Divergence of a vector field.
Object of operation is a vector field Result of operation is a scalar
Designation: div( A) A
In the Cartesian coordinate system:
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div( A) |
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Circulation of a vector field.
curl( A) A
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curl( A) |
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Properties of the differential operators.
Differential operations are linear:
grad( U V ) grad(U ) grad(V ) div( F G) div(F ) div(G)
curl( F G) curl(F) curl(G)
Important Vector Identities:
div curl(F ) 0 |
curl grad(U ) 0 |
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Differential operator Nabla.
In the vector algebra an operator is often used
grad (U ) U div(F ) F curl(F ) F
In the Cartesian coordinate system:
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Integral theorems.
div(E)dV E dS
V S
grad (U )dV dS U
V S
curl(E)dV dS E
V S
Different forms of the Gauss theorem
curl( A) dl A ds
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Stoke’s theorem
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Electrostatic field.
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Variables and units
Variable |
symbol |
Units |
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Charge |
q, Q |
Coulomb |
[C] |
Linear charge density |
τ |
Coulomb / meter |
[C/m] |
Surface charge density |
σ |
Coulomb / meter2 |
[C/m2] |
Volume charge density |
ρ |
Coulomb / meter3 |
[C/m3] |
Electric moment |
p |
Coulomb · meter |
[C·m] |
Displacement |
D |
Coulomb / meter2 |
[C/m2] |
Potential, Voltage |
U |
Volts |
[V] |
Electric field strength |
E |
Volt / m |
[V/m] |
Capacity |
C |
Farad |
[F] |
Electric permittivity |
ɛ |
Farad / m |
[F/m] |
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Coulomb’s Law
Coulomb’s law consists of two statements:
1. The force between two charges q1 and q2 is proportional to both q1 and q2 and also inversely proportional to the square of the distance between them
F ~ q1 q2
r122
2. The axis of the force lies on the direct line between the charges; it is repelling for like charges, and attractive for opposite charges.
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4 0 |
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Electric Field Strength E and
Displacement Field D.
The electric field strength (intensity) is described by a vector quantity represented by the symbol E. It is defined as the force in the field per unit charge
E |
F |
unit - V/m |
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Field induced by the point charge |
E |
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defines a position of the point charge |
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Displacement vector |
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is also called |
D E |
unit |
- C/m2 |
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electric flux density |
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Gauss’ Law.
The electric flux: |
D dS |
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Total electric flux passing any closed surface is equal to the total charge enclosed by that surface
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Integral form: |
D ds q |
Gauss law for the field displacement
Differential form: |
divD |
q
E ds
Gauss law for the field intensity
divE
0
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