Transcript Lecture 1 - UMD Physics

Lab Partner Assignments for Lab 1
0401 (Wed)
Abdulkadir
Fabula
Gima
Loman
Robinson-Tillenburg
Stefany
Brower
Gallagher
Henry
Reiser
Rogers
0301 (Thurs)
Belenky
Fowler
Germain
Quelland
Shetty
Stillwell
Flair
Garhart
Lambert-Brown
San Miguel
Stahl
Whittemore
reminder
No lab next week
Write you name on the white
label on the component box in
your kit
Lab #1: Imperfections in Equipment
• practice with parallel and series circuits, and Ohm’s
law
• Measure the internal resistance of a battery
• Measure the input impedance of the oscilloscope
• Measure the output impedance of the signal
generator
Theme: model the real by an ideal in series
with a resistor.
Basic electrical terms
Make sure you have a solid conceptual
understanding of the following terms, their
relations, and their differences
• voltage
• electric field
• current
• electric potential
• resistance
Current/Voltage
Current
• Current: amount of charge that passes a
point on the wire each second (Ampere =
Coulomb/second)
• Determined by number of charges and by
their speed
Voltage
•Potential energy/charge (normalized potential
energy)
Voltage across something. Current through something.
Basic Electrical Concepts
Conductors
Use a battery or some other emf to set a
voltage across an object. Chemical reaction
in the battery allows rearrangement so as to
maintain an (approximately) constant voltage
difference between the two terminals.
Terminal velocity depends on voltage, the
geometry of the materials, and the
properties of the material Resistivity
Ohmic materials:
A
I = DV where E=DV / d
rl
Resistor code
• Use this to choose the resistance. But never use
this value as the resistance. Always measure it.
• This applies as well to the resisitor box. Never
use the nominal values. Always measure.
Resistance
Material
Insulators
Mica
Glass
Rubber
Semi-conductors
Silicon
Germanium
Conductors
Carbon
Nichrome
Copper
resistivity at room temp (W-m)
2x1015
1012-1013
1013
2200
0.45
3.5x10-5
1.2x10-6
1.7x10-8
Kirchhoff’s Rules
From course work, remember how to use Kirchhoff’s rules to calculate
voltage and currents in circuits? If not, see lab writeup.
• In going round a closed loop, the total change in potential must be zero
• Charge is conserved so that at any junction the current flowing into the
junction is equal to the current flowing out of the junction
Applying these rules to enough junctions and loops generally leads to
enough equations to solve for the number of unknown currents and
voltages.
Effective Resistance
Sometimes you can use these shortcuts instead.
Reff = R1 + R2
1
1
1
= +
Reff R1 R2
When calculating currents and voltages in a circuit, you can
replace these combinations by an “effective resistance”
without altering the current through and voltage across these
“elements”
Circuits
When using these rules, you probably neglected to
take into account the fact that the instruments you
use to measure the circuit can themselves alter the
performance of the circuit.
We will study this in the lab, see how big the effect
is, and from that get an idea of when this needs to
be taken into account when comparing results to
predictions.
Internal Resistance of a Battery
Imagine a simple circuit consisting of a battery and
a resistor.
If you varied the resistance R and
plotted V versus I, what would you
get?
V =e
ε
A horizonal line, independent of R
Simple circuit
A more realistic model
If I plotted V vs I, what would
I get now?
r
ε
I get a straight (not flat) line.
What does the slope represent?
What does the intercept represent?
V = IR
Have 3 variables in the equation
(V,I,R).
Need to get rid of R.
e - Ir - IR = 0
e - Ir e
R=
= -r
I
I
so
V=e -Ir
An even more realistic model
Will get rA and rV from meter manual and estimate effect on
estimate of r and ε
Input/Output Impedences
A function generator, like a battery, is a voltage source.
An oscilloscope, like a multimeter, is a measuring
instrument.
We will measure the “internal resistance” of each of
these devices.
Using the multimeter
• Discuss how to use it for current and voltage
measurements.
• Point out the 400 mA and 10 A terminals for
the ammeter and discuss their purpose.
• Be sure it is set to ‘DC’ not ‘AC’
Never measure the value of a
resistor when it is in a circuit
Multi-meter systematic errors
Estimating Errors: Review
•Systematic errors : sources of error that have the same size
effect on every measurement that is made (or a correlated effect)
• a ruler that was not manufactured correctly
• a consistently delayed reaction when using a stop watch
• your inability to perfectly estimate the size of a stray
magnetic field from your computer that leaks into your
experimental area
• Random errors : sources of error whose effect varies with each
measurement
• precision of your measuring device
• when using a stop watch, a reaction time that sometimes
anticipates the event, some times is in retard of the event.
