Download THE ABSOLUTE DETERMINATION OF εo Maxwell`s unification of

THE ABSOLUTE DETERMINATION OF εo
Maxwell's unification of optics and electromagnetism led directly to the result μoεo = 1/c2
where μo and εo are defined, respectively, by the static magnetic and electric forces and c is the
velocity of light. A determination of εo, since in S.I. μo is a defined constant is therefore equivalent
to a measurement of the velocity of light by purely electrical means. In its present form, the
experiment was first done by Rosa and Dorsey in 1907. It is termed an "absolute" experiment
because an electrical capacitance is determined without reference to other capacitors; from a
knowledge of its geometry εo is deduced.
The capacitor is constructed from two lengths of precision pipe mounted concentrically. The
capacitance per unit length of coaxial cylinders is 2πεo/log(b/a), where b and a are the facing radii. εo
is 8.855 pF/m, and our 1 meter long capacitor is about 1000 pF.
The capacitance is measured using a bridge circuit: 
times a second the capacitor is charged and discharged
by a driven switch. (We use a solid state circuitry
black box to allow much higher working frequencies
than Rosa and Dorsey could attain).
Since the power supply has negligible resistance, the
charging time constant is determined by R4 and the
voltage
at
the
detector
is
V(e t / R 4C  R 3 /( R 2  R 3 )) in each initial half
period.
The discharge occurs very rapidly in the second half periods while that arm of the bridge is
open. If  R4C<<1, the output will be continuous into the second half periods, and at balance the
time average of this voltage will be null. This implies C=R3/  R2R4, so that 1/  C is effectively the
resistance R1 of an ordinary Wheatstone bridge. Note that the balance condition implies that
R3
.
R3<<R2, and more precisely, C =
 R 4 (R 2 + R 3)
It is convenient to use fixed resistances of 100 K and 1 K for R2 and R3 and to balance the
bridge using a 10 K box for R4. The ratio R2/R3 can be determined by substituting another box for
the capacitor and switch, and finding the balance.
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The mains frequency is held by the Hydro at 60 Hz to 1 part in 104 or better, so it makes a
convenient frequency standard. The oscillator driving the switch may be set to a high harmonic of
60Hz by examining the Lissajous figure of its output vs. a small voltage from the mains. Anything
between the 20th and 30th harmonic is a suitable frequency for use with a 10 K box. (The oscillator
will drift badly until it is warmed up).
In assembling the bridge, note that four components are plugged into the mains: the bridge
supply, the switch supply, the oscillator and the scope. You must therefore exercise care in avoiding
pickup. Use an isolation transformer on the oscillator output. Try interchanging connections which
are nominally equivalent to reduce the 60 Hz to a minimum. You may want to use batteries to power
the bridge.
Examine the output of the bridge. You will see the R4C decay and also a smaller blip which
is the switch reversing. Make this stable by adjusting the oscillator level. Note the maximum bridge
voltage permissable before instability spoils the signal. In order to ascertain when the time average
voltage is zero, shunt the scope with about 5 μF. This essentially integrates the voltage. At balance,
the d.c. level will return to zero at the end of each cycle. By varying the bridge voltage quickly you
can establish the null to better than 1 in 104 on R4 with the scope at its highest sensitivity.
The result must be corrected for the capacity in the switch itself and the edge effect of the
fringing field. Dismantle the capacitor and replace the long conductors with the short ones. Find a
balance using a 1MΩ pot for R4. The ohm-meter will read the setting with sufficient accuracy to
make the correction. Measure the dimensions of the long and short capacitors. Calculate εo.
Consider the likely errors. How sensitive is the capacitor to the dismantling? You can
reassemble it and evaluate the reproducibility (use a different frequency to avoid prejudice). Note
that the error in C is only 1/2 that in εo, which is always a good feature.
GMG /82
Rev. 85 GMG
Rev. 86 TED
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