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Question: In an oil drop experiment, the following charges were experimentally determined for five oil droplet...

In an oil drop experiment, the following charges were experimentally determined for five oil droplets 1.2×10−18{\mathbf{1}}.{\mathbf{2}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}, 3.0×10−18{\mathbf{3}}.{\mathbf{0}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}, 6.0×10−18{\mathbf{6}}.{\mathbf{0}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}, 2.4×10−18{\mathbf{2}}.{\mathbf{4}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}and  3.6×10−18{\text{ }}{\mathbf{3}}.{\mathbf{6}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}} units. What should be charged on the electron in this arbitrary unit?

Explanation

Solution

We have to know that the formula for charge can be represented as q=ne{\mathbf{q}} = {\mathbf{ne}}. The oil droplets become charged when they are sprayed through a nozzle via friction. The oil droplets enter the space between the plates and from there can be controlled by changing the voltage across the plates.

Complete answer:
We have to know that the magnitude of charge should be smallest and others should be multiple of this integral value. The smallest charge can be finding out by using the formula q=ne{\mathbf{q}} = {\mathbf{ne}} so in the given question the smallest charge is 1.2×10−18{\mathbf{1}}.{\mathbf{2}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}} and the other values are integral multiple of the smallest charge values.
The values observed from experiment are shown below:
Case 1: 1.2×10−18{\mathbf{1}}.{\mathbf{2}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}
Case 2: 3.0×10−18{\mathbf{3}}.0 \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}
Case 3: 6.0×10−18{\mathbf{6}}.{\mathbf{0}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}
Case 4: 2.4×10−18{\mathbf{2}}.{\mathbf{4}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}
Case 5:  3.6×10−18{\text{ }}{\mathbf{3}}.{\mathbf{6}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}
We must have to know that in the above given values smallest values is 1.2×10−18{\mathbf{1}}.{\mathbf{2}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}} but if we look at all the other values that were observed it is seen that all carries the same charge as that of the smallest value. Thus the magnitude of charge in this experiment will be 1.2×10−18{\mathbf{1}}.{\mathbf{2}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}.
So it is concluded that the charge on the electron is 1.2×10−18{\mathbf{1}}.{\mathbf{2}} \times {\mathbf{1}}{{\mathbf{0}}^{ - {\mathbf{18}}}}.

Note:
We have to know that this experiment is known as Milliken’s experiment of oil droplets which signifies that electric charge is quantized. He believed that there is a smallest unit of charge and he needed to prove it. He succeeds and the charge on electrons can be identified using his experiment. Vacuum pump oil is used in oil droplet experiments.