Demonstrating the Quantization o Electrical Charge Millikan s Experiment Objecties o the experiment To demonstrate that electrical charge is quantized, and to determine the elementary electron charge by obsering the motion o charged drops in an electric ield. Introduction The charge o the electron is one o the most important undamental constants in nature. The ratio o the mass to charge o the electron may be readily determined ia the obseration o the path o a pre-accelerated electron through a magnetic ield. Howeer, the determination o the charge alone is a little more diicult. The irst attempt was perormed by Thomson in 1896, using a cloud chamber, and resulted in a alue close to 1.1 10-19 Coulombs, but with a large error. The more precise measurements o Millikan were perormed in 1911 in his now amous drop experiment. The apparatus he used is shown schematically in Fig. 1. +/- V For each droplet with a charge q there is a graitational orce downwards -/+ V (mg), and a buoyancy orce Fig. 1. Schematic diagram o the Millikan apparatus upwards (b, b is the buoyancy and is the terminal elocity o the drop) which are equal when the terminal elocity is reached (i.e. mg = b). The equation o motion is thus, d m (1) dt mg b = The buoyancy may be obtained rom Stoke s law b = 6πηa (2) where a is the radius o the drop and η is the iscosity o. The terminal elocity o the drop is thus mg mg = = () b 6πη a Now i an electric ield, E, is applied to the two plates shown in Fig. 1, then or a positie oltage applied to the top plate, there will be a corresponding upward motion o the drop which possesses charge q. d qe mg b = m (4) dt [Note the change in sign o the buoyancy orce]. In this instance the terminal elocity is Millikin s drop experiment 1
r Eq mg Eq mg = = (5) b 6πη a In Millikan s experiment the terminal elocities are achieed rapidly, and only the motion o the drops with terminal elocity is obsered. I the drops are obsered to moe a distance L in times T (all) and T r (rise), then it is possible to sole the equations () and (5) or q. mg q = ( + r ) (6) E This is howeer expressed in terms o the eectie mass o the drop moing through the, where 4 m = πa ( ρ ρ) (7) This can be used in conjunction with equation to calculate an expression or a 9 η a = (8) 2 ( ρ ρ ) g and thus m can be calculated and thus the corresponding charge 1 ( + r ) (9) E ( ρ ρ ) g Noting that the electric ield can be expressed in terms o the oltage between the two plates and their distance, then d ( + r ) (10) V ( ρ ρ ) g The two equations (8) and (10) are required or the rise and all method or determining q. There is a second technique or determining q, which is the loat technique. For these measurements the appropriate equations are; 9 η a = and 2 ( ρ ρ ) g d (11) V ( ρ ρ ) g You might attempt deriing the latter two equations (11) yoursel. Millikin s drop experiment 2
Fig. 2. Diagram o the Millikan experimental apparatus Apparatus The equipment to be used in the measurement is shown in Fig. 2. The Millikan equipment should be connected to the oltage control system and timer as shown Fig.. Fig.. The connections and switch unctions or the oltage control system and timer. Millikin s drop experiment
Make sure the equipment is correctly connected beore switching on, i in doubt consult a demonstrator. The atomizer should be illed such that the bent capillary tube stands about 2 mm in the, and the spray nozzle should be positioned such that it points towards the bore holes in the plastic coer. The used has a density o 877 kgm - at 15 o C, and 871 kgm - at 25 o C. The capacitor plates hae a diameter o 8 cm and a separation o 6(±0.05) mm. The telescope itsel is equipped with an eyepiece with a magniication o 10, and the objectie magniication o the telescope is 2(±0.05). The telescope is used to locate the position o the drops by measurements against the scale proided. The length o the micrometer scale is 10 mm, with 0.1 mm graduations. Procedure Setting up the equipment - Turn the lens holder o the micrometer eyepiece until you can clearly see the micrometer scale. - I necessary, turn the eyepiece to orient the micrometer scale ertically. For this purpose you should slightly loosen the astening screw. Since alling droplets are obsered on the micrometer scale as rising droplets due to the reersion o the image in the microscope, the scale start (0) should point upward and the scale end should point downward (10). - Use the knurled knob to push the measuring microscope close to the plastic coer. The illuminated capacitor plates can be seen at the top and bottom in the circular-iewing ield. The