In this lab, you will be plotting out the electric field of two different charge configurations: a set of parallel plates and a dipole.

You will plot out the equipotentials of each configurations, from which you can sketch the field lines.^{Hoveroverthese!}

• 1 Digital Voltmeter
• 2 Voltage Probes (one stationary, one hand probe)
• 1 Platform
• 2 Carbon Acetate Sheets (one parallel plate, one dipole)
• 1 Power Supply
• 1 Meter Stick
• 4 Banana Cables
• Record data in this Google Sheets data table

Electric Field & Voltage

This lab explores the relationship between electric field strength, voltage, electric field lines, and equipotentials.¹

The electric field \(\vec{E}\) is a vector field, meaning it is a vector associated with every point in space. The closely related quantity of voltage \(V\) is a scalar function of position, and is related to electric field.

Take two nearby points in space, \(\vec{x}_1\) and \(\vec{x}_2\), and suppose the voltages at these points are \(V_1=V(x_1)\) and \(V_2=V(x_2)\). With these voltages, we can approximately¹ compute the component of electric field in the \(\vec{x}_2-\vec{x}_1\) direction as:

$$E_{\vec{x}_2-\vec{x}_1\text{ component}}=-\frac{V_2-V_1}{|\vec{x}_2-\vec{x}_1|}\label{EFieldStrength}$$

Field Lines & Equipotentials

While an explicit calculation of electric field is excellent, we are often more concerned about getting a picture of electric field than getting an exact description. A good tool to get this qualitative picture is electric field lines.

Electric field lines start at positive charges (or somewhere infinitely far away) and end at negative charges (or somewhere infinitely far away). In between, they always travel parallel to the electric field. They never cross or make loops.

Once we have these field lines, we can interpret them to get a qualitative understanding of field strength: "bunching" of electric field lines (i.e., where they get dense) indicates that the electric field is stronger there, and "spreading" indicates that the electric field is weaker.

A different part of our qualitative picture is the equipotential lines. These are lines (or, in 3D, planes) on which the voltage is constant. If we space our equipotentials by some fixed voltage intervals (e.g., put an equipotential every 1V), then we can use equation \eqref{EFieldStrength} to show that "closer together" equipotentials mean stronger fields (because \(V_2-V_1\) is the same, but \(|\vec{x}_2-\vec{x}_1|\) decreases.)

Also from equation \eqref{EFieldStrength}, by considering two "nearby" points, we can show that the component of electric field parallel to the equipotential is zero; hence, the electric field is perpendicular to equipotentials. This allows us to make the key relationship between equipotentials and field lines: equipotentials are always perpendicular to electric field lines.

This allows us to understand the qualitative behavior of our electric field lines (and hence our electric field) from our equipotentials, which can be determined by simply measuring voltage. That is what we will be doing in this lab.

To measure voltage, we are going to use a device known as a voltmeter. This device measures the voltage between its endpoints.²

Physics Behind Electric Fields

Electric fields are made by charges. For a single isolated charge \(q\), the electric field is given by Coulomb's law:

$$\vec{E}=\frac{kq}{r^2}\hat{r}$$

Here, \(k=8.99\times 10^9\frac{\text{Nm}^2}{\text{C}^2}\) is Coulomb's constant, \(r\) is the distance between the charge and the point you are measuring, and \(\hat{r}\) is the unit vector pointing from the charge towards the point you are measuring.

If you have multiple charges, you add together their electric fields (with the \(r\) modified appropriately for each charge!) to get the total electric field. This suffices for electric fields made by pointlike charges; for electric fields made by "diffuse" charges (spread out over some region), the same idea holds, but the math gets more complicated.

One key configuration of charges is one or more "plates" (infinite planes) of charge. Each plate makes an electric field that is constant in strength, emitted directly outward or inward (depending on the sign of the charge on the plate). The strength of the electric field depends on the charge density (charge per unit area) of the plate.

Similar to point charges, if there are multiple plates, we add together their electric fields to get the overall electric field. (Same if there is a plate and point charges, or any other combination of chrage configurations.)

