LECTURE NOTES—EXPERIMENT 1

 

PURPOSE

The purpose of this experiment is to study the production, induction, conservation, and distribution of charge.  We will use a very sophisticated electrometer to measure voltage.  The voltage measured will be proportional to the amount of charge induced in, or delivered by touching to, a Faraday ice pail or an isolated conducting sphere.

 

THEORY

Insulators are materials in which all electrons are firmly bound to atoms and are not free to move.  Insulators resist the flow of electrons.  This is referred to as resistance to current flow or, simply "resistance".  In this course, we will usually deal with resistance as a “lumped” parameter confined to circuit components called resistors.  But it is important to remember that all materials, except superconductors, exhibit some resistance as an inherent property of the bulk material.

 

Conductors are materials, usually metals, in which not all electrons are firmly bound to the atoms in the crystalline lattice of the material.  Some are free to move.  The number of free electrons per atom varies with the conducting material, but one per atom is a typical number.  The strength with which a particular (outer) electron is bound to an atom in a conductor is reduced by the influence of many other atoms nearby in the crystalline lattice of the material.  The fact that some outer electrons in metals are relatively easy to remove from the electrical attraction of the nuclei of their individual atoms means that the material exhibits a relatively low resistance (compared to non-metals) to the flow of current (electrons).  Electric fields provide the force that causes charges to move and currents (moving charges) to flow.

 

The key issue is:  Why do charges move at all?  The answer is:  They move in response to a force exerted by an “electric field.”  We will discuss electric fields in detail in Experiment 3.  For now, the following will suffice.

 

An electric field is a model, a mental construct.  Here is how it works.  Many experiments show that if there is a charge q at some location in space and we introduce a “test” charge, q0, there is a force between the charges acting along the line joining their centers given by  

                                                                                                                        (1)

If we return the charge q to its original location after each test and place the test charge successively at many, many points and map the resulting force at each position, we will observe that at any given point in space there is a force with a magnitude given by equation (1) that acts along the line joining the two charges.  This can be thought of conceptually as a “field of force.”  The field of (electric) force per unit test charge generated by the charge, q, is given by

                                                                                                                   (2)

 

Hence, the electric field is defined as the force (magnitude and direction) (in Newtons) experienced by a unit test charge, q0, (in Coulombs) at a distance r (in meters) from a given charge distribution, q. The model we have chosen to use (because it is useful) is that this interaction is mediated by the “electric field” (of force).   

 

Gauss’ Law relates the electric field measured at any point in space to the distribution of charge, q, that generated the electric field.  A qualitative statement of Gauss’ Law is:  The number of electric field lines passing through a closed surface surrounding a collection of charges is proportional to the net charge within the closed surface.  The surface is usually chosen to be a sphere or cylinder to make the math easier—another of those imperfect simplifying models for which physics is famous or infamous.  This hinges on having chosen the convention that we draw a fixed number of electric field lines from a unit charge.  A quantitative statement of Gauss’ Law is:

                                                                                                        (3)

 is a vector perpendicular to a closed surface enclosing charge, q.   is the electric field at the distance, r, from the point charge (or from an arbitrary charge distribution if the distance from the charge distribution, r, is very large compared to the size of the charge distribution—another flawed, but simplifying, model).

 

A conductor is (by definition) a material in which charges are (relatively) free to move.  Thus if there is an electric field in a conductor, due to charges introduced to the conductor, the charges will move until the electric field within the conductor is zero.  I.e., they will move until the net force on all charges in the conductor is zero.  Thus, all points in a conductor under static conditions (no charge flowing) are at the same potential and we say that the conductor is an equipotential structure.  We will see in Experiment 3 that electric field vectors are always perpendicular to equipotential surfaces.  Electric fields are NEVER parallel to equipotential surfaces because the direction of the electric field is along the line of maximum change in electric potential.  The minimum energy arrangement for an isolated charged conductor is when any excess charge is uniformly distributed on the conductor’s surface.  All this means that, under static conditions, excess charges can exist only on the surface of a conductor.  This excess charge may redistribute itself in the presence of an external electric field, as we will see in this experiment. 

