MILLIKAN’S OIL DROP (Exp#1) Introduction The goal of this experiment is to use Millikan oil drop apparatus to demonstrate that electric charge only comes in discrete units – “the quantization of charge“ ; and to measure the intrinsic charge of the electron (the smallest discrete unit of charge) e. Theory The oil drops in this experiment are subject to three different forces: viscous, gravitational and electrical. By analyzing these various forces an expression can be derived which will enable measurement of the charge on the oil drop and determination of the unit charge on the electron. When there is no electric field present, the drop under observation falls slowly, subject to the downward pull of gravity and the upward force due to the viscous resistance of the air to the motion of the oil drop. The resistance of a viscous fluid to the steady motion of a sphere is obtainable from Stokes’s law, from which the retarding force acting on the sphere is given by (1) where is the radius of the sphere, locity of the sphere. the coefficient of viscosity of the fluid, and For an oil drop with mass, m, which has reached constant or terminal velocity, ward retarding force equals the downward force taking buoyancy into account. the ve- , the up- (2) where = density of air. Now let an electric field, E, be applied between the plates in such a direction as to make the drop move upward with a constant velocity, . The viscous force again opposes its motion but acts downward in this case. An oil drop with an electrical charge q, when it reaches constant velocity is again in a state of equilibrium and (3) Page | 1 Solving Eq.3 for mg and equating this to Eq. 2 gives (4) (4) The electric field, E, is obtained by applying a voltage, V, to the parallel plates which are separated by a distance, d. Therefore, (5) From Eq. 5 it is seen that for the same drop and with a constant applied voltage, a change in q results only in a change in and (6) where c is the constant When many values of are obtained, it is found that they are always integral multiples of a certain small value. Since this is true for the same must be true for ; that is, the charge gained or lost is the exact multiple of a unit charge. Thus the discreteness of the electronic charge may be demonstrated without actually obtaining a numerical value of the charge. In Eq. 5 all quantities are known or measurable except a, the radius of the drop. To obtain the value of a, Eq. 2 may be written therefore (7) where is the density of the oil, the density of the air, and as stated previously, is the viscosity of the air. Since the density of the air is very much smaller than the density of the oil, , is negligible and Eq. 7 is solved to give (8) Substitution of this value of a in Eq. 5 gives an expression for q in which all the quantities are known or measurable: (9) Page | 2 Millikan, in his experiments, found that the charge resulting from the measurements seemed to depend somewhat on the size of the particular oil drop used and on the air pressure. He suspected that the difficulty was inherent in Stokes’ law which he found not to hold for very small drops. It is necessary to make a correction, multiplying the viscosity by the factor where p is the barometric pressure (in centimeters of mercury), b is a constant of numerical value and a is the radius of the drop in meters. This correction is sufficiently small that the rough value of a obtained from Eq. 8 may be used in calculating it. The corrected charge on the drop is, therefore, given by (10) Using the MKS system of units, the electrical charge q is expressed in Coulombs. Data & Calculations Table 1.1 ρ ; density of oil = 800 kg / m³ ρ' ; density of air = 1.29 kg / m³ η ; viscosity of air = 1.827x10¯⁵ Ns/m² Vstop (V) 4.0 7.0 4.1 6.0 18.0 15.6 8.7 4.6 x (m) t (s) 0.0045 0.0016 0.0020 0.0030 0.0030 0.0040 0.0030 0.0020 3.86 5.81 10.78 15.38 5.17 12.22 14.76 6.28 vt(=x/t)(m/s) q = 2x10¯¹⁰ vt3/2/ Vstop( C ) 0.00117 0.00028 0.00019 0.00020 0.00058 0.00033 0.00020 0.00032 8.96E-18 2.09E-19 2.47E-19 2.72E-19 4.66E-19 3.04E-19 2.00E-19 4.94E-19 Page | 3 Conclusion The setup of the Millikan Experiment uses macroscopic elements to measure a microscopic quantity. As a logical result, the correct calibration of the used components is essential to get accurate data. Our results show a nearly chaotic distribution instead of discrete charge levels. Sometimes, we observed collisions of different drops. Therefore, it can’t be ruled out that collisions happened we did not notice. Additionally, we observed chaotic movement to the left or right and even movement both up and down during one measurement. This might be explained by the influence of other drops, which carried opposite charge and came close to the observed drops. The observation itself was difficult as well. Sometimes we lost the drop during the measurement or the large number of other drops led to confusion. Although our measurement failed to clearly show the discrete composition of the electric charge, the setup could be improved to provide better results. Page | 4