7. Ohm's Law for a Liquid Belarus IYPT 2002
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Transcript of 7. Ohm's Law for a Liquid Belarus IYPT 2002
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:
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
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
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
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
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
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.
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
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
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.
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).
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.
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.
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.
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.
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.
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.