[PDF] Electromagnétisme : PEIP 2 Polytech

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bx,by? ??bz?M!O z x y Mzr M!O z M z x A( )M a)b) y xy x r(x,y,z) =---→OM=xbx+yby+zbz= x y = (x,y,z).????? M d dx+∂---→OM∂y dy+∂---→OM∂z dz=bxdx+bydy+bzdz .????? dV=dxdydz .????? -→dS -→dS=bxdydz+bydxdz+bzdxdy .????? v(-→r)≡dqdV.????? Q tot=ZZZ V v(-→r)dV.????? ???? ???? ?? ?????a??? ???? ?????? ?? ??????a > x >0?a > y >0? ??a > z >0??ρ0?? Q cube=ZZZ cube

ρ(x,y,z)dV=Z

a 0 dxZ a 0 dyZ a 0 dzρ0a

6xy2z3

ρ0a

6×Z

a 0 xdxZ a 0 y2dyZ a 0 z3dz

ρ0a

6×a22

×a33

×a44

=ρ024 a3.

3xy3? ???????

Q rect=ZZ rect

σ(x,y)dS=σ0ab

3Z a 0 xdxZ b 0 y3dy

σ0ab

3a22 b 44
=abσ08 dΦ =∂Φ∂x dx+∂Φ∂y dy+∂Φ∂z dz .????? dΦ =---→gradΦ·d---→OM.????? grad=bx∂∂x +by∂∂y +bz∂∂z ?? ?????M???-→r? ??? ?????? ??? x=ρcosϕ y=ρsinϕ z=z .?????? b

ρ,bϕ?bz

b

ρ,bϕ?bz

bρ? fO z x y r r M r z f r b

ρ,bϕ,bz

E(M) =Eρbρ+Eϕbϕ+Ezbz??-→E(M) = E E E b

ρ,bϕ?bz

b --→OM∂ρ ∂--→OM∂ρ = cosϕbx+ sinϕby b --→OM∂ϕ ∂--→OM∂ϕ =-sinϕbx+ cosϕby b z≡∂--→OM∂z ∂--→OM∂z bz,?????? b bϕ b cosϕsinϕ0 -sinϕcosϕ0 b x b y b =T b x b y b b

ρ,bϕ?bz

b x b y b =Tt b bϕ b cosϕ-sinϕ0 sinϕcosϕ0 b bϕ b b x= cosϕbρ-sinϕbϕ b y= sinϕbρ+ cosϕbϕ b z=bz.?????? ---→OM=ρbρ+zbz, d ---→OM=∂---→OM∂ρ dρ+∂---→OM∂ϕ dϕ+∂---→OM∂z dz . ---→OM∂ρ =bρ+ρ∂bρ∂ρ =bρ puisque∂bρ∂ρ =0 ---→OM∂ϕ =ρ∂bρ∂ϕ (cosϕbx+ sinϕby) =ρ(-sinϕbx+ cosϕby) =ρbϕ. ---→OM=bρdρ+bϕρdϕ+bzdz .?????? -→dS=bρρdϕdz+bϕdρdz+bzdρρdϕ .?????? cylindre dV=Z R 0 dρZ 2π 0

ρdϕZ

L 0 dz=LZ R 0

ρdρZ

2π 0 dϕ = 2πLZ R 0

ρdρ=πR2L .

Q disque=ZZ disque

σ(ρ)dS=Z

a 0

ρdρZ

2π 0 dϕσ

0ρ2a

2

2πσ0a

2Z a 0

ρ3dρ=2πσ0a

2ρ44

a 0 =πσ0a22 dΦ =∂Φ∂ρ dρ+∂Φ∂ϕ dϕ+∂Φ∂z dz .?????? dΦ =---→gradΦ·d---→OM.?????? gradΦ =∂Φ∂ρ bρ+1ρ bϕ+∂Φ∂z bz??????

E(ρ,ϕ,z) =----→gradV(ρ,ϕ,z)

E=Eρbρ+Eϕbϕ+EzbzE

ρ=-∂V∂ρ

E

ϕ=-1ρ

∂V∂ϕ E z=-∂V∂z E(ρ) =----→gradV(ρ) =λ2πϵ0ρbρ. ???O? ?? ?????M??? ?????? xOy.?ϕ= (-→Ox,---→OM′) x=rsinθcosϕ y=rsinθsinϕ z=rcosθ??????M! fO z x y r M q rf M!! b r,bθ,bϕ

E(M) =Erbr+Eθbθ+Eϕbϕ,

bϕ???-→uϕ? ??? ??????? ??M?? ?????? ?? ??????M′′?? ?? ?????M′′M=OM′? ??????? ????

b r,bθ?bϕ b r≡∂--→OM∂r ∂--→OM∂r = sinθcosϕbx+ sinθsinϕby+ cosθbz b

θ≡∂--→OM∂θ

∂--→OM∂θ = cosθcosϕbx+ cosθsinϕby-sinθbz b --→OM∂ϕ ∂--→OM∂ϕ =-sinϕbx+ cosϕby,?????? b r bθ cosθcosϕcosθsinϕ-sinθ b x b y b =T b x b y b b x b y b =T-1 b r bθ =Tt b r bθ sinθsinϕcosθsinϕcosϕ b r bθ

V(r) =q4πϵ01r

-→E(-→r) =q4πϵ0b rr

2=q4πϵ0-→

rr

3(????-→r=rbr),

V(x,y,z) =q4πϵ01px

2+y2+z2-→E(x,y,z) =q4πϵ0x

bx+yby+zbz(x2+y2+z2)3/2.

OM=rbr.

d ---→OM=∂---→OM∂r dr+∂---→OM∂θ dθ+∂---→OM∂ϕ dϕ . ---→OM∂r =br+r∂br∂r =br ---→OM∂θ =r∂br∂θ =rbθ ---→OM∂ϕ =r∂br∂ϕ =rsinθbϕ. -→dS=brr2sinθdθdϕ+bθrsinθdrdϕ+bϕrdrdθ .?????? ?????? ?? ?????R=ZZZ sph`ere dV=Z R 0 drZ 0 dθZ 2π 0 r2sinθdϕ=Z R 0 r2drZ 0 sinθdθZ 2π 0 dϕ = 2πZ R 0 r2drZ 1 -1du= 4πZ R 0 r2dr=4π3 R3. du=-sinθdθ??? ?? ??????? ? Z 0 sinθdθ⇒Z 1 -1du= 2. 0rR Q ??????=ZZZ sph`ere v(r)dV=Z R 0 drZ 0 dθZ 2π 0 0rR r2sinθdϕ = 4πρ0Z R 0r 3R dr=4πρ0R r 44
R 0 =πρ0R3. Q ???????=ZZ

σ(θ)dS=Z

0 dθZ 2π 0 dϕa2σ0sin2θsinθ = 2πa2σ0Z 0 sin2θsinθdθ= 2πa2σ0Z 1 -11-cos2θd(cosθ) = 2πa2σ0Z 1 -11-u2du= 2πa2σ0 u-u33 1 -1=8πa2σ03 dΦ =∂Φ∂r dr+∂Φ∂θ dθ+∂Φ∂ϕ gradΦ =br∂Φ∂r +bθ1r +bϕ1rsinθ∂Φ∂ϕ E(-→r) =----→gradV(r) =-br∂V∂rquotesdbs_dbs29.pdfusesText_35

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