The Millikan oil-drop experiment was the first experiment that attempted to determine the charge on the electron. Today students perform modern versions of the experiment in various ways, usually involving small latex spheres. The spheres are injected between two conducting plates held at a potential difference \(V\). Assume that the plates themselves are contained in a vacuum chamber. The spheres are of identical mass but may carry different and unknown amounts of charge. The voltage is adjusted until a selected sphere rises a constant velocity and the calculated charge on the sphere is recorded. This procedure is repeated many times for many different spheres. Eventually a graph of the results is produced: a) Write down the condition that the spheres rise with constant velocity as a function of voltage and charge. b) What shape should the theoretical curve of the charge versus the voltage be? c) How might you physically account for the lack of data between \(1000 \mathrm{~V}\) and \(2000 \mathrm{~V}\) ? d) If the distance between the plates is \(5 \mathrm{~cm}\), what is the mass of the latex spheres?

Step by step solution

01

Analyze the forces acting on the sphere

To maintain a constant velocity, the net force acting on the sphere should be zero. Thus, the gravitational force acting on the sphere should be balanced by the electrostatic force between the plates. In other words: $$ F_m = F_e $$ where, \(F_m\) is the gravitational force (force due to mass) and \(F_e\) is the electrostatic force acting on the sphere.

02

Write the forces in terms of variables

Gravitational force can be written as: $$ F_m = mg $$ where, \(m\) is the mass of the sphere, and \(g\) is the acceleration due to gravity. The electrostatic force can be written as: $$ F_e = qE $$ where, \(q\) is the charge on the sphere, and \(E\) is the electric field between the plates. Since \(F_m = F_e\), we get $$ mg = qE $$

03

Electric field in terms of potential difference

Now, we need to find the electric field between the plates, \(E\), in terms of the potential difference, \(V\). The electric field can be calculated as: $$ E = \frac{V}{d} $$ where, \(d\) is the distance between the plates.

04

Writing condition for constant velocity

Using the previous equations, we can write the following condition for the sphere to rise at a constant velocity: $$ mg = q \left(\frac{V}{d}\right) $$

05

Relationship between charge and voltage

To find the relationship between charge \(q\) and voltage \(V\), we can rewrite the above equation as: $$ q = \frac{mgd}{V} $$ This equation represents a linear relationship between charge \(q\) and the inverse of voltage \(\frac{1}{V}\). Therefore, the graph of charge versus the inverse of voltage would be a straight line.

06

Lack of data explanation

The lack of data between \(1000\mathrm{~V}\) to \(2000\mathrm{~V}\) may occur due to the limited range of charges on the spheres. It might be the case that there are no spheres with charges that correspond to the voltages in this specific range.

07

Mass of Latex Sphere

For this step, we will set our equation in terms of the mass of the sphere: $$ m = \frac{qV}{gd} $$ Given the value of the distance between the plates \(d = 5\text{cm} = 0.05\text{m}\). Since we have not been given any specific values for the charge \(q\) and voltage \(V\), we cannot calculate the exact value of the mass \(m\). If specific values for \(q\) and \(V\) are available, one can plug them into this equation, along with the value for \(g = 9.81\mathrm{~m/s^2}\), to find the mass of the latex spheres.

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