|
A1. 1
$$
\vec{B}_{mp} = \frac{\mu_0 q_m}{4 \pi} \frac{(z-h)\hat{z} + \vec{\rho}}{[(z-h)^2+ \rho^2]^{3/2}} $$ |
0.10 |
|
|
A1. 2
$$
\vec{B}{}^\prime = \frac{\mu_0 q_m}{4 \pi} \frac{(z+h)\hat{z} + \vec{\rho}}{[(z+h)^2+ \rho^2]^{3/2}} $$ |
0.20 |
|
|
A1. 3
$$
\vec{B} =\frac{\mu_0 q_m}{4 \pi} \frac{(z-h)\hat{z} + \vec{\rho}}{[(z-h)^2+ \rho^2]^{3/2}}+\frac{\mu_0 q_m}{4 \pi} \frac{(z+h)\hat{z} + \vec{\rho}}{[(z+h)^2+ \rho^2]^{3/2}} $$ |
0.10 |
|
|
A2. 1
$$
\vec{B} = 0 $$ |
0.20 |
|
|
A3. 1
$$
B_z^\prime=0 \text{ при } z=0 $$ |
0.10 |
|
|
A3. 2
$$
\Phi_B=0 \text{ при } z=0 $$ |
0.10 |
|
|
A3. 3
$$
B_z^\prime=0 \text{ при } z=-d $$ |
0.10 |
|
|
A3. 4
$$
\Phi_B=0 \text{ при } z=-d $$ |
0.10 |
|
|
A4. 1
$$
B_\rho (\rho, \, z=0) d \rho = \mu_0 j(\rho) d\rho\cdot d $$ |
0.40 |
|
|
A4. 2
$$
\vec{j}(\vec{\rho}) = \frac{1}{\mu_0 d} \hat{z} \times \vec{B}(\vec{\rho},\,z=0) = \frac{q_m}{2 \pi d} \frac{\hat{z} \times \vec{\rho}}{(h^2 + \rho^2)^{3/2}} $$ |
0.20 |
|
|
A5. 1
$$
\left. \frac{\partial B_z^\prime}{\partial z}\right|_z-\left. \frac{\partial B_z^\prime}{\partial z}\right|_{-d-z} = \mu_0 \sigma(d + 2z) \frac{\partial B_z^\prime}{\partial t} \approx \mu_0 \sigma d \frac{\partial B_z^\prime}{\partial t} $$ |
0.20 |
|
|
A5. 2
$$
\left. \frac{\partial B_z^\prime}{\partial z}\right|_z=-\left. \frac{\partial B_z^\prime}{\partial z}\right|_{-d-z} $$ |
0.20 |
|
|
A5. 3
$$
\frac{\partial}{\partial t} B^\prime_z(\rho,\, z;\,t) = \frac{2}{\mu_0 \sigma d}\times \frac{\partial}{\partial z}B^\prime_z(\rho,\, z;\,t) $$ |
0.20 |
|
|
A6. 1
$$
B^\prime _z (\rho,\,0 ;\, t) = f(\rho,\, z+ v_0 t) \text{ вблизи } z = 0 $$ |
0.40 |
|
|
A7. 1
$$
\text{При } t = 0\quad B_z ^\prime(\rho,\, z \ge 0) \text{ имеет вид } F(\rho,\, z+h) $$ |
0.10 |
|
|
A7. 2
$$
\text{При } t>0 \quad z \to z + v_0 t $$ |
0.10 |
|
|
A7. 3
$$
v_0 = \frac{2}{\mu_0 \sigma d} $$ |
0.20 |
|
|
B1. 1
Положение монополей $q_m$:
$$ (x,\,z) = [-n v \tau, - h - n v_0 \tau], \text{ при } n \ge 0 $$ |
0.40 |
|
|
B1. 2
