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First Task: Derive numbers for
the mobility µ. |
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First we need typical conductivities
and electron densities in metals, which we
can take from the
table in the
link. |
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At the same time we expand the table
a bit |
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Material |
r [W cm] |
s [W1 cm1] |
Density d × 103 [kg
m3] |
Atomic weight w
[×
1u = 1,66
· 1027 kg] |
n = d/w
[m3] |
Silver
Ag |
1,6·106 |
6.2·105 |
10,49 |
107,9 |
5,85 · 1028 |
Copper
Cu |
1,7·106 |
5.9·105 |
8,92 |
63,5 |
8,46 · 1028 |
Lead
Pb |
21·106 |
4.8·104 |
11,34 |
207,2 |
3,3 · 1028 |
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For the mobility µ we have
the equation |
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With q = elementary charge = 1,60
1019 C we obtain, for example for
µAg |
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µ |
Ag = |
6,2 · 105
1,6 · 1019 · 5,85 · 1028 |
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m3
C · W · cm |
= |
66,2 |
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cm2
C · W |
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The unit is a bit strange, but
rembering that [C] = [A · s] and [W]
= [V/A], we obtain |
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µAg |
= |
66,2 |
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cm2
Vs |
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µCu |
= |
43.6 |
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cm2
Vs |
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µPb |
= |
9,1 |
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cm2
Vs |
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Second Task: Derive numbers for
the drift velocity vD by considering a reasonable field
strength. |
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The mobility µ
was defined as |
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So what is a reasonable field strength in a
metal? |
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Easy. Consider a cube with side
lengt l = 1 cm. Its resistance R is given by |
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A Cu or Ag cube thus
would have a resistance of about 1,5 ·106
W. Applying a voltage of 1 V, or
equivalently a field strength of 1 V/cm thus produces a current of
I = U/R » 650 000 A
or a current density j = 650 000 A/cm2 |
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That seems to be an awfully large
current. Yes, but it is the kind of current density encountered in integrated circuits! Think
about it! |
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Nevertheless, the wires in your
house carry at most about 30 A (above that the fuse blows) with a cross
section of about 1 mm2; so a reasonable current density is
3000 A/cm2, which we will get for about U = 1,5
·106 W · 3000 A = 4,5
mV. |
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For a rough estimate we then take a
field strength of 5 mV/cm and a mobility of 50
cm2/Vs and obtain |
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vD |
= |
50 · 5 |
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mV · cm2
cm · V · s |
= 0,25 |
cm
s |
= 2,5 |
mm
s |
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That should come as some surprise! The electrons
only have to move v e r y s l o w l y
on average in the current
direction (or rather, due to sign conventions, against it). |
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Is that true, or did we make a mistake? |
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It is true! However, it
does not mean, that electrons will not run
around like crazy inside the crystal, at very high speeds. It only means that
their net movement in current
anti-direction is very slow. |
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Think of an single fly in a fly swarm. Even better
read the module that
discusses this analogy in detail. The flies are flying around at high speed
like crazy - but the fly swarm is not going anywhere as long as it stays in
place. There is then no drift velocity and no net fly current! |
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© H. Föll