 

Consider an electrically isolated, uncharged, spherical conductor.  That it is uncharged means it has no net charge.  That is, each positive atomic charge is paired with a negative electron and these pairs of charges are uniformly distributed within the conductor. 

 

Consider now an electrically isolated, spherical conductor with a net positive charge.  As we saw above, the net positive charge will, in the absence of an external electric field, be uniformly distributed on the surface of the conducting sphere.

 

Imagine what happens when the charged sphere is touched to an uncharged conducting sphere.  (Remember, both spheres are isolated—they are not in contact with anything but each other.)  Two conductors in electrical contact form a single equipotential surface.  Thus the net positive charge will distribute itself uniformly over the surfaces of the two spheres.  If the spheres are then separated, ½ the original positive charge will be found on each sphere.  This permits obtaining two spherical conductors with identical charges.  Consider what happens if we now momentarily ground one of the spheres.  This leaves us with one sphere with ½ the electrical charge of the original charged sphere and one uncharged isolated sphere.  If we again bring the spheres into contact and then separate them, the result is two spheres with ¼ the charge on the initially charged sphere.  This technique was used repeatedly in early experiments that determined the laws of electrostatics. 

 

Suppose you started with an uncharged, isolated sphere and an identical sphere containing a net charge q.  Suppose that initially the two spheres were so far apart that the 1/r2 dependence of the electric field (of force—see Equation (2)) generated by the charged sphere produced no observable effect on the uncharged sphere. Under these conditions, the paired charges on the uncharged sphere would be uniformly distributed.  What would you expect to happen to the uniform distribution of charges on the uncharged sphere when the positively charged sphere is brought close to it without touching?  Since the second sphere has a positive charge, would you expect it to attract or repel the electrons in the uncharged sphere? What about the positive charges in the uncharged sphere—would you expect them to move?  We will address these questions in this experiment.

 

Consider a long, hollow, conducting, isolated, uncharged, cylindrical shell with open ends.  (This is an approximate model of the Faraday Ice Pail we will use in the experiment.)  If you introduce a positively charged sphere into the center of the cylinder without touching the cylinder, what would you expect to happen?  Consider equation (1).  Electrons on the cylinder would experience an attractive electrical force due to the positively charged sphere.  As negative charges move to the inner surface of the cylinder, they leave behind a net surplus of positive charges which are not free to move.  Within the metal of the cylindrical shell, as we have discussed before, there would be no electric field.  The conducting cylindrical shell will be an equipotential structure and its surface will be an equipotential surface.  In static equilibrium, there will be a net negative charge on the inner surface of the cylindrical shell and a net positive charge on its outer surface.  The electric field, within the material of the cylinder, due to this separation of charges in the cylinder, is exactly equal in magnitude to and in the direction opposite to that of the electric field generated by the charged sphere inside the cylinder.  An observer or detector outside the cylinder would see exactly the same net positive charge on the surface of this cylinder as on the charged sphere.  If I now momentarily ground the cylindrical shell, while the charged sphere is still inside, what do you think would happen?  The connection to ground will provide electrons to neutralize the net positive charge on the outer surface of the cylindrical shell.  It will not affect the net negative charge on the inner surface of the cylinder.  This net negative charge on the inside of the cylinder exists in response to the electrical forces generated by the positively charged sphere.  Note that now the system consisting of the charged sphere and the cylinder is electrically neutral and no observer outside the cylinder would observe any electric field from this system.  Recall what we did.  We used a momentary grounding of the cylinder to add sufficient electrons to the cylinder to exactly neutralize the excess of positive charge on the outside of the cylinder induced by the positively charged sphere.  Thus the added amount of negative charge is exactly equal and opposite to the net positive charge on the sphere and the system (sphere plus cylinder) is now electrically neutral.  I.e., there is no net charge on the system and there is no electric field produced outside the cylinder (the outer boundary of the system).

 

If I now remove the positive sphere from the electrically isolated cylinder without touching the cylinder, would you expect there to be a net charge remaining on the cylinder or would you expect it to be neutral?  Recall that the momentary contact with ground provided a source of electrons to neutralize the positive charges on the outer surface of the cylinder.  It did not affect the negative charges on the inner surface of the cylinder.  Thus, after the positively charged sphere is removed we would expect that the cylinder would have a net negative charge.