Положение монополей $-q_m$:
$$ (x,\,z) = [-n v \tau, - h - (n-1) v_0 \tau], \text{ при } n \ge 0 $$ |
0.40 |
|
|
B2. 1
Магнитный потенциал
$$ \Phi _+= \frac{\mu_0 q_m}{4 \pi} \left[ \sum_{n=0}^{\infty}\frac{1}{\sqrt{(x+ nv \tau)^2+(z+ h + nv_0\tau)^2}}- \sum_{n=1}^{\infty}\frac{1}{\sqrt{(x+ nv \tau)^2+(z+ h + (n-1)v_0\tau)^2}} \right] $$ |
0.30 |
|
|
B2. 2
$$
\Phi _+= \frac{\mu_0 q_m}{4 \pi \tau} \int_0^{\infty}dt^\prime\, \left[ \frac{1}{\sqrt{(x+ vt^\prime)^2+(z+ h + v_0 t^\prime)^2}}- \frac{1}{\sqrt{(x+ v t^\prime + v\tau)^2+(z+ h +v_0 t^\prime)^2}} \right] $$ |
0.20 |
|
|
B2. 3
$$
\Phi_+ = \frac{\mu_0 q_m v}{4 \pi} \frac{1}{(z+h)-v_0 x}\left[\frac{z+h}{\sqrt{x^2+ (z+h)^2}} - \frac{v_0}{\sqrt{v^2+v_0^2}}\right] $$ |
0.20 |
|
|
B3. 1
$$
\Phi_T(x,\,z) = \Phi_+(x,\,z+-\Phi_-(x,\,z) \text{ где } \Phi_-(x,\,z) = - \Phi_+(x, \, z- \delta_m) $$ |
0.20 |
|
|
B3. 2
$$
\Phi_T(x,\,z) = \Phi_+(x,\,z)-\Phi_+(x,\,z-\delta_m)= \delta _m \times \frac{\partial \Phi_+}{\partial z} $$ |
0.20 |
|
|
B3. 3
$$
\Phi_T(x,\,z) = - \frac{\mu_0 m v}{4 \pi} \left[ \frac{v}{((z+h)v- v_0 x)^2 }\left(\frac{z+h}{\sqrt{z^2 + (z+h)^2}}- \frac{v_0}{\sqrt{v_0^2 + v^2}} \right)-\right. $$ $$ - \left. \frac{x^2}{[(z+h)v - v_0 x][x^2 + (z+h)^2]^{3/2}}\right] $$ |
0.30 |
|
|
B3. 4
$$
F_z = - q_m \left. \frac{d}{dx}\Phi_T(0,\,z)\right|_{z=h} + q_m \left. \frac{d}{dx}\Phi_T(0,\,z)\right|_{z=h-\delta_m} $$ |
0.20 |
|
|
B3. 5
$$
F_z = \frac{3 \mu_0 m^2}{32 \pi h^4}\left[ 1 - \frac{v_0}{\sqrt{v_0^2 + v^2}}\right] $$ |
0.20 |
|
|
B3. 6
$$
F_x= - q_m \left. \frac{d}{dx}\Phi_T(x,\,h)\right|_{x=0} + q_m \left. \frac{d}{dx}\Phi_T(x,\,h - \delta_m)\right|_{x=0} $$ |
0.20 |
|
|
B3. 7
$$
F_x = -\frac{3 \mu_0 m^2}{32 \pi h^4} \frac{v_0}{v}\left[ 1 - \frac{v_0}{\sqrt{v_0^2 + v^2}}\right] $$ |
0.20 |
|
|
B4. 1
$$
v_0 = \frac{2}{\mu_0 \sigma d} = \frac{2}{4 \pi \times 10^{-7} \times 5.9 \times 10^7 \times 0.5 \times 10^{-2}} = 5.4 ~м/с $$ |
0.30 |
|
|
B5. 1
$$
\text{В режиме } v< v_c: \quad v_0(v) = v_0 $$ |
