Electric Fields and Potential

Purpose

• •
  To develop an understanding of the relationship between equipotential lines and electric field lines.
• •
  To investigate equipotential lines produced by three different configurations of charge.
• •
  To discover the cause of electric shielding.

Equipment

We'll use the Electric Field Mapping apparatus in this lab activity. You can get quick access to help by rolling your mouse over most objects on the screen. Complete instructions are provided in Explore the Apparatus.

Simulation and Tools

Open the Electric Field Mapping simulation to do this lab.

Theory

Everyone has experienced the fun of rubbing a balloon against a sweater and then seeing it attach itself to nearly anything nearby. And this is sometimes accompanied by a few sparks. The balloon in its uncharged state is made up of equal numbers of negatively charged electrons and positively charged protons. But when it's rubbed against the sweater, it will attract some electrons away from the sweater. When the charged balloon is brought near your body, you can feel its influence when it's still several inches away. When that happens, it's either time to run or, if you're brave, to excitedly ease a finger toward the balloon and feel your hair stand on end, then get zapped, and run for it. Or is it just me? (Nope. Please Google "static charge images.") You're hopefully aware that when a storm approaches it may be time to run for shelter, but not across an empty field, and not to the shelter of a tree (which is shelter from the rain but a dangerous place to be when lightning is imminent). We'll learn what makes a good shelter in this lab. Coulomb's Law tells us that a charge in one location can exert a force on another charge some distance away. And that will pretty well explain the effect of the charged balloon or cloud. But as your finger nears the balloon, it does feel like there's "something in the air" around the balloon. How does the balloon over there act over here?

Electric Fields

The concept of an electric field was hypothesized in the 19^th century as a way of describing this action at a distance. An electric field is said to exist in the space around any charge and the force on a second charge is exerted by the field at its location. We could visualize the field near a charged balloon by drawing many force vectors around it showing the force acting on a test charge moved from point to point. The length of these force vectors would decrease with distance according to Coulomb's Law. If instead the individual vectors are replaced by continuous lines always tangent to the force vectors, it can be seen that the density of the lines serves the same function as the length of the force vectors. The lines actually get further apart for the same reason that the force vectors get shorter—the geometry of the 3D space the forces act in. These continuous lines are called "lines of force" or "electric field lines." In this lab we will observe electric fields in two dimensions.

A Few Observations About the Behavior of Electric Fields

• •
  These electric field lines start on positive charges and end on negative charges. Or they may do only one of these two. For example, the field around a lone positive charge points away from it, and we may not be concerned with where it might end. See Figure 2, and the corresponding field for a negative charge in Figure 3.

• •
  When multiple charges are involved, the total electric field at each point is equal to the vector sum of the individual fields contributed by each charge. For example, suppose the charges in Figures 2 and 3 were moved close together. Their overlapping fields would add vectorially at every point, producing the resultant field shown in Figure 4.

• •
  The net charge on a conductor will redistribute itself on its surface until it reaches static equilibrium. That is, the charges will move about until each has zero net force acting on it. Once each charge is in equilibrium, there would be no field tangent to the surface of the conductor. Thus all electric field lines would be perpendicular to the surface of the conductor in this static case. (Any other direction would result in further motion of the charges. In this lab there will be a very small current flowing through the apparatus to allow us to measure the fields, but this steady current will be small enough that the charges will remain distributed as they would be with truly static charges.)
• •
  Since the force on a test charge can be in only one direction, we know that field lines can never cross.

Work, Electrical Potential Energy (EPE), and Electric Potential, V

It's also useful to attribute the work done by electrical forces to electric fields. In Figure 5, two infinite parallel plates produce a uniform electric field, shown in green. This field will do positive work on a positive charge moving from A to B. Thus the test charge will have a lower amount of electrical potential energy at B than at A and the energy per charge, V, will be less at B than at A. The electrical potential difference,

ΔV,

between A and B is given by where W_AB is the work done on the positive test charge by the field in moving the test charge from A to B. Note the signs involved. The electrical potential drops from A to B, so V_B – V_A would be negative. But positive work, W_AB, is done since the positive test charge moves in the direction of electric force. Thus the negative sign is required to make both sides of the equation negative. Since there is no component of the electric field along line 1, no work would be done by the field on a charge moving between A and C. Thus the electrical potential energy of a charge would be the same at any point along this line. And the difference in electrical potential between these points would be zero. This line and the other three lines parallel to it connect points of equal electrical potential and are called equipotential lines.

• •
  Equipotential lines are always perpendicular to electric field lines.