 

Suppose we repeat the above procedure of inserting an electrically isolated positively charged sphere into the, initially neutral, isolated conducting cylinder.  This time, we touch the positively charged sphere to the cylinder and then remove the sphere.  Would you expect any residual charge on the cylinder?  If you are not sure, review the discussion above about bringing an isolated charged sphere into contact with an isolated uncharged sphere.  Also consider what it means electrically that two conductors in contact form a single equipotential surface.

 

An equipotential surface can only exist when there is no electric field parallel to the surface.  Thus charges re-distribute themselves until the net electric field (net electric force) parallel to the surface is zero.  This can only occur when the distribution of charges is uniform on the surface.  When the distribution is uniform, each charge experiences uniform electric force (Equation (1)) from all sides, hence the net electric force (electric field—Equation (2)) is zero.

 

ELECTRICAL SAFETY

Unless you manage to stick some conductor like your keys into a standard electrical outlet (DO NOT DO THIS) there is no voltage and current that you will work with in this lab that will hurt you in any way.  Please observe the following safety rules in this lab.

 

Rule 1:  Turn off the power supply before handling any connections on, or wires connected to, the power supply.

 

Rule 2:  Never touch two terminals of a power supply together.  Voltages of 5-15V and currents greater than 10A can result in flying, molten metal particles that are, at a minimum, painful and at a maximum can permanently damage your eyesight.

 

Rule 3:  If you must handle wires or connections at a high voltage, use only one hand with the other hand in a pocket or held behind you.  There are no voltages and currents, other than those directly from the wall outlets, in this lab that can create a problem and if you obey Rule 1 the situation will never arise anyway.

 

THE EXPERIMENT

There are three parts to this experiment.  The manual is quite detailed and complete and I will not go into detail on the experiments here.  My goal here is to help you see the “forest” because it is very easy to get lost in the details of the experiment (the “trees”) and lose track of the point of the experiment (the “forest”).  A famous expression that makes this point and is popular in US Air Force circles is:  “When you are up to your behind in alligators, it is hard to remember that you set out to drain the swamp.”  (This is a bowdlerized version, but you get the point.)

 

Part 1.  In Part 1, we will generate charges using what are referred to in your manual as Charge Producers.  By rubbing them together, charge is transferred from one to the other.  This is exactly the process that takes place when you rub a plastic rod with fur or a glass rod with silk or when you scuff your feet on a carpet on a very dry winter day.  We will place these Charge Producers into an isolated, conducting, cylindrical shell as described above in the THEORY section.  This cylindrical shell is called a Faraday Ice Pail or Faraday Charge Pail.  It consists of an electrically isolated wire-mesh cylinder surrounded by another, larger, mesh cylinder that serves to shield the Charge Pail from the influence of external charge distributions (electric fields).  It is very important that the shield be kept grounded and the Charge Pail be grounded when you are directed in the manual to do so.  We will examine charging the Charge Pail through induction by, and by contact with, the Charge Producers.  You will use a very sensitive electrometer to measure the voltage (electrical potential difference) between the Charge Pail and the grounded shield.  The voltage measured using the Electrometer, as we will see quantitatively in Experiment 2, is proportional to the net charge on the pail.  In this experiment, the actual value of the voltage is not important.  What is important is the comparison of the voltage at one point in the experiment to the voltage in another.  These comparisons are a surrogate for comparisons of net charge because voltage is proportional to charge in this experiment.  The constant of proportionality is the capacitance, C, and hence q = CV).

 

Part 2.  In Part 2, we examine conservation of charge.  Charge is conserved and we will attempt to demonstrate this with the Charge Pail and the Charge Producers.  In this experiment, a very large part of the goal is to get the provided equipment to give the expected results.  This will not always happen the first time.  You may have to ground everything and repeat parts of the experiment several times.  If you are taking this course in Fall Semester, you will have to deal with the influence of high humidity that tends to permit charges to “bleed off” readily.  If you are taking the course in Winter Semester, you will have to deal with low humidity that keeps charges from bleeding off readily (good news), but also allows for the build-up of static charges on your body and clothes (bad news).  These static charges can have attention-getting results as when you scuff your feet across a carpet on a very cold day and then touch a metal doorknob.

