L11 Using the Current and Coulomb Balance to Measure the Speed of Light

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L
11

Using the Current and Coulomb Balance

to Measure the Speed of Light



Lab Description

This lab consists of two data gathering sections.

1.

The Current Balance

2.

The Coulomb Balance

Each section may take an entire lab to complete.

When the two data
gathering sections are completed, calculations can be made to
compare with the known speed of light.

Equipment Needed for Part 1: The Current Balance

Calipers, Digital

Coulomb Balance, Sgt.W.

Set up with current rods

Laser, HeNe

Leads, Banana Jack

Mass Set, Fractional

Meter Stick

Power Supply, Elenco

Power Supply, Laser HeNe

Ring Stand, 120cm

Ruler, 15in

Table Jack

Tape Measure

Equipment Needed for Part 2: The Coulomb Balance

Calipers, Digital

Coulomb Balance, Sgt.W.

Set up w/ capacitor plates

Laser, HeNe

Lea
ds, Banana Jack (3)

Mass Set, Fractional

Meter stick

Power Supply,
Pasco SF
-
9585A

Power Supply, Laser HeNe

Resistance Substitution App.

Ring stand, 120cm

Ruler 15in

Table Jack

Tape Measure

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Laser Safety

Although these lasers are not particularly dangerous, we should take a few simple
precautions to prevent the unlikely event of eye damage.

1.

Never look directly into the laser beam.
Laser light has a high intensity and can
also be easily focuse
d. A direct shot of the laser beam on your eye will be
focused by your cornea onto a small spot on your retina and can burn or possibly
detach the retina.

2.

Never hold a reflecting object by hand in front of the laser beam.
This
prevents the possibility of
accidentally shining the light into your eyes.

3.

Keep your head above the plane of the laser beam.

4.

Whenever the light strikes an object, there will be a reflection. At times the
reflections can be almost as strong as the incident beam.
Know where the
refl
ections are and block them if necessary.

5.

The laser has a shutter in front of the beam.
When not taking data, place the
shutter in front of the laser beam.



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Part 1 The Current
Balance

Assembly Note

This is a delicate instrument and particular care should be taken to avoid
bending the
8
1

inch rods of the moving frame and the stationary
conductor. The damping vane should be adjusted so that it will not rub
against the damping magnets or the edges of the slot in the base.

The Current Balance


Significance of the Current Balance

The mks sy
stem of units defines the ampere in terms of the force existing
between two parallel conductors carrying current. The current balance
makes this fundamental measurement. The ampere is defined as follows:
“One ampere is that unvarying current which, if p
resent in each of two
parallel conductors of infinite length and one meter apart in vacuum,
would produce between these conductors a force equal to
7
10
2



newtons per meter of length.”

Method of Operation

Pass the current to
be
measured in
opposite directions through two parallel
horizontal bars that are connected in series. The lower bar is fixed;
balance the upper bar a few millimeters above it by adjusting the
counterpoise. The upper bar supports a small pan into which analytical
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weight
s are placed, thereby causing the upper bar to drop down toward the
lower one.

Typical Setup



When the current is turned on and increased sufficiently, repulsion
between the two bars causes the upper bar to rise to its initial equilibrium
position. Obse
rve the position of the beam by means of the mirror
mounted on the beam and an auxiliary telescope and scale placed 1 to 1½
meters from the mirror. The current is thus measured in terms of the
weight it will lift. Use weights up to 200mg. Plotting weigh
t against the
square of the current should give a straight line, the slope of which may be
varied by changing the initial adjustment of the counterpoise. Currents
above 20 amperes may cause excessive heating. We will be using a DC
power supply.


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Usefulne
ss

The simple, straightforward current balance has been designed primarily
for use as a teaching instrument rather than a high precision measuring or
calibrating device. In the laboratory, it provides a fundamental
quantitative experiment for students, wi
th results accurate to within a few
percent. Like any sensitive balance, it must be used on a solid table in a
draft
-
free location. Its sensitivity challenges the student to use his best
technique, yet operation is rapid enough to provide adequate data i
n a
normal laboratory period. In a demonstration the instrument may be used
to show that wires carrying current attract or repel depending on relative
current direction, and that repulsive force increases with current. For this
lab we will use the appara
tus in a quantitative manner.