beginning and end o the micrometer scale are at a small distance to the capacitor plates. Do not attempt the ollowing without consulting the demonstrator irst. To eliminate disturbing light relections or to correct the obseration region, i you are not satisied with the illumination: - Loosen the astening screw o the capacitor and moe the capacitor. - You can also adjust the lamp with the help o the adjusting screw (recessed head screw). Obsering droplets - Use the rubber ball to spray between the capacitor plates so that droplets can be seen in the entire obseration ield. - By moing the measuring microscope, create a plane, in which a selected droplet is clearly seen as light point. Objectie magniication Due to the objectie magniication M, a all or rise distance s o the droplet between the capacitor plates is represented on the scale section x = Ms I the image o an droplet moes in the time t on the scale oer a distance x, the elocity o the droplet is x = M t The objectie magniication is M = 2 quite accurately. For more exact measurements, you should determine the magniication: Millikin s drop experiment 4
- Remoe the plate capacitor and put a suitable scale ertically on the base plate. - Adjust the microscope so that external scale and micrometer scale can be clearly seen next to one another. By comparing the two scales, determine the exact magniication. - Then, do not moe the eyepiece any more. Timer/Counter Operation o Set mode to t E,F. o Press start until corresponding LED is lit o Cable rom clock1 on Millikan control box should be connected to E. o Cable rom clock2 on Millikan control box should be connected to F. o Zero timer: press 0. o Times can be read out using button t E,F when E LED is lit, irst time is gien i.e. time between start o clock and eent E. When F LED is list this gies the time between eents E and F. Fall/rise method The all elocity and the rise elocity r are determined rom the all time t and rise time t r or a pre-selected distance s. The ollowing equations can then be used or the radius a and the charge q o the droplet (see the introduction). a = 9 2 ( ρ η ρ ) g d ( V + ) r ( ρ ρ ) g - Zero counter press 0. - First turn switch U and switch t downward. - Use switch U to turn on the capacitor oltage and adjust it using a rotary potentiometer so that a selected droplet rises. As soon as the droplet is in the upper area o the capacitor: - Use switch U to turn o the capacitor oltage. As soon as the droplet is next to a pre-selected graduation scale mark: - Use switch t to start measuring the all time. As soon as the droplet has allen oer a pre-selected distance: - Use switch U to turn on the capacitor oltage, to end measurement o the all time and start measurement o the rise time. As soon as the droplet has risen oer the same pre-selected distance s : - Use switch t to end time measurement. - Read all time, rise time (press button t E,F ) and capacitor oltage U and record with all or rise distance s. Float Method The loat potential U and the all speed are determined rom the all time t or a preselected distance s. The ollowing applies or the radius a and the charge q o the droplet: 9 η a = and 2 ( ρ ρ ) g d V ( ρ ρ ) g Millikin s drop experiment 5
- Zero counter press 0. - First turn switch U and switch t downward. - Use switch U to turn on capacitor oltage, then adjust it using a rotary potentiometer so that a selected droplet loats. - Use switch U to turn o the capacitor oltage. As soon as the droplet is next to a selected scale graduation mark: - Use switch t to start time measurement. As soon as the droplet has allen oer a pre-selected distance: - Use switch U to turn the capacitor oltage back on and thus stop time measurement. - Read all time t and capacitor oltage U and record with all or rise distance s. Analysis Use both methods to deduce the charge and radius o the drops. The iscosity o, η, is 1.824 10-5 Nsm -2, and you will need to calculate the density o the. A plot o the charge against radius should resemble that shown in Fig. 4. Fig 4. Measurements o the charge and radius o drops. Cunningham ound that there was a small deiation rom Stoke s equation or the riction or small drops with a radius a (this is the deiation shown or small diameters in Fig. 4). This results is a modiication o the equation or the charge such that q q ' = A 1 + a where the constant A takes the alue 0.07776 10-6 m at standard pressure and at 25 o C. Reanalyse your results using this correction. t Reerences Adapted rom Leybold instruction sheets 559 421, 575 451 and 559 411. Millikin s drop experiment 6
P5.6e Millikin s drop experiment 7