MEASUREMENT OF THICKNESS OF A VERY THIN SAMPLE Purpose

King Fahd University of Petroleum & Minerals
PHYS102
MEASUREMENT OF THICKNESS OF A VERY THIN SAMPLE
Purpose
To study how the resistance of a conductor (graphite) of a uniform cross section varies with
the length and measure its resistivity.
To estimate the number of atoms in the layer of graphite left by a pencil line drawn on a
paper.
Background
Consider a very thin strip of a conducting material (Fig. 1). Its thickness, t, is much less than
a micron (micrometer, 10-6 m) . It is not possible to measure its thickness, even if we try to
use a micrometer. Can you guess why?
l
t
W
Multimeter
Figure 1. Measurement of resistance of a thin metal sheet.
Measurement of thickness of such a thin sample has generally been made by a use of an
indirect method. One of these methods is outlined below.
The resistance, R, of a conducting rod is given by
R=ρ
l
A
…………………….……………..(1)
where ρ is the resistivity, l is the length and A is the area of cross-section of the rod. If we
have a conductor in the shape of a strip, as shown in Fig. 1, then its resistance, R, measured
along the length is given by
R=ρ
l
A
=ρ
l
Wt
…………………………….(2)
where ρ is the resistivity of the strip, l, W and t are shown in Figure 1.
The thickness, t, of the strip can be determined from Eq. (2), if we measure R, l and W, and
use the known value (provided, it is available) of the resistivity ρ, of the strip material.
Resistivities of some pure materials, such as Cu, Al, Au, C, etc. are given in your textbook.
However, for different alloys and impure materials, the resistivities are not known. In such
cases we have to determine ρ first.
© KFUPM – PHYSICS
revised 20/02/2012
65
Department of Physics
Dhahran 31261
King Fahd University of Petroleum & Minerals
PHYS102
l
d
Multimeter
Figure 2. Measurement of resistance of pencil lead.
Study (1) Measurement of resistivity of a pencil lead.
1. Measure the length, l, and diameter, d , of the pencil-lead (Fig. 2). Record these values
with the uncertainty in their measurement, for example you should be writing l = 10.3 ±
0.05 cm and d = 2.31 ± 0.01 cm. Use a vernier calliper to measure the diameter of the
pencil lead.
2. Using a multimeter, measure the resistance, R, of the pencil lead across its length
(Figure 2).
3. Using the above data and Eq. (1), calculate the resistivity, ρ, of the pencil lead.
4. Calculate the uncertainty (∆ρ – the maximum possible error in ρ) in the calculated value
of ρ. Consider the value of R from the digital multimeter to be accurate to 3% , i.e. ∆R/R
= 0.03.
5. Explain why using a simple ruler with a mm division (with an uncertainty in the
measurement of 0.5 mm) is more than enough for the measurement of the length of the
pencil lead in this experiment.
6. In reality the measurement of resistance could be significantly more than the actual
resistance between the ends of the pencil lead; this introduces a systematic error into the
experiment. Could you explain the source of this error?
© KFUPM – PHYSICS
revised 20/02/2012
66
Department of Physics
Dhahran 31261
King Fahd University of Petroleum & Minerals
PHYS102
Study (2) To estimate the thickness of a pencil line and the number of atoms in
the thickness of the pencil line.
1. On a sheet of graph paper, using the pencil provided, draw a line 4.0 mm wide and 160
mm long with a 10 mm square at one end. Shade in the line and square so that they are
dense black (see Fig. 3). Take care to make the line to be of uniform thickness
everywhere.
Fig. 3 Measurement of resistance
Multimeterat different distances (measured from x=0)
along the pencil line.
The square (Figure 3) is to act as an electrical contact at the zero end of the line.
2. Keep one connector from the multimeter firmly pressed in the square and press the other
firmly at different distances along the pencil line (Fig. 3) in order to find the resistance, R,
for several lengths l (= 1.0, 2.0, ... to 15.0 cm) of the pencil. Record all your results in a
table.
3. Plot a graph of R against l, and draw a straight line of best fit.
4. Find the slope of the straight line. Use this value together with the known values of ρ
(found in study 1) and W (= 4 mm), to find the thickness, t, of the pencil line, from Eq.
(2).
5. Assuming the size of an atom of carbon in the pencil lead to be 2 × 10-10 m, estimate how
many atoms thick your pencil line is.
6. Discuss the sources of errors in this experiment.
© KFUPM – PHYSICS
revised 20/02/2012
67
Department of Physics
Dhahran 31261
King Fahd University of Petroleum & Minerals
PHYS102
7. If you were to continue to draw this pencil line beyond the 160 cm limit using a new
pencil how long (in km) would this line get before you run out of pencil lead? Assume
you have large enough paper (!) and you don’t get tired. Take density of graphite to be 2
g/cm3. Show all your steps and reasoning.
© KFUPM – PHYSICS
revised 20/02/2012
68
Department of Physics
Dhahran 31261