Vector magnetic potential
Main equations:
divB 0 |
B H |
curl H J |
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Consider a vector A satisfying a relation: |
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B curl A |
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div curl A 0 |
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The equation for the flux density will be |
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automatically : |
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div B 0
A- Vector magnetic potential
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Magnetic flux
Definition of the flux
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B ds |
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S |
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curlA |
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ds |
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S |
(Stokes theorem) |
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Adl
l
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Differential equation for the
Vector magnetic potential
Ampere’s Law: |
curl H J |
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1 |
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Magnetic field intensity and flux density are related by: |
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B |
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Taking into account B curl A , |
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1 |
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we get: |
curl |
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curlA J |
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For the constant magnetic permeability: |
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curl curlA J |
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Identical transformation: |
curl curlA grad divA 2 A |
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grad divA 2 A J
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Gauging of the vector magnetic potential
The vector potential defined by the relation B curl A is not unique
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Adding a term of F grad to the value of the vector potential does not change the flux density, because
curl grad 0
The choice of the exact adding is called gauging
The most often ‘Coulomb gauging’ is used
divA 0
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Integral presentation of the vector magnetic potential
For the Coulomb gauging of the vector |
grad divA 2 A J |
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potential we get |
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2 A J |
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In Cartesian coordinate system this vector equation results in 3 scalar ones:
Ax J x |
A J |
y |
Az J z |
y |
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compare: |
U |
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Integral presentation of the vector magnetic potential
Each scalar equation has an integral solution of:
Ax |
0 |
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J xdV |
Ay |
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J y dV |
A |
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0 |
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J |
dV |
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z |
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4 |
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4 |
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r |
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r |
z |
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r |
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compare:
We can unite these expressions into one vector formula:
U |
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dV |
4 |
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r |
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0 |
JdV |
A |
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r |
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Inductance.
Inductance is a coefficient between the current and the flux linkage.
L i
Units: Henry [Hn]
The magnetic energy stored in a coil is derived as
Wm L i2
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Mutual inductance.
A coefficient between the current in one coil and a flux (flux coupling) in another coil is called a ‘mutual inductance’
21 M 21 i1
Reciprocity principle |
(принцип взаимности): |
M12 M 21
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Inductance of thin contours
The inductance may be split in two parts – external and internal
External flux definition: |
ext ext A2dl2 |
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l2 |
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Field intensity inside a cylindrical conductor
Infinitely long cylindrical wire with the radius of R, and the
current of i
The field intensity inside the conductor at the point |
i |
r |
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H |
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with the radius of |
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satisfies |
the Ampere Law |
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Hdl i |
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Directions of the intensity vector and dl vector are he same. |
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H 2 r J r 2 |
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Current density: |
J |
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Field intensity: |
H i r |
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R2 |
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2 R2 |
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