Electric currents.
11
Variables and units
Variable |
symbol |
Units |
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Current |
i, I |
Ampere |
[A] |
Current density |
J |
Ampere/ meter2 |
[A/m2] |
Conductivity |
σ, γ |
Siemens/meter |
[S/m] |
Resistivity |
ρ |
Ohm / meter |
[Ω/m] |
Conductance |
G |
Siemens |
[S] |
Resistance |
r, R |
Ohm |
[Ω] |
12
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Basic definitions |
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Current: |
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Q |
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I t |
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I J S |
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Current density |
Jn |
I |
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S |
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Current density depends on the |
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velocity and density of moving charges |
J ve e |
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13
Conductivity and resistance
J ve e
ve e E
J Ne e ve conductivity
Ohm’s law:
U R I
e |
is the mobility of electrons |
JNee e E
Nee e
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J E |
- differential form |
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R |
l |
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- Integral form |
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S |
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Power and Joule’s Law
Consider a charge Q moves at a velocity v by an electric field E to a distance l. In this case, the expression of the work done is:
W f l QE l |
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W |
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Power: |
P t |
QE v |
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V |
so: |
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Total charge: Q edV |
P e E vdV E JdV |
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V |
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V |
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For a constant cross-section conductor, the expression of the volume is |
dV dl dS |
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P E JdV J dS E dl
V |
S |
l |
Expression for the power |
P I U I 2 R (Joule’s law) |
15
Continuity Equation
Consider a closed surface ,where a wire carries a
current Iin in the surface and the current Iout goes out from the surface.
General definitions: |
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Iin dQ |
Iout |
Iin dQ |
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dt |
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dt |
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Iout Jout dS |
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edV |
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Jout dS dQ |
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J |
dS |
d |
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S |
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S |
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dt |
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S |
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dt |
V |
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J dS JdV |
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d e |
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Gauss theorem: |
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JdV |
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dV |
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dt |
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S |
V |
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V |
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V |
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Continuity equation |
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d e |
for steady currents |
J 0 |
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J |
dt |
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16
Continuity Equation
Important!: J is the current of conductivity
If displacement current are taken into account:
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dD |
J |
dt |
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Q dV |
divD |
The Gauss Law for the field displacement |
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V |
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d |
d |
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d |
dD |
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Time derivative: |
dt |
dt divD |
dt div dt |
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Continuity equation |
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div 0 |
dS 0 |
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J d e 0 or |
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dt |
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S |
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17 |
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