In Figure 5, the electric field between the infinite parallel plates is constant, so the work done by the field to move a charge from equipotential line 1 to line 2 is the same as the work to move the charge between line 2 and 3 or 3 and 4 or any two other equally spaced lines since the field, and thus the force, is constant. Thus we can see that the potential difference from line 1 to line 2 is the same as that between line 2 and 3, etc. Also, a charge moving from any point on line 1 to any point on line 2 would drop in potential by the same amount regardless of the starting and ending points. From equation 1

E =

F

q0

.

we can then see that the work done by the field would be the same in each case. This also means that the change in kinetic energy would be the same regardless of the starting and ending points on the two lines. This behavior is typical to all conservative forces.

• •
  For a constant field, equipotential lines representing uniform increments of potential will be uniformly spaced.

Figure 6 shows the radial field of a positive point or spherical charge. In order for the equipotential lines to always be perpendicular to the field lines, they must be circular. In three dimensions they would be spherical shells.

• •
  For the field produced by a point charge, equipotential lines representing uniform increments of potential will be increasingly farther apart with increasing distance from the charge according to the inverse square law.

We won't pursue a mathematical description of these shapes in this lab. We'll instead discover and investigate their shapes experimentally. There are two different ways of mapping the fields and equipotential lines produced by charged objects.

• •
  We could devise a way to measure the direction of the force on a small test charge at various points, and then draw a series of smooth curves tangent to the force vectors at each point. The vagueness of the term "smooth curve" makes this a difficult task.
• •
  We could devise a way to measure the electric potential at many points in the field. If we found many points at the same potential, we could connect these points with a smooth curve passing through each point at that potential and then repeat for another potential, etc. Electric field lines could then be created by drawing smooth lines passing at right angles through the equipotential lines. The direction of the electric field would always be from high to low potential. Our apparatus uses this technique.

Explore the Apparatus

Introduction to the Apparatus

Figure 7 shows the tools you'll be working with in the space between two parallel plates. Let's first look at the wiring of the circuit. The DC power supply is set at 40 volts. The positive terminal, which is at 40 volts, is connected to the right plate. One wire from the negative terminal, which is at 0 volts, is connected to the left plate. When the connections are made, electrons at the negative terminal of the power supply travel into the black wire and down to the left plate giving it a negative charge. The larger the voltage of the power supply, the more electrons will flow. At the same time, electrons leave the right plate and travel to the positive terminal of the power supply. This sets up a potential difference of 40 volts between the plates as discussed earlier. If the plates were in a vacuum, the system would remain in this condition. But the plates are attached to a weakly conductive sheet of material, which conducts a flow of electrons from the negative left plate to the positive right plate. By convention, we would say that a (conventional) current flows in the opposite direction. If the sheet was an excellent conductor, we wouldn't have the buildup of charge on the two plates to produce our fields. A third short wire connects the negative, 0-volt terminal of the power supply to the negative terminal of the voltmeter. This keeps that terminal of the voltmeter at 0 volts. The left, red wire of the voltmeter has a probe on the free end that can be dragged to any part of the conducting sheet or either plate. When this is done, the voltmeter displays the potential difference between the 0-volt side of the meter and the point touched. We'll say that the voltmeter's reading at that point is the potential or voltage at the point. It's really the potential difference between that point and the zero voltage side of the meter, but the potential is defined as the potential difference between a point and another point defined as zero potential. In Figure 7 the probe is shown touching the left plate, which is kept at 0 volts since it's attached to the negative side of the power supply. (For your convenience, a small voltmeter display follows the probe wherever it goes.) Between the two plates you see the probe when it was just to the right of the negative plate. When it was placed there, the voltmeter read 10.02 V. The probe readings are shown at two other points along a line between the centers of the plates. At each of those three points, you also see two circles connected by a blue line. There's a third smaller circle too, but it's covered up by the probe. This is a segment of an equipotential line. You'll learn how to work with them shortly. If you move the probe along one of these lines you'd find that the voltage reading stays almost constant, hence the name equipotential line. As discussed earlier, electric field lines are always perpendicular to equipotential lines. The horizontal, orange line is an equipotential line. Note the arrow indicating the direction of the electric field—away from the positive plate and toward the negative plate—which is the direction of the force on a positive test charge. These voltmeter readings are measures of the change in potential as our weak current flows through the conductive sheet. Placing the probe at any point will provide the potential at that point relative to the negative plate. As a result, we can produce a detailed map of the electrical potential between our plates. Here are the steps you'll follow.