 

Part 3.  In Part 3, we use two isolated, conducting spheres.  We will charge, then isolate, one (the Charged Sphere) and then momentarily ground (and hence neutralize all excess charges on) the second (the Sampling Sphere).  We will then observe what happens to the charge distribution on the Sampling Sphere when the Charged Sphere is brought close to the Sampling Sphere.  We will sample the charge distribution on the Sampling Sphere with what is called in your manual a Proof Plane.  It is a plastic wand with flat paddle on one end.  The surface of the paddle is aluminized.  When held so the surface of the paddle is tangent to the surface of the sphere and brought into contact with the sphere, the surface of the Proof Plane and the sphere form an equipotential surface and the Proof Plane acquires a charge density representative of that on the surface of the sphere at the point of contact.  Recall the observation above that two conductors in contact comprise a single equipotential surface. Because the Proof Plane removes some of the charge from the sampled sphere, it is important to not ground the Proof Plane between contacts with the sphere being sampled.  This prevents depleting the net charge on the sampled sphere.  By not grounding the Proof Plane between contacts with the Sampling Sphere, each time the Proof Plane contacts the sphere, any charge removed previously is returned to the sphere.

 

These are not high-precision or high accuracy experiments.  The charge bleeds off the Charge Producers and Proof Plane relatively quickly when there is significant water vapor in the air.  Hence these experiments work better in January than in September.  Take your Electrometer readings quickly.  Move the Charge Producers into and out of the Faraday Charge Pail slowly.

 

Let’s look at what happens physically when you introduce a positively charged Charge Producer into the Faraday Charge Pail.  Note that the pail and shield are each connected to one of the two plates on the capacitor in the Electrometer.  This capacitor is not shown in Figure 1.  As shown in Figure 1, the positively charged Charge Producer induces a negative charge on the inside of the Faraday Charge pail.  It does this by attracting negative charges from the outside of the pail and from the wire connecting the pail to one plate of the Electrometer Capacitor.  This leaves a net positive charge on the plate of the Electrometer capacitor connected to the pail.  The other plate of the Electrometer Capacitor is connected to earth ground and draws from the earth ground enough electrons to balance the positive charge on the plate connected to the pail.  The positive charge on the outside of the Faraday Ice Pail induces a negative charge on the inside of the shield.  This negative charge on the inside of the shield comes from electrons attracted from the outside of the shield.  If the shield were not connected to earth ground, the outside of the shield would be left with a net positive charge.  But the shield is connected to earth ground and the positive charges on the outside of the shield attract sufficient electrons from ground to neutralize the outer surface of the shield.  The shield now has a net negative charge.  The potential difference between the two plates of the Electrometer Capacitor is proportional to the charge on the Charge Producer and appears on the Electrometer meter.  The voltage recorded by the Electrometer is related to the charge by

 

                                                                                                     (4)

 

QElectrometer is the magnitude of the charge on the capacitor in the Electrometer and CElectrometer is the capacitance of the Electrometer capacitor.  We will explore equation (4) in Experiment 2.

                       

                                    Figure 1.  Charge distribution resulting from

charge induced on the Faraday Charge Pail.

 

When you connect something to ground, the ground may drain off an excess of negative charges or act as a source of negative charges to neutralize a surplus of positive charge.  NOTE:  This surplus of positive charge may be only local.  That is, a conducting body may have zero net charge (e.g., as a result of having been momentarily grounded), but may, as a result of some external distribution of charges, have a non-uniform distribution of charge on its surface.  Electrons are free to move.  Under the influence of the electric “force” field produced by an external charge distribution, the electrons in a conductor may redistribute themselves.  The positive charges of the atoms in the metal lattice are not free to move.  As a result, one portion of the conductor may have a net negative charge while another portion may have a net positive charge.  The entire conductor, however, may be electrically neutral—i.e., it may have no net charge on it.

