Millikan oil drop experiment: Determination of elementary charge of electrons

Transcription

1 Millikan oil drop experiment: Determination of elementary charge of electrons Mahesh Gandikota Y Semester IV Integrated MSc National Institute of Science Education and Research Experiment completed: 13/1/2012 Report submitted:2/2/2012 1

2 1 Objectives Measuring the rise and fall times of oil droplets with various charges at different voltages. Calculating the radii and the charge of the droplets. Calculating the charge of the electron. 2 Theory When Millikan did this experiment, he observed that all the charges on the droplets were found to be integral multiples of some charge. He proposed that this charge should be the charge of the electron as it was considered to be the fundamental unit of charge and not to be composed of any further electric charges. 1 The oil drop which acquires charge due to X ray radiation which ionizes the air and the electrons attach themselves to the oil drop. The oil drops (some of them charged, some not) fall down through a hole from the top chamber to the bottom one and attain a terminal velocity due to the air drag 2. Assuming the oil drop to have a spherical shape, the viscous force on it is given by Stokes law. F = 6πrηv (1) The gravitational force on the oil drop is, F = mg = ρ 1 V g (2) The buoyant force on it is, Force due to electrical field is, F = ρ 2 V g (3) F = QE = Q U d (4) When the oil drop attains the terminal velocity, the net force on it is zero. The fall and rise velocities of the oil drop is given respectively by, v 1 = 1 [ QE + 4 ] 6πrη 3 πr3 g(ρ 1 ρ 2 ) (5) 1 Now, on the advent of the quark model, the quantization of charge has gone down to e 3 from e. However, electron being a lepton is not made of any quarks and is a fundamental particle still. 2 The experiment can t be done in vacuum. We need a viscous fluid like air. 2

3 v 2 = 1 [ QE 4 ] 6πrη 3 πr3 g(ρ 1 ρ 2 ) The above equations are simply derived by equating the forces on the oil drop. 6πrηv 1 = QE + mg Buoyant force By substituting for the above forces in terms of volume, equation (5) is directly obtained. For deriving equation 6, the gravitational force opposes it s upward motion and buoyant force enforces it. Thus the sign of the second term changes. From equations (5), (6), the below equations are obtained. (6) where v 1 + v 2 Q = C 1 v1 v 2 (7) U C 1 = 9 2 πd η 3 g(ρ 1 ρ 2 ) where C 1 = kgm(ms) 1 2 r = C 2 v2 v 1 (8) C 2 = 3 η 2 g(ρ 1 ρ 2 ) ρ 1, ρ 2 are the densities of the oil drop and the air respectively. C 2 = (ms) 1 2 The derivation for equation (7), is given thus: Adding equations (5), (6), 2QE = 2Q U d = 6πrη(v 1 + v 2 ) Q = 6πrηd(v 1 + v 2 ) 2U Subtracting equation (5) by (6), 6πrη(v 1 v 2 ) = 8 3 πr3 g(ρ 1 ρ 2 ) r = 9η(v 1 v 2 ) 4g(ρ 1 ρ 2 ) 3

4 Substituting this expression for r in the expression obtained for Q, Q = 6πrηd 2U (v 1 + v 2 ) 9η(v 1 v 2 ) 4g(ρ 1 ρ 2 ) = 9πηd(v 1 + v 2 ) 2U η3 (v 1 v 2 ) g(ρ 1 ρ 2 ) QED Q = 9 2 πd η 3 g(ρ 1 ρ 2 ) (v 1 + v 2 ) (v 1 v 2 ) U The expression (8) was also derived on the way as seen. 3 Apparatus Millikan apparatus, power supply, stage micrometer, stop watch, commutator switch. 4 Data, plot analysis and discussion 4.1 Calculations 1. Voltage (V) 2. t 1 (s) 3. t 2 (s) 4. v 1 (ms 1 ) 5. v 2 (ms 1 ) 6. Q (As) - charge on drop. 7. r (m) - radius of drop. 8. n - The estimation of number of electrons on the oil drop. 9. e (As) - charge of electron found. 4

5 Table 1: Observations e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e For 290 V, Average charge of e = C. For 400 V, Average charge of e = C. For 500 V, Average charge of e = C. Total average charge of e = C. 5

6 7 Measurement of charge of oil drops 6 Charge on the oil drops (x 10^{-19}C) Dummy axis Figure 1: Data points of observation 4.2 Some other points to discuss We can observe from the data, that a bigger radius of the oil drop does not necessarily have to carry higher charge. It usually happens that, we should be changing the focus on the oil drop when we are following it s movement. This shows that the oil drop has some other components of velocity too other than only the up or down velocity. That s the reason we have to keep changing the focus plane to exactly point out its position. Most of the oil drops will not be charged. From the data we can find that, we also got oil drops on which there was only one electron! The oil drops having horizontal velocities other than the vertical velocities does not affect the values obtained as the vertical velocity is 6

7 governed only by the vertical forces and is not influenced by the presence of other vertical velocities. The velocities of the oil drop as we see is very small. This is kind of expected because, even in the microscope which magnifies distances, we see the oil drop to be moving so slowly, it has to be that in reality, it has to be moving very slowly. 5 Error analysis 5.1 Standard deviation The standard deviation of the estimation of the charge of the electron about the mean value ( ) was found to be Propagation of errors From equation (7), we get the propagation error formula as Q Q = v 1 + v 2 + U v 1 + v 2 U v 1 + v 2 (9) v 1 v 2 I made a C++ program to calculate the above propagation error and I obtained, Propagation error = C. We can see that the experiment has a very small propagation error inbuilt in it. This is quite a precise method to calculate the charge of the electron. 5.3 Percentage error Accepted value of electron charge is C. Obtained value = C Percentage error = = 0.31% Sources of error The reaction time of the observer while handling the stop watch. This can be avoided by making use of a video camera. We can analyse the video later to get a much more exact value of time intervals. We estimated the shape of drop to be spherical in order to use the Stokes formula. 7

8 6 Conclusions and remarks The charge was observed to be quantized. The least charge observed was near to the value of the electron. Assuming the least charge observed is the charge of the electron (from previous knowledge), the experimentally calculated electron charge was found to be = C with a propagation error of 0.052C and with a percentage error from the accepted value was 0.31%. The standard deviation was quite disappointing Bibliography 1. wikipedia 2. { dir/ryan_mcallister/slide3.htm} 8