Download - 7. Ohm's Law for a Liquid Belarus IYPT 2002
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It is often said that electric current „flows“. Is this the only analogy between electric current and the flow of a liquid? Investigate theoretically and experimentally other analogies between these two things.
Condition of the problem:
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0
zj
yj
xj zyx
Where j – current density (it equals to a charge, which is transferred through unit of cross-section in time unit).
movement of some quantity of substance in a space in time. :
0
zyxzyx
Value equals to liquid mass, which is transferred through unit of section in time unit.
The first Kirchgof's law (law of conservation of energy)
Electric current and a liquid flow.
Law of mass conservation:
–+
––
+
movement of charged particles (electrons or ions)
m q
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Electric current and a liquid flow.
Ohm's law: xjx
yj
y
yjy
- electric potential,
- conductivity of given medium
Liquid movement.
tzyx ,,,
xx
yy
zz
Components of velocity:
Current.
02
2
2
2
2
2
zyx
02
2
2
2
2
2
zyx
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Similarities:
• .Electric potential is analogous to potential of velocities .
• .Value corresponds to value .
• .Equipotential surfaces in electric current (x, y, z)=const are analogous to equipotential surfaces in liquid flow (x, y, z)=const.
j
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Determinate conditions.
1. Liquid flow should be irrotational.
zv
yv yz
x 21 0
yzzy
2. Mentoined formulas model fluid nonviscous flow.
Newton’s formula:
n
where η - dynamic (absolute) coefficient
of velocity; - velocity gradient on normal to
ground, where stress acts;
vn
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Movement of liquid in a tube.
ll
Re
Poiseuille’s equation (11):
421
128d
lppV
where p - pressure in the given point of a tube; V - discharge of liquid; l - length of the corresponding section of a
tube.
Reynolds number:
For laminar flow Re crRe<
Critical Reynolds number is 2300.
2212121
4d
lS
lRI
Corresponding express for the part of electric Corresponding express for the part of electric circuitcircuit::
p
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Experimental setup
Pressure was determined by height of liquid lifting in a glass tube. Reynolds number calculated by experimental dates was not higher than some hundreds. From this follows, that we can consider flow as laminar.
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0 100 200 300 400 500 600 700
05
1015202530354045505560657075808590
H, m
m.
L, mm.
0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,11 0,12
0,02
0,04
0,06
0,08
0,10
0,12
0,14
dp/d
l, re
l. un
.
dm/dt, rel. un.
Dependence of height of liquid column in glass tube
on lenght of conductor L for different liquid velocities.
Dependence of ratio on
discharge of liquid
dpdl
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Tubes with different diameters and lengths.
Mariotte’s vessel. This setup is guaranteed constant pressure difference and, therefore, constant discharge in the process of experiment.
"Subcurcuit"Tube №1Tube №2Series connectionParallel connection
"Resistance"
Experimental setupExperimental setup
160420590125
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0,0 0,1 0,2 0,3 0,4 0,5
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
experiment theory
Dis
char
ge o
f wat
er (k
g/m
in)
Pressure difference (atm)
The dependence of discharge on pressure difference in the ends of one conductor.
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Dependence of resistance on
number parallel tubes.
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,60
10
20
30
40
50
experiment theory
"Res
ista
nce"
(m2 s-1
kg-1)
Length of pipe (m)
1 2 3 40
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
theory experiment
"Res
ista
nce"
(m2 s-1
kg-1)
Number of equal resistors
Dependence of resistance on length of tubes (series connection).
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Bridge circuit
• Condition of current absence in a conductor R5:
4
3
2
1
RR
RR
Experimental setup.Experimental setup.Liquid in connection element does not move, when lengths
of corresponding tubes are proportional or equal.
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Alternate currentAnalogues to an induction coil and a condenser.
Condenser properties:• It doesn’t let current pass, but collects
charge on its plates until breakdown.• Creates electrostatic field between plates.• Charged condenser has energy.
Induction coil properties:• Coil prevents from sharp change of electric current.• It also has energy if electric current flows through it.
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Field transistor
• This element, depending on potential of bibb, conduct or not electrical current.
In the point A constant head pressure is created. Depending on the pressure in the point B plunger let or not pass water.
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Oscillatory circuit.
• “Transistor” plays a role of feed-back. Current in the induction coil corresponds to the liquid movement in section CD. Energy of charged condenser is replaced by potential energy of liquid.
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Logical elements:Element Element ““AndAnd””.. Element Element ““OrOr””.. ““XorXor”” element element..
Only in the presence of overpressure in each of cross-sections 1, 2 liquid flow is taken place.
Liquid will move from point A to point B in the presence overpressure in one or in both of the cross-sections 1, 2.
Liquid moves from point A to B in the presence overpressure only in one of cross-sections 1, 2, not in both.
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In fact the flow of a liquid and electric current are different. They are different in
their physical meaning at first. But if try we can find out some similarities which appe-
ars after some analogues were made.