In this problem, we will go through the famous experiment led by Robert A. Millikan. The charge of the electron that he calculated by this experiment is 0.6% off from the currently accepted value, that too due to the imprecise value of viscosity of air known at the time. This experiment demonstrates that the electric charge of the oil droplet is some integer multiple of electron charge - thereby establishing charge quantization as an experimental fact. It's a free-body diagram. Here, we depict an oil droplet that is falling downwards due to gravity in an air medium. The droplet experiences an upward force due to air friction. When the two forces are on the droplet balance, the droplet falls steadily with velocity vd. Find the friction coefficient k. Given the symbolic expression for the mass of the oil droplet m, acceleration due to gravity g, downward terminal velocity of the droplet vd. Give the answer in terms of these variables. a) Write the mathematical expression for the friction coefficient k. Now, we negatively charge the oil droplet and place it in between the charged plates. There is a voltage V=11.42Volt between the plates and the separation between the plates is d=3.09mm. Previously we have seen the droplet was steadily falling downwards. Now, due to the electric force on the droplet, it starts to move upwards, towards the positive plate. Hence, there's a force downwards on the droplet due to the air friction, as we can see from the free body diagram above. When all the forces acting on the droplet balance, the droplet steadily moves upwards. Use the symbolic expression for the mass of the oil droplet m, acceleration due to gravity g, upwards terminal velocity of the droplet vu, downwards terminal velocity of the droplet vd, potential difference V, separation between the plates d. Give the answer in terms of these variables. b) Write the mathematical expression for charge q. To proceed further, we need to know the mass of the oil droplet. So, Millikan turned off the electric field by taking the plates away. Hence, the droplet falls freely due to gravity. The viscous drag force Fd due to the air acts on the droplet against its weight Fg. The droplet soon reaches the terminal velocity when the forces balance. Stoke's law gives the drag force on the spherical droplet as it moves with the terminal velocity in air. If the viscous coeffecient is denoted by η, terminal velocity by vu, radius of the spherical droplet a and the density of oil ρ - the viscous drag force is given by :- Fd=6×π×a×η×vu. In equilibrium, Fg=Fd - from which Millikan was able to calculate the radius of the oil droplet using the known values of air viscous coeffcient and the density of oil. In our problem, we are given the radius a=0.5μm found from the experimental procedure described here. The density of the oil is ρ=824kgm−3. We have to calculate the mass of the oil droplet. c) Find the mass of the oil droplet. Now, we can calculate the charge using the equation in (b). Because the value of velocity of the droplet (∼10−5) is very small compared to the values of other parameters, we make the approximation vd+vu≃vd in the numerator of equation in (b). d) Find the charge of the oil droplet. Now, we have the charge of the droplet. To test the charge quantization hypothesis, we'll have to consider the charge of the oil droplet is some integer multiple of an electron charge. Consider the numerical value of electron charge 1.6×10−19 without the sign e) Find the integer multiple of an electron charge for the charge of the oil droplet.
Stuck in part (d) and (e)
In this problem, we will go through the famous experiment led by Robert A. Millikan. The charge of the electron that he calculated by this experiment is 0.6% off from the currently accepted value, that too due to the imprecise value of viscosity of air known at the time. This experiment demonstrates that the electric charge of the oil droplet is some integer multiple of electron charge - thereby establishing charge quantization as an experimental fact.
It's a free-body diagram. Here, we depict an oil droplet that is falling downwards due to gravity in an air medium. The droplet experiences an upward force due to air friction. When the two forces are on the droplet balance, the droplet falls steadily with velocity vd. Find the friction coefficient k.
Given the symbolic expression for the mass of the oil droplet m, acceleration due to gravity g, downward terminal velocity of the droplet vd. Give the answer in terms of these variables.
a) Write the mathematical expression for the friction coefficient k.
Now, we negatively charge the oil droplet and place it in between the charged plates. There is a voltage V=11.42Volt between the plates and the separation between the plates is d=3.09mm. Previously we have seen the droplet was steadily falling downwards. Now, due to the electric force on the droplet, it starts to move upwards, towards the positive plate. Hence, there's a force downwards on the droplet due to the air friction, as we can see from the free body diagram above. When all the forces acting on the droplet balance, the droplet steadily moves upwards.
Use the symbolic expression for the mass of the oil droplet m, acceleration due to gravity g, upwards terminal velocity of the droplet vu, downwards terminal velocity of the droplet vd, potential difference V, separation between the plates d. Give the answer in terms of these variables.
b) Write the mathematical expression for charge q.
To proceed further, we need to know the mass of the oil droplet. So, Millikan turned off the electric field by taking the plates away. Hence, the droplet falls freely due to gravity. The viscous drag force Fd due to the air acts on the droplet against its weight Fg. The droplet soon reaches the terminal velocity when the forces balance. Stoke's law gives the drag force on the spherical droplet as it moves with the terminal velocity in air. If the viscous coeffecient is denoted by η, terminal velocity by vu, radius of the spherical droplet a and the density of oil ρ - the viscous drag force is given by :- Fd=6×π×a×η×vu. In equilibrium, Fg=Fd - from which Millikan was able to calculate the radius of the oil droplet using the known values of air viscous coeffcient and the density of oil. In our problem, we are given the radius a=0.5μm found from the experimental procedure described here. The density of the oil is ρ=824kgm−3. We have to calculate the mass of the oil droplet.
c) Find the mass of the oil droplet.
Now, we can calculate the charge using the equation in (b). Because the value of velocity of the droplet (∼10−5) is very small compared to the values of other parameters, we make the approximation vd+vu≃vd in the numerator of equation in (b).
d) Find the charge of the oil droplet.
Now, we have the charge of the droplet. To test the charge quantization hypothesis, we'll have to consider the charge of the oil droplet is some integer multiple of an electron charge.
Consider the numerical value of electron charge 1.6×10−19 without the sign
e) Find the integer multiple of an electron charge for the charge of the oil droplet.
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