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Millikan's oil drop experiment
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Aim

 

  1. To experimentally demonstrate the concept of Millikan’s oil drop experiment.
  2. To find the terminal velocity of the drop.
  3. To find the charge on a drop.

 

Apparatus

 

Millikan’s oil drop apparatus,oil, Dc supply.

 

Construction

 

Oil drop experiment was performed originally by the American physicist Robert A. Millikan in 1909. It measures the size of charge on a single electron.

 

Apparatus consist of an atomizer, which helps to spray tiny droplets. By means of a short focal distance telescope, the droplets can be viewed. There are two plates, one positive and the other negative above and below the bottom chamber.dc supply is attached to the plates. Some of the oil drops fall through the hole in the upper plate.

Using X-rays the bottom chamber is illuminated causing the air to ionize. As the droplets traverses through the air, electrons accumulate over the droplets and negative charge is acquired. With the help of dc supply a voltage is applied. Speed of its motion can be controlled by altering the voltage applied on the plates. By adjusting the voltage applied, drop can be suspended in air. Millikan observed one drop after another, varying the voltage and noting the effect. After many repetitions he concluded that charge could assume only certain fixed values.

He repeated the experiment for many droplets and confirmed that the charges were all multiples of some fundamental value and calculated it to be 1.5924(17) ×10−19 C, within one percent of the currently accepted value of 1.602176487(40) ×10−19 C. He proposed that this was the charge of a single electron.

 

 Theory

 

Initially the oil drops are allowed to fall between the plates in the absence of electric field. Due to gravity they accelerate first, but gradually slowdown because of air resistance.

 

The terminal velocity v1 in the absence of an electric field is calculated as

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»v«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mfrac»«msub»«mi»l«/mi»«mn»1«/mn»«/msub»«msub»«mi»t«/mi»«mn»1«/mn»«/msub»«/mfrac»«/math»

where 'l1' is the distance travelled by the oil drop and ’t1’ is the time taken.

 

The drag force acting upon the drop is calculated from stokes’s law and is given as

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»F«/mi»«mi»v«/mi»«/msub»«mo»=«/mo»«mn»6«/mn»«mi»§#960;«/mi»«mi»§#951;«/mi»«mi»r«/mi»«msub»«mi»v«/mi»«mn»1«/mn»«/msub»«/math»

The apparent weight (true weight minus up thrust) for a perfectly spherical body is given by,

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»F«/mi»«mi»G«/mi»«/msub»«mo»=«/mo»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«mi»§#960;«/mi»«msup»«mi»r«/mi»«mn»3«/mn»«/msup»«mi»g«/mi»«mo»(«/mo»«mi»§#961;«/mi»«mo»-«/mo»«msub»«mi»§#961;«/mi»«mrow»«mi»a«/mi»«mi»i«/mi»«mi»r«/mi»«/mrow»«/msub»«mo»)«/mo»«/math»

At terminal velocity the oil drop is not accelerating, so the total force acting on it must be zero

                           FV-FG=0.

i.e.,

                           «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»F«/mi»«mi»V«/mi»«/msub»«mo»=«/mo»«msub»«mi»F«/mi»«mi»G«/mi»«/msub»«/math»
                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mfrac»«mrow»«mn»9«/mn»«mi»§#951;«/mi»«msub»«mi»v«/mi»«mn»1«/mn»«/msub»«/mrow»«mrow»«mn»2«/mn»«mi»g«/mi»«mo»(«/mo»«mi»§#961;«/mi»«mo»-«/mo»«msub»«mi»§#961;«/mi»«mrow»«mi»a«/mi»«mi»i«/mi»«mi»r«/mi»«/mrow»«/msub»«mo»)«/mo»«/mrow»«/mfrac»«/math»

r-radius of oil drop
η-viscosity of air
V1-terminal velocity
g-acceleration due to gravity
ρ-density of liquid
ρair-density of air
 


Now a field is produced in the bottom chamber with the supply voltage. A likely looking drop is selected and kept in the middle of the field of view by adjusting the voltage.

If the electric forces Fe, balances the gravitational force FG, the drop suspends in the air.
Then,

 

                                «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»F«/mi»«mi»e«/mi»«/msub»«mo»=«/mo»«msub»«mi»F«/mi»«mi»G«/mi»«/msub»«/math»

                                 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»q«/mi»«mi»E«/mi»«mo»=«/mo»«mi»m«/mi»«mi»g«/mi»«/math»

                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»q«/mi»«mi»V«/mi»«/mrow»«mi»d«/mi»«/mfrac»«mo»=«/mo»«mi»m«/mi»«mi»g«/mi»«/math»

 

where V is the balancing potential and d is the distance between the plates.

 

If the applied electric force Fe is greater than the downward forces, some of the drops (the charged ones) will start to rise. Now the electric force will act upwards,gravity and viscous forces acts downwards.

Corresponding terminal velocity v2 is calculated as, 

                 «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»v«/mi»«mn»2«/mn»«/msub»«mo»=«/mo»«mfrac»«msub»«mi»l«/mi»«mn»2«/mn»«/msub»«msub»«mi»t«/mi»«mn»2«/mn»«/msub»«/mfrac»«/math»        

where l2 is the distance travelled by the oil drop and t2 the time taken.

Now the total force acting on drop is Fe-F’v-FG=0.

                                       Fe=F’v+FG

 

F'V is the new viscous force under the action of electric field.

                           «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi»q«/mi»«mi»E«/mi»«mo»=«/mo»«mn»6«/mn»«mi»§#960;«/mi»«mi»§#951;«/mi»«mi»r«/mi»«msub»«mi»v«/mi»«mn»2«/mn»«/msub»«mo»+«/mo»«mn»6«/mn»«mi»§#960;«/mi»«mi»§#951;«/mi»«mi»r«/mi»«msub»«mi»v«/mi»«mn»1«/mn»«/msub»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»

                             «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»q«/mi»«mi»V«/mi»«/mrow»«mi»d«/mi»«/mfrac»«mo»=«/mo»«mn»6«/mn»«mi»§#960;«/mi»«mi»§#951;«/mi»«mi»r«/mi»«mo»(«/mo»«msub»«mi»v«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi»v«/mi»«mn»2«/mn»«/msub»«mo»)«/mo»«/math»      

    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»q«/mi»«mo»=«/mo»«mn»6«/mn»«mi»§#960;«/mi»«mi»§#951;«/mi»«mi»r«/mi»«mo»(«/mo»«msub»«mi»v«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi»v«/mi»«mn»2«/mn»«/msub»«mo»)«/mo»«mfrac»«mi»d«/mi»«mi»V«/mi»«/mfrac»«/math»

Millikan repeated the experiment no. of times, each time varying the strength of X-rays ionizing the air. As a result no. of electrons attaching to the oil drop varied. Then he obtained various values for q, and is found to be a multiple of 1.6 x 10-19C.

 

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