Angular function
Equation for the angular function: |
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2 S( ) |
k 2 |
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S( ) |
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or: |
2 S( ) |
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S( ) 0 |
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k |
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Solution: |
S( ) g cos(k ) hsin(k ) |
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0 360 |
Evidently k is an integer number. |
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Radial function
Equation for the radial function: |
r |
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r |
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R(r) r |
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R(r) k 2
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R(r) |
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R(r) |
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or: |
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r |
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k |
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Let us try to find a solution of this equation by substituting: R(r) r
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r k 2 |
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2 k 2 |
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r |
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The solution: |
R(r) crk dr k |
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General solution of the Laplace’s equation in a cylindrical coordinate system
Combining the solutions we will get:
Uk (r, ) crk dr k g cos(k ) hsin(k )
This function is known as circular (angular) harmonic of order k.
General solution:
U (r, ) ck rk dk r k gk cos(k ) hk sin(k )
k 1
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Application of the variable separation method for the magnetic field modeling
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(2) |
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i |
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C |
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A |
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The scalar magnetic potential exists in the domain (1), but does not exist in the domain (2).
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Reduced scalar magnetic potential
(редуцированный скалярный магнитный потенциал)
Magnetic field intensity may be presented as
M |
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J |
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H (r ) Hc (r ) |
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Hm (r )
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is the field intensity induced by the current sources ( J ) |
Hc (r ) |
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is the field intensity induced by the magnetized objects (M ) |
Hm (r ) |
Hm (r )
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is the potential field: |
Hmdl 0 |
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A special potential may be introduced: |
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r0 |
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Um (r ) Hm |
dl |
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r |
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Hm (r ) Um
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Combination of scalar magnetic potential and the reduced magnetic potential
M |
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J |
Reduced magnetic potential exists in the whole space
Scalar magnetic potential exists only in a simply connected
domain with no currents inside
Inside the M |
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U is the scalar magnetic potential |
- domain: H (r ) U |
Outside the M - domain: |
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Hc |
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H (r ) Um |
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J (r |
(r |
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The field induced by current |
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dr |
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sources may be calculated |
Hc |
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by Biot – Savart Law: |
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4 J |
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r |
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Combination of scalar magnetic potential and reduced magnetic potential
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J |
(1) |
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Inside the magnetized domain (1) the scalar magnetic potential satisfies the differential equation
(1) : U 0
If const |
U 0 |
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In the domain (2) the reduced magnetic potential satisfies the differential equation
(2) : |
Um |
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This equation is valid in |
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the domain (1) as well. |
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Combination of scalar magnetic potential and reduced magnetic potential
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- is the border of the domain |
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with the magnetized |
matter |
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J |
Inside the magnetized domain |
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Hc dl |
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(2) |
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Scalar magnetic |
Uc |
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potential of the currents |
(r ) Hc |
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r |
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0dl
Solution of the problem should provide boundary conditions on
1) H (1) H (2) 2) Bn(1) Bn(2) Hn(1) 0 Hn(2)
H (1) |
H ( 2m) H (2)c |
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Hn(1) 0 Hn(2m) 0 Hn( 2c) |
U (1) |
Um(2) Uc(2) |
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Magnetic field of the line current near a magnetized cylinder
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0 |
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(2) |
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Um , Hc |
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C |
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b |
U |
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The final goal is to express the potentials by expansions
U rk gk(1) cos(k ) hk(1) sin(k )
k 1
Um r k gk(2) cos(k ) hk(2) sin(k )
k 1
Uc
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The scalar potential induced by the current line
0 |
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Hc |
Uc Hdl |
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2 r |
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r |
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i
Let us choose a point (line) of zero potential here
Arc length = |
r |
The field intensity along the arc = const = |
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2 r |
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The potential in the point |
r : |
Uc |
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2 r |
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The dependence |
Uc is anti-symmetric |
with respect to the angular coordinate: |
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Uc Uc |
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