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  You'll first create a set of equipotential lines connecting points of approximately equal potential.
• •
  Then you'll use these lines to guide you in producing a map of the electric field. Your electric field lines will either be straight or smoothly curve so as to cross equipotential lines at right angles as shown in Figure 10.

Using the Apparatus

In this exploration you'll be creating your own version of Figure 10, so refer back to it as needed for help and clarification.

Measuring Potential (Voltage), Setting Voltage Markers, Creating Labels

Creating Equipotential Lines

Creating Electric Field Lines

Some Points about Electric Field Lines and Equipotential Lines

Charged Conductors in Electrostatic Equilibrium

Procedure

Please print the worksheet for this lab. You will need this sheet to record your data.

We now want to investigate electric fields in three different configurations: the field between parallel plates, the field of a dipole, and the field inside a hollow shell located in an electric field. You should use a different station for each of these. This will allow you to work on more than one configuration without having to erase what you've done. You have five stations to choose from. For each part of the procedure, you will use a power supply voltage, V_ps, assigned to you in WebAssign. In each case, you'll follow a similar procedure. You'll set the voltage to your assigned voltage. (Part III is an exception.) You'll adjust the positions of the electrodes if necessary. You'll then begin creating a series of equipotential lines. Using the fact that electric field lines are always perpendicular to equipotential lines, you then create a set of electric field lines to illustrate the field produced by your charged electrodes. You'll always want to identify the voltage of each equipotential line using labels. Always use whole numbers such as 10 V,

20 V,

etc., even though your measured values of the voltage will vary from point to point along the equipotential lines. (In Part III, we'll actually use tenths of volts, but these values will be specified as needed.) Always be sure that your field line direction arrows are in the correct direction.

I. The Field Produced by a Pair of Equally but Oppositely Charged Parallel Plates

You now need to add the electric field lines. Remember there are two important points that will guide you in adjusting the curvature of your lines.

• •
  By convention the direction of these vectors is in the direction of the force on a positive test charge; that is, from the positive plate to the negative plate.
• •
  Electric field lines always cross equipotential lines at right angles. Since our conductors are in electrostatic equilibrium, their surfaces are equipotential surfaces. So our electric field lines will intersect them at right angles.

This second point is problematic since charges are not uniformly spaced along the plates, but instead they're concentrated more toward the ends of the plates. This causes some strong curvatures of lines near the plates that we can't detect with our bulky probe. Figure 12 shows how the lines will start out perpendicular to the plate but then quickly change to the direction predicted by our field lines. If we could have measured potentials very close to the plate, we would have been able to predict this quick change in direction. This is the reason for starting our field lines a quarter to a third of a centimeter from the plates. See Figures 10 and 13 for any needed clarification.

A 3D Perspective of Electric Potential–Parallel Plates

It's useful to think about the way that potential varies with position in the same way that altitude varies with position (latitude and longitude) on a hilly terrain. A topographical map represents the third dimension with curves connecting points of equal altitude. Moving across a map between two topographical, aka "topo," lines corresponds to moving across the terrain while rising or falling a given vertical distance. The closer together the topo lines, the shorter the horizontal distance is between points differing in altitude by a given amount; that is, the steeper the terrain in that area. Our equipotential lab represents the same sorts of steep and gentle rises and falls as a charge moves across the grid. But in this case the rises and falls represent changes in electrical potential. Figure 15a is just such a map for our parallel plates. The power supply was set to

40 V

when this data was taken. Compare it to your sketch above. The two long spikes represent the two plates. They're pointed because data was taken only every centimeter, so the plates are effectively lines. Notice the steep drop in voltage between the plates and similar drops, or rises, to the 20-volt plane everywhere else. A voltmeter doesn't actually measure voltage, but rather the difference in voltage (potential difference) between its two terminals. So the plates could be at 1,040 V and 1,000 V and the results would be identical. In Figure 15b this is illustrated by decreasing all the voltage readings by 20 V. As a result, the left part of the graph drops down into negative voltages and the right side drops from a maximum of 40 V to 20 V. So the voltage range is –20 V to +20 V. As far as q₀ is concerned, the fall looks just as scary. Only the potential difference matters. Figure 15a was created in Microsoft Excel. Rather than finding points of equal potential and connecting them as you did in this lab, the potential was measured at each 1 cm intersection on the grid and recorded in an Excel spreadsheet. Here's how to create the graph in Excel. (The method varies with Excel version. Google awaits your inquiries.) Note: this is not a part of your assignment. I just thought you might find it interesting.