 

I have not told you exactly what will happen and why when, in Part 3, you look at the charge distribution on a conducting sphere in response to the proximity of a charged conducting sphere.  However, I have given you the information in these lecture notes you need to figure it out.   Note that the most significant difference between the two spheres in Part 3 and the Charge Producer and Pail in Figure 1 is that while the Charged Sphere (analogous to the Charge Producer in Figure 1) is connected to the power supply and held at a fixed voltage, the Sampling Sphere (analogous to the Pail in Figure 1) is isolated except when momentarily grounded as directed in the steps in your manual.

 

The Electrometer we will use is state-of-the-art for teaching-labs.  It has an internal resistance of 1014Ω.  As you will see in later labs when we work with other voltage measuring devices, this is a HUGE resistance.  For all practical purposes, it is an open circuit.  The key point is that the internal resistance of the meter is so large that effectively no current flows through the meter and hence it does not change the charge distribution on the Charge Pail.  As we will see in great detail later, nearly all electrical measuring devices perturb the circuit being measured by drawing current from it.  By Ohm’s Law, the voltage across a resistor is given by the current through the resistor times the resistance,

 

                                                V = IR.                                                                               (5)

 

A measuring device that draws current from the circuit will reduce the current through the resistor and as a result the device will measure a voltage smaller than actually exists across the resistor when the meter is not connected.  Don’t get hung up on this right now.  Just keep in the back of your mind the truism that nearly all measuring devices perturb the circuits being measured and hence produce measurements that are not truly representative of the parameters in the unmeasured circuit.  We will explore this in great detail.  This is one of the major unifying themes of this course.

 

TIPS

-          The Electrometer is so sensitive you must discharge it before each use.

-          The Electrometer is so sensitive that you must regularly ground yourself by touching the outer shield on the Faraday Charge Pail, e.g., with the edge of your hand.  To convince yourself of the importance of this, without touching the shield, move your hands and arms around over the Faraday Charge Pail—you will see the needle on the Electrometer move.  This is particularly important in January when the air is so dry that you can easily build up thousands of volts in static charge just by moving around.

-          Keep the Electrometer as far as possible from the Faraday Charge Pail.

-          Do not touch anything connected to the Electrometer unless you are grounded (e.g., by touching the shield on the Faraday Charge Pail, or by touching the outer sheath of a BNC connector at the end of a coaxial cable).  On a dry day, you can easily acquire a charge sufficient to “peg” the Electrometer.  To peg a meter means to make the needle hit one stop or the other with great force.  This can easily damage an analog meter such as the Electrometer.

-          It is vital the Electrometer and the Electronic Voltage Supply be grounded to a single “earth ground” such as the green female banana plug receptacle on the side of the power block above the lab benches.

-          Set the Electrometer initially on the 100V scale so you will not peg the meter.  Move to a more sensitive range as needed.

-          The total charge on an isolated body is constant.  But it may not be uniformly distributed if the body is in an external electric field.

-          DO NOT rub the aluminum-surfaced Proof Plane on the blue or white Charge Producers.  The white or blue surface becomes contaminated and the Charge Producers no longer work properly.

-          DO NOT touch the blue or white Charge Producers with your fingers.  Oils from your hands will interfere with the operation of the Charge Producers.

-          Do not touch the conducting spheres with your skin.  Skin oils will damage them.

-          Some electrons in metals are free to move.  The positive charges are fixed.  Hence a net positive charge in a indicates of an absence of electrons.  A net negative charge indicates an excess of electrons beyond what is needed to neutralize the positive charges.

 

QUESTIONS AT THE END OF THE EXPERIMENT

In answering the questions, keep in mind the following points.

-          Two conductors in contact comprise a single equipotential surface.

-          Charge is conserved.

-          When you rub fur on plastic or silk on glass, the magnitude of the net charge on the fur and plastic or silk and glass is zero.

-          Gauss’ Law

-          These notes.

-          I will take off points for things that are blatantly wrong.

-          If an answer shows a sincere effort, but lack of understanding, I will comment, but be merciful.

-          Your answers must be legible.

-          You must attempt to answer all questions and do all exercises even if you did not complete the relevant portion of the experiment

 

If you send me an e-mail indicating you have read these lecture notes, I will raise your final semester grade by 1%.