Щекинов Ю.А. Задачи по электродинамике

div rot A = 0 rot grad φ = 0

div ([AB]) = B · rot A − A · rot B

rot ([AB]) = AdivB − BdivA − (A · ∇)B + (B · ∇)A

divr

grad r

grad(p · r) grad

p·r

r

3

(p∇)r div[pr] rot[rp] gradA(r) · B(r) divφ(r)A(r) rotφ(r)A(r)

p =

div r grad r

2

rot r div r/r

2

rot(a · r)b div(a · r)b rot(a · r)r div(a · r)r rot[r[ar]] div[r[ar]]

a, b =

R

r(a · dS)

R

(a · r)dS a

R

r

2

dS

Q F

q r

Q a |a|  |r|

a Q

q q

b q

Q F

q r

OX q h  r

x r

a q

r  2a F

R Q

∗

a b

c b + c < a

a < b Q −Q

χ

R

σ

σ a

∗

div A = lim

∆V →0

I

AdS/∆V,

dV = r

2

sin θdθdφdr, dS = r

2

sin θdθdφ,

div A =

1

r

2

∂

∂r

(r

2

A

r

) +

1

r sin θ

∂

∂θ

(sin θA

θ

) +

1

r sin θ

∂A

φ

∂φ

.

R

χ

R Q

ρ = αr

β

β r

−1

r

−2

d σ

d σ = 2σ

0

|x|/d − σ

0

x

q

Ox a

q

a

q −q Ox

x = −b x = b

Ox

a

q h

R  h

R χ

z  R

R

1

R

2

σ R

1

→ 0 R

1

→ 0 R

2

→ ∞

z  R

2

a Q

∗

h r = h

σ

ρ = −

e

πa

3

exp(−2r/a).

p

φ =

p · r

r

3

.

E =

Ae

−kr

r

2

e

r

.

+σ

−σ

+σ

+Q L+a

L

Q −Q m

+Q L

L

Q

+Q m

R +Q

−Q

m

σ a

a < b χ −χ

φ E 2a

ρ 

ρ R

R ρ

∗

Q n T

+q −q

2a

+q +q

φ =







−ax, x > 0

ax, x < 0

R

1

ρ

R

2

a R

2

+ a < R

1

φ E

φ τ

φ =

Z

τ · r

r

3

dS,

r

(x, y) τ

q a

τ σ τ σ

q

R

1

< R

2

ρ

∗

V

∗

V

δ

δ(x) = lim

α→∞

sin αx

πx

.

δ

0

(−x) = −δ

0

(x),

δ(ax) =

1

a

δ(x),

δ(x

2

− a

2

) =

1

2a

[δ(x − a) + δ(x + a)],

+∞

Z

−∞

δ(a − x)δ(x − b)dx = δ(a − b).

+∞

Z

−∞

f(x)δ

0

(x)dx = −f

0

(x).

δ

(x) = 0, x < 0,

(x) = 1, x > 0.

Щекинов Ю.А. Задачи по электродинамике

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