(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm doing Millikan's oil drop experiment in a lab. I've got a set of measurements and all that, but in analysis of the data I can't seem to get values for the charges on individual drops that seem reasonable. I find that each drop is carrying between 10 and 300 elementary charges. I've gone through my work over and over, and I'd appreciate if anybody here could point out if I'm doing something wrong.

2. Relevant equations

The equations given for an oil drop falling through air and rising in an electric field are, respectively:

[tex]6 \pi r \eta u_g = \frac{4}{3}\pi r^3 (\sigma - \rho ) g[/tex]

[tex]6 \pi r \eta u_E = qE - \frac{4}{3} \pi r^3 (\sigma - \rho ) g[/tex]

where r is the radius of the drop, η is viscosity of air, u_{g}is the speed at which the drop falls under the influence of gravity, u_{E}is the speed at which the drop rises in an electric field, σ is the density of the oil, ρ is the density of air, q is the charge on the drop, and E is the electric field strength. From these two equations are derived expressions for the radius and for the charge on a drop:

[tex] r = \sqrt{\frac{9 \eta u_g}{2 (\sigma - \rho ) g}}[/tex]

[tex] q = 6 \pi \frac{d}{V} \eta (u_E + u_g ) r [/tex]

Where d is the distance between capacitor plates and V is the voltage applied.

Also used is Millikan's correction to Stokes' law for particles at small velocities, which involves multiplying the speed by [itex] (1 + \frac{k}{r P})^{-1}[/itex], where P is air pressure and k is a (supplied) constant. This gives a value slightly smaller than one, and the final calculation for q looks like:

[tex]q = 6 \pi \frac{d}{V} \eta (u_E + u_g ) \sqrt{\frac{9 \eta u_g}{2 (\sigma -\rho ) g}}\left( 1 + \frac{k}{r P} \right) ^{-\frac{3}{2}}[/tex]

The procedure, having made various measurements, is to first find the radius r, then use that to calculate the correction and from that the charge.

These equations are all directly out of the instruction sheet for the experiment. I didn't have to derive anything myself.

3. The attempt at a solution

I have a whole bunch of data points for different drops, so I'll use two here and go through the working. If anything looks like it's the wrong order of magnitude or something, please point it out!

The distance the droplets were measured falling/rising over was 5x10^{-3}m. Viscosity of air η was found (using an empirical formula based on room temperature) to be 1.808x10^{-5}N m s^{-2}. Density of the oil σ was 860 kg m^{-3}. Density of air ρ was 1.2101 kg m^{-3}The distance d between capacitor plates was measured as 7.53x10^{-3}m. Applied voltage V was 500 V. Air pressure P was 103,058 Pa. k was given as equal to 8.226x10^{-3}Pa m.

I measured one drop (D1) falling freely over the 5 mm distance in 3.99 seconds, and rising in the electric field in 3.09 seconds (D1 was the quickest drop both falling and rising). Another (D2) fell in 14.82 seconds and rose in 9.37 seconds (D2 was more typical).

Dividing distance by time, I found that D1 fell at u_{g1}= 1250x10^{-6}m s^{-1}and rose at u_{E1}= 1620x10^{-6}m s^{-1}, while D2 fell at u_{g2}= 338x10^{-6}m s^{-1}and rose at u_{E2}= 530x10^{-6}m s^{-1}.

I plugged the u_{g}values into the equation for r above, and found that D1 had radius r_{1}= 34.8x10^{-7}m and D2 had radius r_{2}= 18.1x10^{-7}m. Then the Stokes correction values were calculated to be c_{1}= 1.0229 and c_{2}= 1.0442.

Then using the formula for charge, I found the charges on the drops q_{1}= 495x10^{-19}C and q_{2}= 75.7x10^{-19}C. Letting q_{0}= 1.6x10^{-19}C, the elementary charge, then q_{1}≈ 309q_{0}and q_{2}≈ 47q_{0}. This is much larger than I expected, and furthermore, they aren't all that close to being integer multiples of q_{0}(it's more like 309.4 and 47.3 than 309 and 47). Note that q_{1}is the largest value for the charge I found, and the next-largest in my data set is only ≈77q_{0}(with the smallest being ≈12q_{0}). Can anybody see if there's any mistake here?

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# Homework Help: Millikan oil drop experiment lab help

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