Physics 204:  Spring 2008

 

Static Electricity and the Coulomb Force

 

On a dry day electrostatic effects are easy to see.  We will make use of hard rubber rods, which can be charged negative by rubbing them with fur, and glass rods, which can be charged positive by rubbing with silk.  No simple theory says why this should work, but it is a fact known since the early 18th century or so.  The dependence of these effects on atmospheric conditions, and on the surface condition of the rods, is also somewhat mysterious.  (Everything works best if the rods are clean, not oily from handling, for example.)  It is really quite amazing that anything simple and comprehensible comes out of this!

 

The most noticeable effect is the Coulomb force of repulsion or attraction between charges.  To see it we will use little pith balls painted on the outside with a conducting paint.  They are very light, so that even a small force can disturb them noticeably.  The conducting paint is there so that if charge is placed on the pith balls, by touching with a charged rod, for example, the charge will spread around on the surface, possibly uniformly, but also possibly non-uniformly, with more charge on one side than the other (see the first suggested exercise below).  The pith balls hang on insulating threads, making them, in effect, little pendulums.

 

You can drain charge off a pith ball by touching it to a large conductor.  The charge will spread out over the entire conductor, leaving only a negligible amount on the pith ball.   The charge distributes itself in proportion to the capacitances:   most of the charge is on the big capacitance, i.e., the big conductor, with only a negligible charge left on the pith ball, with its negligibly small capacitance (small size).

 

The lab has three parts,

 

(1)    Free (directed) play, to see how charge can be manipulated

(2)    A quantitative look at the electrostatic force

(3)    Force as a vector

 

There is also an appendix on vector equilibrium of forces, for you to check out and ask questions about, if you like.

 

I.                   Free play

 

Charge up a rod and see if it exerts any force on an uncharged pith ball.  You might expect no force, but I think you will be surprised!  What is going on? 

(Recall a remark above that referred to this situation.)  See if you can move charge around in a way that makes sense.  Can you demonstrate that hard rubber charges up oppositely to glass, for example?  Can you transfer positive charge onto hard rubber, or negative charge onto glass?  Describe some of this in your final lab summary.

 

II.                A quantitative look at the electrostatic force

 

When we see how a pith ball on a thread is displaced by a nearby rod, we can actually estimate the force on the pith ball.  Suppose the pith ball moves over a small distance x from its old equilibrium position (hanging straight down) to its new equilibrium position (in the presence of the rod).  If we remember that a mass m hanging on a string of length L (i.e., a pendulum) is in effect on a spring with Hooke’s law constant k=mg/L, we see that moving it over by x requires a force kx=mgx/L.  This must be the magnitude of the Coulomb force that displaces the pith ball.  Estimate it quantitatively for a case typical of what you have been looking at.  To determine the mass m of a pith ball, it might be best to weigh a lot of them together and divide by the total number to get an average mass –  good enough for the purpose of this estimation.  Notice that for a given force on the ball, smaller m means bigger x, i.e. easier to see and measure.  That is why the balls are made of pith and not, say, steel!

 

Auguste Coulomb used this method to measure the force and to determine that it is proportional to the inverse square of the distance.   His method is illustrated below for two pith balls each positively charged.  You can slide the threads on the supporting rod, thereby changing D, and hence the electrostatic force (as revealed by the displacement x).

 

 

This is a difficult measurement to do, for many reasons, but see if you can determine, even approximately, how the force falls off with distance.   If you are brave, look for a power law connecting x and D (i.e., make a log-log plot).  I would be very interested to see even an attempt at this!

 

III.             Force as a vector

 

We have not called too much attention to the vector nature of force, but this is a very good place to see it.  Suppose you had positive charges A, B, and C, as shown below, and you ask about the net force on C due to the other two.

 

 The repulsion forces are along the lines connecting the charges, so the force on C is the sum of two forces in the directions shown.  What does this mean?  The net force is the vector sum, found by placing the individual vectors to be added head to tail, as illustrated below.

Use a pith ball as charge C and let charged rods be the charges A and B.  Is the force on C in the direction you expect?  You should also try letting some of the charges be negative and some positive:  that will produce Coulomb forces that are attractive instead of repulsive, but they still add as vectors.

 

 

       IV.   Homemade Xerox

           

In a Xerox copier tiny  plastic spheres, negatively charged, (and covered with a layer of ink), are attracted to positively charged paper.  We can do this in a crude way ourselves.  The inky spheres (toner) are sprinkled onto a metal tray connected to the positive side of an electrostatic generator.  Being in contact with this tray, they will pick up positive charge.  Now we lay a sheet of paper over the tray, with a cut-out stencil between the paper and the toner.  We touch a grounded electrode, attached to the other side of the electrostatic generator,  to the paper.  The positively charged toner particles fall through the potential difference, meaning they jump up and adhere to the paper.  With the stencil in place, only an image of the cutout region will receive the toner.  After the toner is  deposited onto the paper, we can fix it permanently by heating the paper on a hot plate.  This melts the plastic and the ink goes into the porous paper.  Try creating such a Xerox image!

 

 

 

V.                 Appendix on Equilibrium of Forces

 

In Section II we considered a pith ball hanging on a thread of length L, displaced a small horizontal distance x by an electrostatic force.  Our method for estimating the force relied on remembering that a pendulum is like a spring, with spring constant mg/L, but suppose you had forgotten that, or never knew it?  A more straightforward way to this result is to consider that the net force on the pith ball in equilibrium is zero.  This vector sum is comprised of the three forces on the ball:  the tension T in the string, which pulls along the string, the weight mg of the pith ball, which is down, and the electrostatic force F, which is horizontal.  These are

shown below, added head to tail, with the angle  also indicated, the angle at which the pith ball hangs away from the vertical.

 

Because the vector sum is zero, the three vectors form a triangle, a closed figure.  It is, in fact a right triangle since mg is vertical and F is horizontal.  Note that  the angle , from the original picture of the pith ball hanging, is x/L, in small angle approximation.  We will give an exact argument to find F, and then express F in small angle approximation at the end.  From the figure we can see that

                                                                                                                   

Thus, if we were trying to determine F, we would use T=mg/cos to eliminate T, finding F=(mg/cos)sin=mg tan.  (More simply, this relationship can also be read off the figure!)  Then in small angle approximation tanx/L, so Fmgx/L, as we said above in Section II.