Principle of Operation

The force between two long parallel conductors carrying equal current is
given from the definition of the ampere by

2
7
10
2
I
d
L
F











Equation 1

where:

F

is the force in newtons (i.e., the weight in the pan converted from
milligrams to newtons. By definition, a Newton (N) is the force
that will impart to a mass of one kilogram an acceleration of one
meter per second per second. Thus,
dynes
N
000
,
100
1

, and a
weight of one milligram
N
5
10
98
.
0



)

L

is the length of the conductors in cm (the length of the shorter
bar on which the weight pan is mounted)

d

is the distance between the conductors in cm when in the zero
current equilibrium positio
n (center to center, to be measured by
optical deflection, as described later)

I

is the current in amperes.

Since this amounts to

2
kI
F



Equation 2

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where the constant

d
L
k
7
10
2





Equation 3

the apparatus may be used as follows

A.

When
F

is plotted against
2
I
, a straight line should be obtained.
The value of
k

may be determined from the slope of this line, or it
may be obtained by taking the mean value of the ratio
2
I
F

from
the observed data,
which is equivalent to the slope.

B.

The value of
k

may be computed independently, using Equation 3,
and compared with the first value.

The counterpoise may be readjusted to give a slightly different zero
-
current equilibrium position; then
k

will have a new v
alue to be
determined by methods A and B and the results compared.


Figure 1

Schematic Block Diagram



Procedure

Note:

Much of the setup is already done by the lab tech or your instructor.
You should, however, do a quick check to make sure it has been
done.

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It is very important that lead wires connected to binding posts on the
balance leave them at right angles with the

conductors that are part of the
frame. The circuit in Figure 1 provides repulsion between the two parallel
bars as required in the experiment. By rewiring the circuit so the current
goes the same way in the two bars they may be made to attract each othe
r
for demonstration purposes.

After placing the balance on a firm table and adjusting the leveling screws
to position the base firmly, replace the frame and adjust the counterpoise
behind the mirror until the frame oscillates freely and comes to rest with
the front horizontal bar a few millimeters above the stationary bar. Adjust
the counterpoise below the mirror until the period of oscillation of the
frame is 1 to 2 seconds. It should come to rest in 10 to 15 seconds when
the poles of the damping magnets

are about 2mm apart.

To check the two conducting bars for alignment, place a coin on the scale
pan to bring the bars into contact without distortion. Thumbscrews on
each front post permit either end of the lower bar to be raised or lowered.
Similar thum
bscrews on each block at the rear permit either end of the
upper bar to be moved forward or backward. The two bars should be
aligned as accurately as can be determined by the unaided eye when
viewed from the front and from the top. When viewed from the f
ront,
with a white paper behind the bars, the two bars may appear to be slightly
bent. If this very serious, let your instructor or the lab tech fix it for you.
It is almost impossible to get them so straight that no light may be seen
between them, but t
his is not essential for good quantitative results.
Nevertheless, the bars are rather easily adjusted and this inspection should
be made before every trial. In general, the bars should always be handled
gently and as little as possible.





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Block Diagram

of Setup


After setting up the laser and scale 1 to 1½ meters from the mirror, remove
the coin from the weight pan and record the rest point indicated by the
position of the laser spot on the scale. Engage the beam lift gently; then
release it and again

record the rest point. If it deviates from the first
observation, the knife
-
edges may not be clean, the base or table may be
unsteady, or the balance or laser may have been jarred.

Measure the length
L

of the upper front bar from center to center of its
two
supporting bars extending from the knife
-
edges. See Figure 2. Also
measure the lever arm (a) or distance from knife
-
edge to center of the
front bar at each side, and take the mean.

The separation of the two bars at equilibrium is determined in the
fo
llowing manner. The scale reading at equilibrium is noted. Then the
upper bar is depressed (by placing a coin on the scale pan) until it is in
contact with the lower bar, and a new scale reading is noted. Simple
geometry will show that the separation

b
aD
d
2
0



Equation 4

where

D

is the difference in readings

a

is the mean distance from knife
-
edge to bar

b

is the distance from mirror to scale

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Center to center distance,
d
, is the sum of the radii of the bars and
0
d
. Bar
d
iameters may be measured with a digital caliper.

Figure 2
: Critical Measurements


To begin a trial, add a weight to the pan, increase the current until the
beam returns to its equilibrium position, and read the ammeter. Repeat
using successive
increments of weight; a range between 20 and 200 mg is
recommended. The ratio of weight to current squared may be calculated
for each measurement; this ration should be constant. Another method is
to plot weight vs. current squared and find the slope of
the best
-
fit line
(which should pass through the origin); erratic readings may thus be easily
detected.

