ Uz
m q
x= 0
E(t)
E(t)

−→
v0−→ B=B0−→ uz
E
q√m
q
e=
∆E=αem α = 13 MeV mm−1kg−1
Ri=
Rf= 1.7cm Ef= 23MeV
∆E m q
dvx
dt =qB0 mvy
dvy
dt =−qB0 mvxu=vx+ivydu dt =−iqB0 mu u =u0e−iqB0
mt
vx=v0cos qB0
mtvy=−v0sin qB0 mtx=mv0 qB0sin qB0 mt+x0y=mv0 qB0cos qB0 mt+y0
dEm
dt =PF=q−→
v∧−→ B.−→ v= 0
R=mv0
qB0E=1 2mv2
0R=√2E
B0
√m
q
q
√m=√2Ef
RfB0Ei=B2
0R2 i
2
q2
m=B2
0R2 i
2
2Ef
R2
fB2
0
=E1
R2
i
R2
f
∆E=EfR2
i
R2
f−1= 702.10−31M eV
m=∆E
αe = 9.10−31