Ohm’s Law
This lab is created to test Ohm’s Law which is a statement that current is proportional to
the voltage difference across a “consumer” of electric energy which we call a resistor. In
particular, the constant of proportionality is 1/R, where R is the resistance, so
I = (1/R)V
where I is the current (measured in amperes or “amps” A), V is voltage (in volts V) and R is the
resistance (measured in ohms ). Resistance is a related to the rate of conversion of electrical
energy (i.e., “consumption”) to other forms of energy like heat, light, torque (in an electric motor
like a propeller) and so on.
Consuming object is called a “resistor”. For it to consume electric energy a current must
flow through it. That requires a power source (e.g., a battery or power supply) connected in a
closed circuit through the resistor like that shown below. The rate of energy conversion by the
resistor in the closed circuit is the power (in watts) and P = IV , which becomes P = I 2R
The power supply is the is the short-long line with V beside
it. The circled “A” is the ammeter, the circled “V” the voltmeter. A
wire lead comes out from the “+” terminal of the power supply, is
connected in “series” with the ammeter and the resistor R (the sawtooth symbol) and back to the power supply on its “-“ terminal. That
is a closed circuit, and current flows out through A and R and returns
back to the power supply.
I
The voltage difference across the resistor is measured by the voltmeter, and proportional to
the electric field in the conducting element which points from the high voltage side to the lower
voltage side of the resistor. The electric field in the conductor forces charge through the resistor.
Note, a voltage difference is only possible in this “dynamic” state – where charges are moving – a
static equilibrium state results in constant voltage (and no electric field) inside a metal.
It may be noted here the moving charges encounter lattice sites in the metal and scatter,
ultimately resulting in a constant “drift velocity” of the current carriers (e.g., electrons). Note, the
current I is in the direction of “positive” charge carriers (this standard came from Benjamin
Franklin); we now know electrons are negatively charged and they carry the current, so they flow
opposite to I.
Ohm’s law is tested by measuring the current I as the voltage V is increased across a
“decade resistance” box having resistance R. Current and voltage are measured using digital
multimeters switched to “amps” and “ohms”, respectively. The measurements are in the
“ohmsLawVideo” in the first 24min 30s. First a 100 resistance is chosen and a voltmeter on the
right and an ammeter on the left is used to display volts and amps as the voltage is incremented
from zero through 0.5 volt increments. Next, a 200 resistance is used. A summary is provided
on the next page. The key result is a graph of data points vs Ohm’s law, and a comparison of the
error and uncertainty ratio for both the 100 and 200 curves.
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Summary of the experiment
1) The circuit as shown in the schematic, starting with a 100 resistor. The resistance is to
be directly measured. Use that value for R which is the “observed” resistance, and any
variation (and/or its scale limit) as its uncertainty.
2) The voltage is varied on the power supply. Values on the digital multimeters should be
used directly and entered into Excel, say. For voltages, the values roughly 0.5, 1.0, 1.5
volts and so on while the ammeter reads roughly 5, 10, 15 milliamps. Estimate the
uncertainty of each measure from variability or scale. [Note: it is very hard to pick up the
decimal point on the meter display. But, milliamps (mA) are two places from the right,
volts are three places from the right. For example, the reading at 10:23min there are
readings of 1027 (ammeter) and 1037 (voltmeter). A closeup of the display shows that
yields 10.27mA and 1.037V.]
3) Read and enter data in both amps and volts where the power is varied from 0 to 10V.
4) Plot all the data as “dots” or “circles”- I suggest using Excel. Do not “connect” the data
points. Instead, Use curve fitting to plot the theoretical curve I vs. V (through the origin) to
estimate the slope and infer the resistance.
5) Use LINEST to get the slope and the uncertainty of the slope (see the guide on the Home
page). Use that to get the observed resistance and its uncertainty.
6) Compare to the observed value from the slope to the resistance already measured and get
the error. Determine the ratio |error|/(total uncertainty). Is it less than “1”? If so, we often
might say the prediction (in this case, of Ohm’s law) is verified.
7) Repeat using a 200 resistance. Then plot I vs V on the same graph as the 100, with
both data points and fit curve. Again, use LINEST to compare |error|/(total uncertainty)
and state whether the prediction is “verified”.
8) Do NOT do the light bulb part – a suitable lightbulb could not be located that day.
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A guide to lab reporting and grading
A. Format in a full lab writeup
In this course we require a specific format (see figure on the next page). It basically consists of
the following form:
Author
Lab partners …
Title
I. Introduction
…
II. Data and Analysis
…
III. Summary and Conclusions
…
These three areas are summarized below:
The “Introduction”
• Describes the purpose of the lab
• Summarizes expectations (numbered equations that summarize the theory)
• Describes the procedure used for observation. (A picture of the apparatus is almost
always helpful)
The “Data and Analysis” section
• Begin with discussion of measurements and uncertainties, plus discussion of any
uncertainty propagation, error or uncertainty propagation formulas, etc.
