a class of analytic solutions for force-free electromagnetic fields
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7/28/2019 A class of analytic solutions for force-free electromagnetic fields
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F
F F(F)
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A
F = dA
d A F
A = Adx
F = ,,
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F F = 0, (F)F = 0
K
K (A F)
K = 0, d K = 0;
F = s1s2 s2s1
s1 s2
F = ds1 ds2
A = s1ds2 + d
ds1 ds2
J =1
4dh d1 d2
h, 1, 2
dF = 0, d F = dh d1 d2
F = hd1 d2
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F = h (d1 d2)
dF = d (h (d1 d2)) = 0
R = (d1 d2)
R =1
2i
(1 + )2
R = 1
2i
1 + 2
dR = 0 d R = 0
R = d1 d2
R = du1 du2
= Sexp(i2) = Sexp(i2)
1 =1
1 + S2, 2 = , u1 =
1
1 + S2, u2 =
h = u1
F = u1R
F = u1 R = u1du1 du2 = d (u1)
2
2 du2
F
f = FJ = F
1
3! (J)
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f = (u1du1 du2)1
3!
1
4du1 d1 d2
f = u1u1u21
3!
1
4
du1 d1 d2 f = u1u1
1
3!
1
4u2u112
f =1
12u1u1FF
FF = 0
RR
= 0
J F = 0;
h = f(u1, u2)
JJ = 0
S3 S2
(r
, t) =
(Ax tz) + i(Ay + t(A 1))
(Az + tx) + i(A(A 1) ty)
(r, t) =(Ay + t(A 1)) + i(Az + tx)
(Ax tz) + i(A(A 1) ty)
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A =x2 + y2 + z2 t2 + 1
2
1 = 11 +
, 2 = 14i
ln(
)
u1 =1
1 + , u2 =
1
4iln(
)
2 u2
h = u1
F = R
1 +
(n)(r, t) =
(Ax tz) + i(Ay + t(A 1))
(Az + tx) + i(A(A 1) ty)
n
(n)(r, t) =
(Ay + t(A 1)) + i(Az + tx)
(Ax tz) + i(A(A 1) ty)
n
RR = R(R) = FF = F(F) = 0
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