0.10 |
|
|
B5. 2
$$
\text{В режиме } v> v_c: \quad v_0(v) = \frac{2}{2 \sigma \delta} = \frac{2}{\mu_0 \sigma}\sqrt{\frac{\omega \mu_0 \sigma}{2}} $$ |
0.10 |
|
|
B5. 3
$$
\omega = \frac{v}{h} $$ |
0.10 |
|
|
B5. 4
$$
v_0 (v) = v_0 \sqrt{\frac{d}{h}} \sqrt{\frac{v}{v_0}} $$ |
0.10 |
|
|
B6. 1
$$
\delta = d $$ |
0.10 |
|
|
B6. 2
$$
v_c = \frac{2 h}{d^2 \mu_0 \sigma} = v_0 \frac{h}{d} $$ |
0.20 |
|
|
C1. 1
M1
$$
\Phi_T (x,\,z) = - \frac{\mu_0 q_m}{4 \pi} \frac{1}{\sqrt{x^2+ (z+h)^2}}+\frac{\mu_0 q_m}{4 \pi} \frac{1}{\sqrt{(x-\delta_m)^2+ (z+h)^2}} $$ |
0.30 |
|
|
C1. 2
M1
$$
F_{z}^{\prime}=\left.\left(-q_{\mathrm{m}}\right)\left[-\frac{\partial}{\partial z} \Phi_{\mathrm{T}}\right]\right|_{x=0,}+\left.q_{\mathrm{m}}\left[-\frac{\partial}{\partial z} \Phi_{\mathrm{T}}\right]\right|_{x=\delta_{\mathrm{m}}} $$ |
0.30 |
|
|
C1. 3
M1
$$
F_{z}^{\prime}=\frac{3 \mu_{0} m^{2}}{64 \pi h^{4}} $$ |
0.40 |
|
|
C1. 4
M1
$$
h_0 = \left[ \frac{3 \mu_0 m^2}{64 \pi M_0 g}\right]^{1/4} $$ |
0.20 |
|
|
C1. 5
M2
$$
F_{z}^{\prime}=2 \frac{\mu_{0} q_{\mathrm{m}}^{2}}{4 \pi}\left[\left(\frac{1}{2 h}\right)^{2}-\frac{2 h}{\left(\delta_{m}^{2}+(2 h)^{2}\right)^{3 / 2}}\right] $$ |
0.60 |
|
|
C1. 6
M2
$$
F_{z}^{\prime}=\frac{3 \mu_{0} m^{2}}{64 \pi h^{4}} $$ |
0.40 |
|
|
C1. 7
M2
$$
h_0 = \left[ \frac{3 \mu_0 m^2}{64 \pi M_0 g}\right]^{1/4} $$ |
0.20 |
|
|
C2. 1
$$
\frac{d F_{z}^{\prime}}{d z}=-k=-M_{0} \Omega^{2} $$ |
0.50 |
|
|
C2. 2
$$
\Omega = \sqrt{\frac{4 g}{h_0}} $$ |
0.30 |
|
|
C3. 1
$$
h_0 = \left[ \frac{3 \mu_0 \left( \frac{4}{3}\pi R^3 M\right)^2}{64 \pi \left( \frac{4}{3}\pi R^3 \rho_0g\right)}\right]^{1/4} = \left[ \frac{R^3 M^2 \mu_0}{16 \rho_0 g}\right]^{1/4} $$ |
0.30 |
|
|
C3. 2
$$
h_0 = \left[ \frac{10^{-18} \times75^2\times10^{-4}}{16 \times 7400 \times 9.8 \times \mu_0}\right]^{1/4}~м $$ |
0.20 |
|
|
C3. 3
$$
h_0 = 25~мкм $$ |
0.20 |
|
|
C4. 1
$$
\Omega = \sqrt{\frac{4 g}{h_0}} = \sqrt{\frac{4 \times 9.8}{25 \times 10^{-6}}}~ с^{-1} = 1.25~кГц $$ |
0.30 |
|