• Refers to data and calculation results contained in tables
o tables are numbered and have captions
o tables contain uncertainty and relative uncertainty, error and relative error (if
applicable);
o compares predicted vs observed, error vs uncertainty, for purpose of verification
•
Contains a description of:
o what equations were used for what table/column
o what model was used to draw a “theoretical curve” on a graph
If there is a list of values expected to be repeated, an average should be calculated
•
•
The “Summary and Conclusions” section
• State the key results.
• Determine verification/non-verification of theory predictions. This is done by comparing
the error and uncertainty; if the error < uncertainty the theory is verified, otherwise it is
not verified.
• If non-verification of theory is found, include specific sources of “systematic error”
which would cause inaccuracy (or how uncertainty might have been under-estimated)
A sample report is in the Appendix.
1
B. Helpful features of MS-Word
For “subscripts/superscripts”: “CTRL-ALT-(SHIFT)-+” or “CTRL-ALT- =”. Do not use “^” for
powers. For symbols: “Insert”→”Symbol” and scroll to desired row, select item and press
“insert”.
C. Examples of grading deductions (out of 10.0 max score)
Below is a list of common errors and point deductions used in grading the lab report:
reason
missing section
missing derivation if it is required in the lab specification
results (value, uncertainty) too far off what is expected
missing model equations used for prediction
missing title
missing error
missing partner list
missing uncertainty
missing relative uncertainty/error
missing average for repeated measurements
missing comparison of error and uncertainty to “verify” or “not verify”
“apples” and “oranges” in representing uncertainty and error
incorrect number of significant figures
incomplete or incorrect explanation or conclusion
inconsistency in results among various sections (e.g., stating one value in
one place, and another value in another place for the same measurement)
missing table number or caption (description/title)
missing figure number or caption (description/title)
missing reference to table or figure from text
missing units
missing labels on a graphical axis
missing acknowledgement of partners
formatting issues
attaching lined pages for results (must be unlined white)
table running on two pages
deduction
1.0
1.0
0.5 – 1.0
0.5
0.5
0.5
0.25
0.5
0.5
0.5
0.5
0.25
0.25-0.5
0.25-0.5
0.25-0.5
0.25
0.25
0.25
0.25
0.25
0.25
0.1
0.25
0.25
Note: this should not be considered a complete list.
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Appendix: Sample Lab Report: (adapted from Project Caliper, Ted Walker, DVC)
Building a Simple Telescope with Lenses in Combination
I. Introduction
In this laboratory exercise observed how lenses could be used to create images and how lenses in
combination could be used to make telescopes. We measured the focal lengths of two converging
lenses using two different methods. We then constructed two telescopes and measured the
magnification of each telescope experimentally, and compared these values to the theoretically
expected magnifications.
We measured the focal lengths of the two converging lenses using sun light (see Fig.1). We took the
two converging lenses outside and used each one to focus the sun’s light rays to a point on a screen.
The distance from the lens to the focused sunlight is defined to be the focal length f of the lens. We
measured this distance for each lens using a meter stick, obtaining the first measurement of the focal
lengths.
f
sunlight
lens
screen
Figure 1: Optical system to measure f with an sunlight
We then calculated the focal lengths using an optical system in a dark room consisting of a light
source object, one of the lenses and a small projection screen (see Fig.2, below).
p
q
object
lens
screen
Figure 2: Optical system to measure f with an object in a dark room
We moved the components back and forth until a sharp inverted image of the object was projected on
the screen. We then measured the distance from the light object to the lens (p) and from the lens to
the projection screen (q). From the measurements of p and q we were able to calculate the focal
length, f obtained from
1 1 1
+ =
(1)
p q f
3
We used this procedure to determine a second measure of the focal length for each of the converging
lenses.
We then set out to make a telescope. Theory tells us that two lenses held the sum of their focal
lengths apart should form a telescope (see Fig. 3). We simply held the converging lens with the
smaller focal length close to one eye (eyepiece) and the other lens the sum of their focal lengths
further out from the first lens.
fo + fe
eyepiece
objective lens
Figure 3: Telescope schematic. fo and fe are the focal lengths
of the objective and eyepiece lenses, respectively.
We then measured magnification m for this telescope. To do this we looked at a series of chalk lines
on the blackboard from across the room with the telescope. The lines were drawn every 10cm. We
looked at the lines through the telescope with one eye and simultaneously looked at the same lines
with the other eye. What we saw was one set of lines magnified by the telescope superimposed over
the same lines unmagnified as seen by the other eye. Since the method used to measure m is so
subjective, each partner in our lab group measured m and we averaged the three numbers.
