Δ q > 0 (. 1.7.1). , . , q, Δ q
W C, Q, 0 Q:
1.7.1. |
, , , Q = CU.
W , . W E (. . I, 2.4)
k , x , F = kx .
, , . . .
E = U / d,
V = Sd , . ,
() , . .
, , w , .
16
, , ρ(r→). dV
dq = ρ(r→)dV,
(39′)
W = 1 2 ∫ ρ(r→)ϕ(r→)dV. | (16.1) |
(39′)→(42). , ,
ρ(r→)ϕ′(r→),
ϕ′(r→) , ρdV. ρdV δ r→ ρ(r→). , = 3 2 q δ = 3 2 1 δ ⋅4 3πδ3ρ = 2πδ2 ⋅ ρ(r→), ,
|
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ϕ′(r→) = ϕ(r→) − 2πρ(r→)δ2.
, δ → 0 ϕ′→ ϕ(r→) ϕ′(r→) ϕ(r→), , .
(42), ρ, (13), −1 4πΔϕ
div(ϕgradϕ) = ϕΔϕ + gradϕ)2;
W = − 1 8π ∫ div(ϕgradϕ)−gradϕ)2]dV = 1 8π ∮ SϕEndS+ 1 8π ∫ V E2dV,
S , V. , S R, R →∞
∮ SR → 0,
ϕ En , 1 R 1 R2 (, , ), R2.
, (42)
W = ∫ E2 8πdV | (16.2) |
, , (39) , , , ,
W = E2 8π. | (16.3) |
(39) (i≠j), (42) (43) . , (42), (43) , (39) - .
, (44) . (44) . , (44) . , (44) .
, . , .
, q E→
F→ = qE→,
E→ , , q. , , , , E→ , . , , , σ (. . 34). , , , , . , , . - . , , , δ, , , , , , .
|
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, . 34 dS . , , .
x , , [0,δ](. 35). , E→ y,z x- Ex(x), ρ(x). ( ). , Ex(x)
dEx dx = 4πρ(x),(∗)
E(0) = 0
Ex(x) = 4π ∫ 0xρ(ξ)dξ.
, ,
f→ = fxe→x,fx = ∫ 0δρ(x)E x(x)dx,
. ρ(x) (*),
fx = 1 4π ∫ 0δE x(x)dEx dx dx = 1 8π ∫ 0δ d dx[Ex(x)]2dx,
..
fx = 1 8πE02,
E0 = Ex(δ) = 4π ∫ 0δρ(x)dx = 4πσ .
,, , σ = ∫ 0δρ(x)dx, , ρ(x). , σ, .. E→0, f→ , ..
f→ = E02 8π n→. | (16.4) |
, (45) , .
,
W = 1 8π ∫ E2dV.
1. R q. R + dR .
Er = q r2 r > R 0r < R
Er = q r2 r > R + dR 0r < R + dR
36.
fr = 1 8πE02,E 0 = q R2.
δA = fr ⋅ 4πR2dR = 1 8πE02 ⋅ 4πR2dR.()
r > R + dR , (R,R+ dR) , ..
dW = −W ⋅ 4πR2dR,()
W .
δA = −dW,
.. δA . () (), 4πR2dR W = 1 8πE02 , .
. : , W , fr, . .
|
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a
Er = q r2 r ≥ R q a3 r r < a
W = 1 8π ∫ 0aq2 a6r2 ⋅ 4πr2dr + 1 8π ∫ a∞q2 r44πr2dr = 3 5 q2 a.
.
W m = W∕c2. , m q ,
rq = q2 mc2,
.. ( 3/5 ).
,
re = e2 mc2 ≃ 2,8 ⋅ 10−13.
, , .