For a second trial, change the counter poise slightly to make the separation
of the conducting bars slightly different. Record the scale readings at
co
ntact and at equilibrium, and take another series of readings. When
setting the counterpoise and positioning the stationary conducting bar, be
sure that the equilibrium distance between the bars is great enough to
prevent them from contacting under the he
aviest weight, but not so great
that excessive current (over 20 amperes) will be necessary to restore the
balance to equilibrium. Higher current readings will tend to drift due to
ohmic heating.

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The purpose of the beam lift is to relocate the knife
-
edges
to their proper
positions on the bearing posts so that the front movable conductor will
always be in the same vertical plane as the fixed conductor below it.
Lifting the beam each time a weight is added or removed avoids the risk of
jarring the beam. Ope
rate the lift mechanism very carefully when raising
and lowering the beam. Using the beam lift may cause a slight change in
equilibrium position (less than a 0.1 mm change in distance between
conducting bars), since microscopic irregularities on the surfa
ces of the
knife
-
edges and bearing posts contribute to a slight change in friction on
the system. The slight experimental error that results may be avoided by
changing the weights without lifting the beam. When using this method,
handle the weights caref
ully with forceps; if the beam is jarred, the
alignment procedure must be repeated and the trial started over. Avoid
vibrations, wind currents, and other such disturbances whenever possible.

The earth’s field will cause erroneous results when using direct

current.
This problem can be avoided by orienting the conducting bars parallel to
the earth’s field. The effect can also be eliminated by inserting a reversal
switch in the circuit and taking two current readings for each load (one for
each direction) a
nd using the mean current value in computations. If good
technique is used, the experimental error should be no more than 5%.

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Part 2 The Coulomb Balance

Description

The Coulomb Balance is designed to determine the electrostatic constant
(permittivity in
free space,
0

) by measuring the electrostatic force
between two oppositely charged plates.

The Coulomb Balance consists of two flat aluminum plates with attached
aluminum mounting rods. Two short rods are also supplied for attaching
t
he fixed lower plate to the balance assembly.

The Coulomb Balance


The Plates replace the conductor rods on the Current Balance and are used in
measuring electrostatic forces in a manner similar to the use of the conductors in
measuring electromagnetic fo
rces.

Theory

Maxwell predicted that electromagnetic fields should travel with the speed

0
0
1



or

0
0
1



This value has been proven identical to the value obtained by Michelson for the
speed of light, there supporting the th
eory that light is an electromagnetic
radiation, and suggesting that measuring
0


and
0


is an indirect way of
determining c, the speed of light.

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This electrostatic constant is the constant of proportionality in the
relationship for
expressing the force between electrostatic charges, just as the gravitational
constant is the constant of proportionality in the similar relationship for
expressing the force between masses. Thus, the Coulomb Balance experiment is
the ele
ctrical analog of the Cavendish gravitational experiment.

The force of attraction between two oppositely charged, parallel plates of equal
area is given by

2
2
0
2
d
AV
F




Equation 1

Note:

For a full explanation and derivation of Equation 1 see p
age 16.

where

F

is the force of attraction in Newtons

0


is the electrostatic constant in farads/meter

A

is the area of plates in
2
m

V

is the potential
difference between the plates in volts

d

is the distance between plates in meters

The value of
d

should be small. If
d

becomes large relative to the area of the
plates, edge effects become significant, lack of parallelism between the
plates
increases, and Equation 1 is no longer valid.

Equation can be written

2
0
2
2
Fd
AV
 


Equation 2

Hence, with
d

and
A

remaining unchanged, by measuring
V

for various values of
F
, and average value of
0


can be determined.

In the experiment with the Current Balance which follows, the data obtained are
used to compute an arbitrary constant,
c
k
, which can be shown to be related to the
electromagnetic constant,
m
k

(MKS system) by

c
m
k
L
d
k
2



Equation 3

since
7
10


m
k
. The terms are defined in the Current Balance experiment.

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Since

m
k


4
0


Then, from Equation 3,
0


is seen to be

c
k
L
d


2
0



Equation 4

With the data from the two experiments, the two constants,
0


and
0

, can be
computed, and from their values, the speed of light can be determined.

Set Up

Carefully determine and mark the center of the upper plate.

M
easure the dimensions of the plate and compute its area. Attach the two plates
to the balance. The plates must be attached so that the horizontal supporting rods
are in the outside surfaces of the plates and not between them. Your lab tech or
instructor

will explain this.

