Transcript Measuring the exponent in Coulomb’s law

Measuring the exponent in
Coulomb’s law
2
)/r
Fc= k (q1q2
k = 1/(4πεο)= 9.0X109 Nm2/C2
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Coulomb’s original Torsion Balance
Coulomb’s work was
published in 1783, but
earlier work was done by
Priestly, who first proposed
the inverse square
relationship between
charge separation and
force.
Cavendish also worked
in this field but didn’t
publish his work.
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A More Recent Apparatus
Lab apparatus at Delaware Technical and Community College
• Charged Pith balls
repel as A is forced
towards B.
Deflection angle
measures force
A
B
• Plotting log(Force)
versus log(spacing)
gives exponent of
spacing
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A different approach
• Balloons are used as the charged object since:
– they stay charged for a long time
– they can hold a lot of charge with very little mass
• In a “Pendulum” arrangement, the Electric,
Gravitational and Tension forces balance.
• Balloon displacement (from vertical) measures
Electric Force
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Basic Geometry (to determine q alone)
Coulomb’s Law:
Fc= kq1q2/S2
θ θ
L
Fc
Fg=mg
Since ΣF=0 ,
tan θ= Fc/Fg
q1q2= S3(2Lk/mg)
Fc
S
Small angle
approximation
is in use
Fg=mg
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To measure the charge separation
exponent , one of the experimental
parameters must be varied in
controllable amounts
• Possible choices:
q1q2= S3(2Lk)/(mg)
– Charge
(Wiley & Stutzman,1978)
– Thread length (Akinrimisi, 1982)
– Balloon mass
• Graphing the separation versus variations in the
parameter allows for the determination of the
exponent
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Using a variable thread length
• Starting with: q1q2= S3(mg /2Lk)
• Gives :
L= S3(mg /2q1q2k)
Exponent in
Coulombs law
• S3= SP x S1
Slope= P+1
Log(L)
Difficulties:
-Both S & L must be
measured carefully
.
-thread
management
Intercept= Log(mg/2kq1q2)
Log(S)
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Balloon mass as a variable
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Using a variable mass
• Starting with: q1q2 = S3(mg /2Lk)
• Gives :
m-1= S3(g/2Lq1q2k)
• S3= SP x S1
Slope= P+1
-Log(m)
Difficulties:
Center of mass
and center of
charge no longer
coincide
Intercept= Log(g/(2Lkq1q2))
Log(S)
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Introducing a new Variable in the
Geometry (connection point spacing, W)
θ
W
Now, ΣF=0 gives:
θ
(S-W) -1 = S2[mg/2kLq1q2]
(varies)
L
Fc
Fc
(S-W)/2
Fg=mg
S
A plot of
-Log(S-W) versus
Log(S) should have a
slope equal to the
exponent in
Coulomb’s law
Fg=mg
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Data Acquisition
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Hoped for Results
Slope= 2
Intercept= Log(mg/(2Lkq1q2))
Log(S)
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Typical Results
Coulomb's Exponent
1.3
1.2
Correlation
Coefficient
= .995
1.1
-Log(S-W)
1
Intercept
indicates:
q= 170 nano
Coulombs
0.9
0.8
0.7
0.6
Typical range of values:
Exponent: 1.8- 2.2
0.5
Charge: 75- 200 nC
0.4
0.3
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
Log(S)
-0.15
-0.1
-0.05
0
Correlation Coefficient:
.98-1.0
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Key factors in making this experiment
reasonable accurate
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Symmetrical charging
Photographic balloon spacing measurements
“Staggered” sequence of spacing measurements
Quick data taking procedure (1 to 2 minutes)
Nearly spherical and equal size balloons
Minimal air drafts
Balloons attached “upside down”
Maintain about 1 meter elevation above floor
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Some interesting Calculations
• Initial Discharge rate= Δq/Δt ≈ .1 nano-amp
balloon spacing rate of decrease (~1cm/minute)
• Electric field at the balloons surface
E= kq/r2 ≈ 5X105 Volts/meter
field for air)
(about 10% of breakdown
• Electric Potential at the surface
V= kq/r ≈ 30,000 Volts
• Ratio of atoms to excess electrons on the balloon
– # atoms in 1 gram of rubber ≈ 1022
– # electrons in 150 nCoulombs ≈ 1012
– Therefore, only one atom in ten billion has an extra electron
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Additional fun with balloons
• H20 molecule dipole with (+) and (–) sides
• Candle experiment with metal screen
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