Multi-meter systematic errors
Will assume that the systematic error due to the factor calibration is in
the form
𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝜆𝑉 𝑉𝑡𝑟𝑢𝑒 + 𝑏𝑉
𝜆𝑉 = 1 ± 𝜎𝜆𝑉
𝑏𝑉 = 0 ± 𝜎𝑏𝑉
𝐼𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝜆𝐼 𝐼𝑡𝑟𝑢𝑒 + 𝑏𝐼
𝜆𝐼 = 1 ± 𝜎𝜆𝐼
𝑏𝐼 = 0 ± 𝜎𝑏𝐼
Systematic Errors and fits
Last week, we learned
• how to propagate errors in measured quantities to errors
in quantities calculated from them via a simple algebraic
formula (both random and systematic are handled the
same way).
• how to calculate the uncertainty on the fit slope and
intercept from a linear fit due to random errors in the x y
variables
This week we’ll learn how to calculate the uncertainty on
the fit slope and intercept from a linear fit due to systematic
errors in the x-y variables.
Error on slope and intercept
due to statistical error
sb = s
åx
2
j
N å x - (å x j )
2
j
Note error on
intercept
scales with
1/root(N)
Fitting and systematic errors
Suppose you are measuring V using a meter that has infinite
accuracy and that has no random errors, but that always reports a
voltage that is always off by 0.25V?
Adding more measurement points does not reduce the error.
Previous formula can not work for systematic errors
Systematic Error in Slope
How can slope be changed? If voltage is always off by a scale
factor, or if current is always off by a scale factor, slope is off by the
same factor. The error in the offset (b) does not cause an error in
the slope at all.
xmeasured = lx xtrue + bx
ymeasured = l y ytrue + by
s = (m × s l x ) + (m × s l y )
2
m
2
2
Systematic Error in Intercept
What if the voltage is always off by a fixed,
constant amount?
xmeasured  x xtrue  bx
ymeasured   y ytrue  by
  (b    y )  ( by )  ( m   bx )
2
b
2
2
2
(see “Propagation of Systematic Errors” on the class
web site, for a more complete, rigorous derivation of this
result.)
Random and Systematic errors
• first, fit to a straight line using only random
errors
• get the error on the fit m and b due to random
errors from the spreadsheet
• calculate the errors on m and b due to
systematic errors as shown on previous 2 slides
• take the error on m due to random errors and
the error on m due to systematic errors and add
them in quadrature
• ditto for b
Fitting and Systematic Errors
If you don’t understand this (how to calculate the
systematic error on slope/intercept and then
combine with the statistical error), don’t leave the
room today until you do! It’s important for this and
future labs!
linearizing
This semester, we will often do a variable transformation in
order to get a linear dependence that we can easily fit.
1
1
1
=
RS +
VS VB RIN
VB
feed to fitter:
1
y®
VS
x ® RS
get from fit m and b.
1
m®
VB RIN
1
b®
VB
Please show to yourself that
b
R IN =
m
Linearizing
When we transform variables, we also need to
recalculate the errors.
In this lab:
1
y
x
y 1
 2
x x
y 
x
x
2
Rounding Uncertainties
If your digital voltmeter says 3.02 V, the real measurement could be
between 3.015 and 3.025V with equal probability. What is the
uncertainty? -> want +- 1 sigma to include 68% of the measurements.
D /2
1=
ò
A dx where D is the least
lsb significant bit (lsb)
- D/2
® A =1/ D
choose "s " so
s
1
.68= ò dx
D
-s
2s
.68 =
D
s = 0.34D
Sqrt(12)
When you have an LSB, what is the random error?
Imagine a step with width D centered at zero.
Remember:
RMS  x  x
2
x0
D/2
x =
2
ò
-D/2
x
3
3
x dx
2
D
=
D/2
-D/2
D
RMS = D / 12 = .29D
D2
=
12
2
Notes on lab
• check that probes are not set on x10
• check that ammeter is set on DC, not AC
•Never use the nominal value of a resistor. Always measure the
resistance using an ohm meter. Always remove the resistor from the
circuit before measuring its resistance (why?)
• All numbers should have units and be carefully labeled.
•Some of the resistors have values that drift with temperature. It is
important to measure V&I simultaneously. If you measure one, wait a
minute, then measure the other, you’ll get a bad result. There will be a
random error from your ability to read the 2 meters at the same time.
How will you estimate this random error? (Drift is biggest when using
smallest resistor. Why?)
•Be careful with grounds when measuring the output impedance of the
signal generator.
Be Careful With Grounds (Shown in Red Here)
Be Careful on This Step!
Start with the voltage set to 0
on the variable-voltage power
supply, and slowly increase, keeping
current I < 400 mA
I
Power
Supply
Voltage knob
… otherwise the fuse is blown in the ammeter
reminder
No lab next week