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