Schematic Diagram


With the upper and lower plates aligned and with the knife edges properly
positioned on the bearing surfaces, adjust the counterpoise behind the mirror until
the frame with upper plate oscillates freely and comes to
rest with the upper plate
a few millimeters above the lower one. Adjust the sensitivity by means of the
counterpoise below the mirror until the period of oscillation
is lessened as much
as possible.

(You shouldn’t have to do these adjustments because the
y have been
done previously. There shouldn’t be much oscillation in the system after it has
been set up.)

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You should make
sure

the balance is on a solid table in a draft
-
free location.
Connect the high voltage DC terminals of the power supply to the term
inals on
the balance that connect to the plates. Connect the megohm resistor in one branch
of this circuit to limit the current to prevent possible damage to the power supply
in case the two plates accidentally touch. The power supply we are using in our

setup has the voltmeter on it.


The Coulomb Balance
: Typical Setup



The precise alignment and parallel positioning of the plates are carried out by the
same procedure described for aligning and adjusting the conductors in the
instructions for the Current Balance. The setup and the use of the telescope and
the scale to det
ermine the separation of the plates are also described in those
instructions. For convenience, the formula is repeated here. The separation of the
plates,
d
, is given by

b
Da
d
2



Equation 5

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where

D

is the difference in readings of the scal
e for the contact and equilibrium
position of the plates

a is the perpendicular distance from the line of the knife edges to the center of
the plates

b

is the distance from the mirror to the scale

Diagram

for measurements.


Procedure

1.

Measure and record th
e values of
a

and
b
.

2.

Place sufficient weight (a small coin) anywhere on the upper plate to bring it into
contact with the lower one.
Note:

Be sure the power supply is turned off during
any time the plates are in contact. Record the reading of the scale.

3.

Place a 50 mg weight on the upper plate at the exact center (which was
determined and marked earlier) and record the scale reading.

4.

Remove the 50 mg weight and let the plate come to rest. Turn on the power
supply and gradually increase the voltage until
the plate separation is reduced to
the same value (determined by watching the scale reading) as it was with 50 mg
on the upper plate.

5.

The electrostatic force of attraction between the plates is


newton
g
mg
4
10
9
.
4
50




.

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6.

From the scale readings obtained in steps 2 and 3, compute
d
, the separation of
the plates.

7.

Compute the electrostatic constant
0


using Equation 2.

8.

Make several trials using the same or slightly different masses. Compute the
average value of
0


and compare with the accepted value. Discuss the possible
reasons for lack of agreement.

9.

Use this value of
0


and the

value of
0


obtained from the data of the Current
Balance experiment to compute the speed of light.

Value of Constants

1.

meter
farad
or
newtonm
coulomb
2
2
9
0
10
98776
.
8
4
1






2.

meter
henry
or
meter
ampere
weber




7
0
10
4



3.

sec
10
9979
.
2
8
m
c



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Derivation of Equation 1

(from Page 11)

Excerp
ted from Larry Smith’s website at
http://www.snow.edu/~larrys/PHSX222L/PHSX222LLabs/ElectrostaticForce.html

Theory:

Consider two parallel metal plates of area A and separation d. If these plates
carry equal and opposite charges of magnitude q, then it can
be shown that the electric
field between the plates is

0
tot
q
E
A




Equation 1

When an electric charge is brought into an electric field produced by another charge it
experiences a force given by

one
F qE



Equation 2

Thus, the force of attraction between two parallel plates is the charge on one plate
multiplied by the field due to the other plate (which is one
-
half the value given by the
above formula). The force attracting the plates together can be found by solving f
or q in
equation
1

and substituting in equation
2
.

2
0
2
E A
F




Equation 3

The electric fiel
d E can be calculated from the potential difference
Δ
V:

V
E
d




Equation 4

The force attracting the plates together can thus be written as

2
0
2
( )
2
A V
F
d




Equation 5

All quantities in this expression except

o

are susceptib
le to measurement. Thus

o

is
determined.

There are two ways to obtain the plate separation d: direct measurement with a ruler, and
using the formula

2
Da
d
b



Equation 6

where D is the distance between the laser beam spots for the contact and equilibrium
positions of the plates, a is the perpendicular distance from the line of the knife edges to
the center of the plates, and b is the distance from the mirror to the laser b
eam spot on the
wall.

The value of d should be small; if it becomes large relative to the area of the plates then
edge effects become significant and lack of
parallelism

increases (and then equation
5

isn't valid).