# How many electrons must I rub off a ruler to pick up a piece of paper from 2cm (1 inch) away. Show thinking?

Obviously F=mg, but how would I determine the force on the paper if the particles are uniformly distributed in the ruler/paper? Can I idealize the paper/ruler as point particles? Or will I need to use some integration to get the electric fields. Also, polarization of the (neutral?) paper comes into the picture... I'm more comfortable with math than physics, mostly unsure about what approximations are reasonable, and there's so much going on in my head with this. Please help!

### 3 Answers

- CwCcLv 79 years agoFavorite Answer
That depends on a lot of things. In particular, you would need to find the electric susceptibility of the paper. The field produced by static charge is inverse-square. The polarization of the paper varies directly with the external field, with a proportionality constant equal to the electric susceptibility. The dipole field created by the polarized paper is inverse-cube. Overall, the force between the ruler and the paper will vary with the inverse fifth power of distance. To get any more detailed, you would need to know the geometry of the situation, and the charge distribution.

- AenimaLv 69 years ago
You'll need to use newton's laws. You'll also have to make many suppositions such as: how you orient the ruler with the paper, surface charge density of paper, and other things.

Fnet = ma

F(electric) - F(gravity) = ma.

If you want to know the minimum amount of electric force to pick the paper up, you'll want that

F(electric force)>= Fg

Once you have that, you'll need to use a single charge on the tip of the ruler dq, and find the force dF on every charge on the paper of magnitude dQ.

dF = kdqdQ/r^2, where r will not be constant, dQ will be a very small amount of charge per unit volume of the paper, dq a very small charge per unit area of the ruler (if metal), if an insulator you'd have that dq is a very small charge per unit volume.

Once you solve for the force in terms of the surface charge densities and dimensions between the paper and the ruler, you'll plug F(electric) in the formula, solve for little q, and use the charge per electron to substitute q for electron charge. That will tell you the minimum amount of electrons needed so that F(electric) = F(gravity). More electrons on the ruler will give an upward acceleration. less electrons on paper will do the same.

- zeimusuLv 79 years ago
There is no simple answer as it depends on the dielectric properties of the paper. It is possible to model the paper as as single dipole with charge only on its top and bottom surfaces. However the size of this dipole depends on the paper, so is ultimately the result of experiment. Initially you should model the force as a uniform field.

Short answer It is a massive number, but a tiny fraction of the electrons that are present.

If you want to keep yourself busy over the summer, work through the MIT 802 course on electrostatics.