II. The Field Produced by an Electric Dipole

An electric dipole is made up of two point charges of equal magnitude but opposite signs. This is an arrangement that happens a lot in nature on the molecular scale. In a water molecule, H₂O, the two hydrogen atoms are at a 105° angle from one another, resulting in a constant positive side to the molecule with the oxygen providing the negative side to the molecule at the third corner of the triangle. When a negatively charged object is brought near a falling stream of water, the water molecules rotate with their positive, hydrogen side closer to the negatively charged object. This results in a net attraction because of the difference in distance between the negative object and the positive and negative sides of the dipole. To be point charges, the objects holding the charges must be far enough apart so that the charges on one object are all very nearly equidistant from the charges on the other. Ours aren't really point charges, but they're a pretty good approximation. And the excess charges are symmetrically arranged around the electrodes, which makes the result more like that of a real dipole. From your work with parallel plates, you should have a good idea of how the apparatus is used. You'll find a series of points that are at the same potential and then connect them with smooth curves—equipotential lines. You'll then draw a collection of electric field lines that start out near the positive electrode and end up at corresponding points near the negative electrode. Since our electrodes are circular now, we'll not draw our lines starting and ending at certain y-values. Instead we'll use clock positions such as 8 o'clock and 2 o'clock as end points.

A 3D Perspective of Electric Potential–Electric Dipole

The 3D potential vs. position graph for the dipole, shown in Figure 18, looks similar to the parallel plates except for there being a larger space between them which resulted in a more of a gentle slope. Note that charges released away from the line connecting the two points would always move in a sweeping arc and the particular path would be highly dependent upon the charge's exact starting position. Thus the uniform field of a parallel plate system is always used for accelerating charges for scientific work, old model televisions, and any other situation where predictability is required. This is much like the behavior of an antenna except that the charges in the antenna are flowing back and forth along the antenna wire between the two ends. This change in field over time produces a changing magnetic field over time which produces a changing electric field, etc. These varying electric and magnetic fields propagate away from the antenna as an electromagnetic wave. If the frequency of this oscillation happens to be between 4.3 × 10¹⁴ and 7.9 × 10¹⁵ Hz, your eyes will be able to detect it.

III. The Field Inside a Circular Conducting Ring between a Pair of Equally but Oppositely Charged Parallel Plates; Shielding

Pencil required: . Your final task will test your skill and patience, and let you discover the reason why you're provided some degree of protection from lightning by staying in your car. That is, if your car has a metal body. This arrangement uses two oppositely charged parallel plates. They're farther apart this time to make room for a small, uncharged conducting ring in the center of the grid area. Let's think about what difference the ring might make. You should have noticed that the 18-V line bent back away from the ring at its center. Just as a surveyor would do, you'll need to take more than five voltage readings for each of your next few lines because the terrain is getting more complicated. Your labels will start to get crowded, but you can move them around as needed. You might want to keep them all above the center line of the grid and use the space below the center line for the lines you'll create on the right side of the grid. We won't draw a complete set of electric field lines. You should be able to see the "terrain" pretty well with your experience with the two other configurations. What's more interesting is what's happening in the center. The field lines coming from the positive plate are going to start out swerving outward a bit but will then swerve back in toward the ring because of the negative charge on its right side. It would take a lot of trial and error to put together a convincing series of field lines to show what's happening. Instead we'll just work in the center third of the screen. Now for the big punch line. The ring is in electrostatic equilibrium just as the plates are. And since the ring is not connected to either electrode, it remains neutral. But it can be polarized by the field it's immersed in. Remember that we said that electric field lines start on positive charges and end on negative charges. Now imagine overlaying your plus and minus charges from Figure 20 onto the ring in your apparatus. Notice how nicely the field lines diverted to the ring can leave the positive plate and arrive on the negative side of the ring. And they can leave the positive side of the ring and end on the negative plate. This effectively stops the field at the outside of the ring. This effect, by conductors, of preventing electric fields from penetrating their surfaces is called "shielding." It's what the outer mesh of wire on a coaxial cable is for—to keep out stray fields. It's what keeps you fairly safe inside a closed conducting shell such as a car when you're in a lightning storm. It also reduces your radio reception inside your car. It gets even weaker if you drive along a steel bridge with girders above, below, and on each side of you. So even a non-solid shell can provide shielding. A 3D Perspective of Electric Potential–Shielding This one pretty well speaks for itself. I'd say it looks like a crazy new design for skee ball, but there's currently a bit of a legal battle going on with that sport. Seriously. Look it up. I'm going with "corn hole for terminally lazy."

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