We then made another telescope using the diverging lens as the eye lens. We again determined the
magnification of this telescope by looking at the lines on the chalk board. We noticed that this second
telescope had the advantage of not inverting the image.
The measurements are compared to the theory
m = − fo / fe
(2)
where fo and fe are the focal lengths of the objective and eyepiece lenses, respectively.
II. Data and Analysis
Table 1-2 contain the focal length data. The uncertainties are based on determining where the
sunlight was well focused. The 5mm uncertainties in Table 2 are determined as follows: 1mm for the
meter stick, 1mm for the lens thickness, 1mm for the position of the object and 2mm for
determination of the best possible focus.
Table 1. Focal lengths observed using sun light
Lens
focal length (cm)
Lens 1
10.2 ± 0.6
Lens 2
19.7 ± 0.4
4
Table 2. Focal lengths calculated using dark room images
Lens
p(cm)
q(cm)
f(cm) = pq/(p+q)
Lens 1
27.2 ± 0.5
14.4 ± 1.5
9.42 ± 0.19
Lens 2
32.4 ± 0.5
61.3 ± 0.5
21.20 ± 0.10
The uncertainty in the predicted f is obtained by propagation of the measured uncertainties in p
and q:
2
2
2
2
??
??
?
??
?
??
√
√
?? = ( ??) + ( ??) = {[
−
] ??} + {[
−
] ??}
??
??
? + ? (? + ?)2
? + ? (? + ?)2
which for Lens1 becomes
?? = √{[
27.2
27.2+14.4
−
(27.2)(14.4)
(27.2+14.4)2
2
] 1.5} + {[
14.4
27.2+14.4
−
(27.2)(14.4)
(27.2+14.4)2
2
] 0.5} cm = 0.19cm
The focal lengths (observed and calculated) are compared in Table 3. The error in the object
observation is again taken by comparing to the focal length with sun-light, and the total uncertainty is
the sum of observed and predicted uncertainties. Lens1 prediction is marginally verified, while the
Lens2 prediction is not verified.
Lens 1
Lens 2
Table 3. Error, total uncertainty and verification
error (cm) in f
total uncertainty (cm)
verification
-0.8
0.8
yes
+1.5
0.5
no
This probably means that the uncertainties in the image distance were underestimated due to not
having the image in clear focus. This would mean that p and q for the second method weren’t
actually measured correctly. We probably needed more than 2mm uncertainty to obtain the best
possible focus.
Telescope 1 was constructed using lens 1 as the eye lens and lens 2 as the objective. Telescope 2 was
constructed using a –2.5cm focal length lens as the eye lens and lens 1 as the objective lens. We can
calculate the expected theoretical magnifications using equation (2) (see Table 4). Since we trust the
focal lengths determined with the sun light the most, we chose to use those numbers to calculate the
expected magnifications.
Table 4. Predicted magnifications
f (cm)
f (cm)
m = -f /f
o
Telescope 1
Telescope 2
19.7 0.1
10.2 0.1
e
10.2 0.1
-2.5 0.1
o e
-1.931 0.014
+4.08 0.05
The Table 4 values of m represent the “known” values which can be compared to observed
magnifications (Table 5). The + refers to upright images while the – refers to inverted images.
Uncertainties are calculated using the half-range plus the uncertainty in the predicted m. Error is
computed using the predicted values from Table 4.
5
Telescope 1
Telescope 2
Ted’s
(m)
-2.1
+3.6
Table 5. Observed magnifications
Annie’s
Joe’s
Average
(m)
(m)
(m)
-1.6
-1.8
-1.8
+4.2
+3.8
+3.9
Uncertainty
(m)
0.3+0.14 = 0.4
0.2+0.05 = 0.3
Error
(m)
– 0.13
+ 0.18
This shows that errors are within uncertainty, verifying the predictions of equations (1) and (2). The
telescope magnifications measured were both about 5% lower than the magnifications predicted by
theory. This is reasonable since the theoretical equation assumed that the object was very far away
(our object was only about 15 feet away) and that the eye was relaxed (something that’s hard to
control).
III. Summary and Conclusions
In this lab we measured the focal lengths of two converging lenses using two methods: sunlight
focusing on a point and an object’s relative size upon magnification. As Table I shows, the
prediction of theory was marginally verified by Lens1 using an object image, as the error magnitude
0.8cm is equal to the uncertainty 0.8cm but failed when using sunlight for Lens2 which is about
200% outside the uncertainty (as stated above, is probably due to poor determination of what is
“focused” in the object measurement) thus failing to verify the theoretical prediction. On the other
hand, measurement confirms theoretical predictions of magnification (as shown in Table 5), as |error|
uncertainty for both Telescopes. We then built two telescopes and determined their magnifications
both experimentally and theoretically, getting consistent results within uncertainty.
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R= 1
slupe
OR – S slope
R Salope
–
R
slope
slope=R
SR = R² Sslope
2
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