Going beyond charges themselves, one of the key electrical objects we use is a conductor. These materials (usually metals) allow electrons to move around freely (contrasted with insulators) whenever there is an electric field.

After a short time, a conductor's electrons will have rearranged themselves so that the electric fields inside are zero. This means that the entire conductor is a solid equipotential; the voltage is constant everywhere within it.

Ultimately, all of the net charge ends up on the edge of the conductor, and thus field lines end there. The field lines at the edge of a conductor also always enter the conductor perpendicularly, because the conductor is an equipotential.

In front of you, you should have two sheets of black carbon paper with a shape on each side. The in-lab goal is to form a map analogous to the following image (but for the configurations we have):

One sheet will have a T-shape on each side; this is the "parallel plates" configuration. The other will have a line ending in a circle on each side; this is the "dipole configuration."

All of the remaining instructions in this section should be performed for each sheet.

Setup

First, place the carbon sheet on the platform. Place the clamps such that they are contacting the silver conductive strips.

Wire a red wire from the red terminal of the power supply to one end of the platform, and a black wire from the black terminal of the power supply to the other end of the platform.

Wire a black wire from the black (-) port on the voltmeter to the stationary probe (the one with the stand). Wire a red wire from the red (+) port on the voltmeter to the hand probe (the one with the handle).

Plug in the power supply, turn it on, and set it to 7.5V. Turn on your voltmeter, and ensure that it is set to a 20V maximum.

Place the stationary probe on the silver conductive surface that is connected to the black port on the power supply. For optimal results, place it near where you put the clamp.

Place your other probe on the other silver conductive surface (anywhere, doesn't matter). Your voltmeter should read approximately +7.5V when you do so. (It's probably a bit less, perhaps as low as 6.5V; much lower than that, consult your TA.) If you see -7.5V, you have one of your sets of wires the wrong way around.

Now: take the printout that matches the carbon sheet you are working on. All data in this lab will be recorded on this printout for the moment (although there will be a digital component to your final report).

Measuring Voltages

Leave the stationary probe where it is. We will now move the hand probe to various points on the sheet to measure voltage at that location.

First, place the hand probe on the same conductor as the stationary probe. Record the voltage displayed by your voltmeter onto your sheet as the low plate voltage, as displayed in the above sample image.

Then, place your hand probe on the opposite conductor, and record similarly.

Now: place your hand probe in sequence on each dot going from one conductor to the other down the center row, so from (8,3) to (8,9). Record these voltages on your sheet by the relevant dot.

Measuring Equipotentials

Now: move your stationary probe to the one of those central dots [say (8,3) for instance], and leave it there. Set your voltmeter maximum down to 200mV.

Move around your hand probe on your sheet and find various points (which may not be on the dots) where the voltmeter measures "zero."¹ Mark these points with an "X" - somewhere around 5-7 such points, distributed over the entire width of your sheet (to the edge of the dots).

Your goal here is to have enough points that you can smoothly "connect the dots" from one to the other. Exactly how many dots that is is left up to your judgement (and that of your TA); just make sure you don't miss any important details (especially, say, out near the edges of the parallel plates...).

Once you have enough data points, smoothly connect the dots (pencil recommended). This line is your equipotential, with the voltage of the point you measured at the center.

Then, move your stationary probe up to the next central dot [say (8,4)], and repeat this procedure for each central dot until you have a full set of equipotentials across your sheet (seven total).

Once you have all that done for your first sheet, proceed to your second sheet; once you have it done for both sheets, you are done.

Electric Field Sketches

We now want to add electric field lines to our sketch, in addition to the equipotentials you've already drawn.

Recall that electric field lines are everywhere perpendicular to equipotentials, that conductors are also equipotentials, and that electric field lines go from high voltage to low voltage.

With all of that in mind, draw your electric field lines, including arrows indicating their direction. Please make sure it is clear which lines are your field lines and which lines are your equipotentials using whatever means you (or your TA) find appropriate - a legend, different colors, etc.

In your report, in addition to including these sketches themselves, you should write a brief description of them in your results section: how do the field lines go, for each configuration? Are there any notable features? What physics do these features indicate?

