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∂
2
U
∂r
2
+
2
r
∂U
∂r
+
1
r
2
∂
2
U
∂θ
2
+
cot θ
r
2
∂U
∂θ
= 0
U = (Ar
m
+
B
r
(m+1)
)P
m
(cos θ).
A B m = 0, 1, 2..
f(r) r < R,
f(r) = A
0
+ A
1
r + A
2
r
2
+ . . . + A
n
r
n
+ . . .
r > R,
f(r) =
B
1
r
+
B
2
r
2
+
B
3
r
3
+ . . . +
B
n+1
r
(n+1)
+ . . . .
U
θ=0
= f(r).
R
υ(r, θ)
r = R R
θ,
U
r=R
= f(θ), 0 ≤ θ ≤ π.
137. Äîêàçàòü, ÷òî óðàâíåíèå Ëàïëàñà
∂ 2U 2 ∂U 1 ∂ 2U cot θ ∂U
2
+ + 2 2
+ 2 =0
∂r r ∂r r ∂θ r ∂θ
äîïóñêàåò ÷àñòíîå ðåøåíèå
B
U = (Arm + )Pm (cos θ).
r(m+1)
A è B -ïðîèçâîëüíûå ïîñòîÿííûå, m = 0, 1, 2..
138. Äàííàÿ ôóíêöèÿ f (r) ïðè r < R, ìîæåò áûòü ðàçëîæåíà
â ðÿä âèäà:
f (r) = A0 + A1 r + A2 r2 + . . . + An rn + . . .
à ïðè r > R, â ðÿä ñëåäóþùåãî âèäà:
B1 B2 B3 Bn+1
f (r) = + 2 + 3 + . . . + (n+1) + . . . .
r r r r
Íàéòè òàêîå ðåøåíèå óðàâíåíèÿ, ïðèâåäåííîãî â ïðåäûäóùåé
çàäà÷å, êîòîðîå óäîâëåòâîðÿëî áû óñëîâèþ: Uθ=0 = f (r).
139. Êðóãîâîå ïðîâîëî÷íîå êîëüöî ðàäèóñà R çàðÿæåíî åäè-
íèöàìè ñòàòè÷åñêîãî ýëåêòðè÷åñòâà. Íàéòè ïîòåíöèàë υ(r, θ)
íàýëåêòðèçîâàííîãî êîëüöà íà ëþáóþ òî÷êó ïîëÿ.
140. Íàéòè òàêîå ðåøåíèå óðàâíåíèÿ çàäà÷è 137, ÷òîáû ïðè
r = R ( íà ïîâåðõíîñòè ñôåðû ðàäèóñà R ñ öåíòðîì â íà÷àëå
êîîðäèíàò) îíî îáðàùàëîñü áû â äàííóþ ôóíêöèþ óãëà θ, ò.å.
Ur=R = f (θ), 0 ≤ θ ≤ π.
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