Parallel Plate Analysis

Take your voltages for the center column, and make a voltage vs. position plot.

For position measurements, note the sketch is not to scale (it's a bit bigger). The actual grid on the carbon sheet is a cm grid, so use that to determine your measurements.

As for uncertainties: for voltage, use the last digit on the voltmeter; e.g., if you measured 2.54V, your uncertainty would be 0.01V. For position, use 1mm unless you have reason to think it would be higher [e.g., if your hand-probe had an unusually broad tip].

Take note of your slope, and in your report, state this and explain what it means, physically. [I.e.: what is this quantity, physically? It has a simple interpretation.]

Dipole Analysis

For the dipole, we will do some quantitative analysis to attempt to compute the charge on each pole (and then see if these charges are, as we expect, the same in magnitude but opposite in sign). This analysis will be done in the Google Sheets data sheet.

Consider first your high-voltage pole. Take the two closest voltages on the center column; call the one further from the pole \(V_2\) and the one closer to the pole \(V_1\), and record them on your data table.

These two points are 1cm apart, so record that as your position difference. As with the plot described above, take 1mm as your uncertainty unless you have reason to think it should be higher.

Using those two voltages and the position difference between them, compute the component of electric field that goes radially outward from the charge. [This may be positive or negative; think carefully!]

Now: we want to take this electric field, and use it to compute the charge. We're going to neglect the influence of the other charge in this calculation (since we're "far away" from it), so we'll just use Coulomb's law.

To use Coulomb's law, we need to know the distance between the charge (i.e., the center of the pole) and the point where we measured the electric field (which we'll take to be halfway between the two voltage points we took), called \(r\). With our 1cm scale, you can take \(r\) to be 1.5cm, since the charge is centered on [as an example, depending on the orientation of your sheet] (8,2) and we're considering our measurement of electric field to be halfway between (8,3) and (8,4).

Since our "actual electric field point" could really be anywhere between the two spots we measured, take the uncertainty in \(r\) to be 0.5cm. [This is a bit high, but we would need more information for a more precise measurement of r.]

Now: take the electric field, the radial distance from the charge, and Coulomb's constant and use that to compute the charge, \(Q\), on that pole.

Propagate uncertainties all the way through, then repeat the series of calculations for the other charge [again, being careful with your signs!].

Your TA will ask you to discuss some of the following topics (they will tell you which ones):

• Parallel Plate Linearity: Your plot of voltage vs. position for the parallel plates should have come out looking reasonably linear. What does this tell us, physically, about the electric field?
• Dipole Field Strength: Based on the ways we qualitatively understand "electric field strength" using equipotentials and field lines, circle the place on your dipole configuration where the electric field is the strongest.² (It may be handy to do this in a different color than your field lines/equipotentials, to prevent your sketch from getting too jumbled.)
• Systemic Errors: What errors could have impacted the qualitative behavior (i.e., the shape) of our electric field lines, compared to the ideal field lines for a dipole/plates? (Hint: what makes electric fields? How could such things have been present in the ambient environment?) Note: we're looking for things that could have impacted our actual, physical field lines here, not just altered our measurement of them.
• Alternative Configurations:
  • Suppose you swapped positive and negative charges in the dipole arrangement. What would happen to the electric field lines (both shape and direction)? Would they be the same or different?
  • What if you made both charges positive, or both negative? Would the electric field line configuration be the same or different (and if different, how so)?
• Deviations from Expected Behavior:
  • Conductors:We expect an ideal conductor to be a constant voltage throughout. The conductive surface (silver marks) is mostly a constant voltage, but sometimes you may observe a millivolts-ish difference between different points on it. Explain, if you can.
  • Third Dimension: We call the first configuration a "parallel plates" arrangement, but technically, they're not "plates," because they don't extend vertically, in the third dimension. But if it's not parallel plates of charge, we don't expect the field lines to all be parallel, which is what we observe. What is going on here?³

Guide to Uncertainty Propagation & Error Analysis (Quick Reference)

Spare Parallel Plates Sketch Paper

Spare Dipole Sketch Paper