The Electric Field of a Dipole

Photo shows the power supply, a voltmeter, and the special conductive
paper with the electrical probes

In this lab we will investigate the electric field of a dipole configuration and calculate the charge of the dipole itself.

We know that the electric field due to a single charge is kq/r2. We also know that for most charge configurations we expect the formula for electric field to approach the formula for a point charge if we view it from very far away. If we have two charges of equal magnitude, one positive and one negative, we might expect them to cancel each other out as we move far away from the charges. To first approximation this does happen, but a weak field does exist because the charges aren't at the exact same position. This situation is known as an electric dipole. Instead of decreasing as the square of the distance from the center of the dipole, the electric field decreases as the cube of the distance.

From electrostatic theory we can calculate that at distances much greater than separation of the dipole charges the electric field is given by:

equations for voltage, as the lab may be
ahead of lecture, instructor will go over these in class

We also know that for small distances (d), we can claim that

average electric field is equal to change in voltage divided by distance

In this lab we will determine the electric fields and with that information find the charge of the dipole (the total charge will be zero because it is made up of two equal but opposite charges, but by convention we refer to the charge of dipole as the value on the positive charge). We can define 3 cos2 θ - 1 as X1 and Sin(2θ) as X2, and this will given us Ex as a linear function of X1 and Ey as a linear function of X2. How does this help? If we use X1 as our variable then we can plot Ex on the y-axis and X1 on the x-axis and we should get a straight line. The slope of that line can reveal information about the charge on the dipole.

Dipole Measurement Procedure:
1. Pin down the dipole conductive sheet with a pin at each corner.

2. Connect the wired pins to the power supply (red positive, black negative). Use the DMM to set the output voltage to about 20V. Do not change the voltage for the rest of the experiment.

3. Push the pins into the center of each electrode spot on the dipole sheet.

4. Start collecting data in Microsoft Excel. You should have four columns. The first column should be labeled point, the second X1, the third Vx and the last Ex.

5. Go to point 1 on the corner of the dipole sheet marked X-Component. Use the probes to measure the voltage. The tips of the probes should straddle the point marked on your paper. Make sure you measure in the X direction and not the Y direction!

Closeup shows that the points on the probe fall
along a line parallel with the x-axis rather than the y-axis

6. In your table, this corresponds to point 1. X1 has a value of -1 (at point 1 θ is at 90, hence the value of the cosine function is zero, this is why X1 is -1 at point 1). X1 increases by steps of 0.3, so X1 will be -0.7 for the second entry. Vx is whatever value you just measured. Ex is simply Vx divided by the spacing of the probe tips (10.3mm).

7. Repeat until you have completed all 11 points in the X section.

8. Do the same for the Y section of the paper, this time being careful to measure in the Y direction. In your lab notebook, plug in the initial and final angles and argue that the values of the X2 variable will run from 0 to 1 in increments of 0.1.

Data Analysis
We'll be using this method many times this quarter, so make sure that both you and your lab partner understand what you are doing. If you have questions about this, it is important that you talk to your instructor about how to do this

In Excel, select the two two columns X1 and Ex. We will next want to make a graph. Pick the Chart Wizard option from the toolbar. Pick X-Y Scatter Plot and take the default option. Add a title, labels and units to your chart. Click finish and Excel will generate a graph.

Under the Chart menu, click on "Add Trendline". For options you want to display the equation and the R-squared value. This will put a least-squares fit on your graph and give the slope. The R-squared number should be close to one. If it is not, it means that some of your data may be non-linear. Note that if your R-squared number is equal to one, this often is a sign that you have made some kind of error as it implies straight-line data with no "noise".

From the slope you can go back and calculate the charge. Note that

Ex = (kQa/R3) X1

Therefore the slope of the Ex vs. X1 graph should be equal to the quantity in the parenthesis. Solve for Q. Note that a = 10mm and R = 100mm.

Repeat this process for Ey and X2 and compute a value of Q from that series of measurements.

At this point you will have two values for Q, they should be the same (we have measured the same physical quantity in two different ways). Because no measurement is perfect, these numbers will differ. Can we claim that within the uncertainty of measurement that these two numbers are the same? Here we need to use some error-analysis tools we learned in the first lab.

Before you do the next step, save your data and graphs. Excel is a little quirky, and can freeze while using the following function. So make sure that you don't have to redo all of your work.

We derived Q from the slope of the graph, what's the uncertainty in that slope? Excel can tell us! We will use the LINEST feature in Excel. Go to an empty cell in Excel and type

=LINEST(known Y value cells, known X value cells, 1, true)

where "known Y value cells" are the data entries that correspond to the Es and "known X value cells" are the Xs. For example, if we had column A as X1 and column B as Ex, and we had eight rows of data, we would type the following:

=LINEST(B1:B8, A1:A8, 1, true)

When we type this in Excel will return a single value, the slope. LINEST is an array function, which means that even though it is trying to give back a lot of values, it only displays one per cell. What Excel is expecting you to do is highlight a 2x2 set of cells with the LINEST formula in the upper-left cell. Once this is done, go up to the formula bar and highlight the entire LINEST equation you typed in and hit Ctrl-Shift-Enter (on a Mac use Apple-Return). This will fill in the 2x2 square. Upper left will be the slope, lower left will be the uncertainty on the slope. Upper right will be the intercept and lower right will be the uncertainty on the intercept. Remember how to do this, you will do it many times this quarter

The next question is "given the uncertainty on the slope, how do we calculate the uncertainty in Q?" Take the uncertainty in the slope and divide it by the slope, that gives the percentage error of the slope. Since Q is the product of a constant times the slope, the percentage uncertainty in Q is the same as the percentage uncertainty in the slope. So the actual uncertainty in Q is Q itself multiplied by the percentage uncertainty.

So we should now have two values of Q and each will have an uncertainty associated with it. Again, ideally the value of Q from the x measurements should be the same as the value of Q derived from the y measurements. Hence the difference between these two results should be zero. Of course, since these are experimental measurements, the difference between the two will be non-zero, but the difference should also be within the uncertainty of our measurements. First check to see if the difference is smaller than the larger of your two uncertainties. If it is, then the values of Q are smaller than the uncertainty involved, and you have shown that the two methods give the same value of Q. If your difference in Qs is larger than the bigger uncertainty, then you will have to add the two uncertainties. How do we add uncertainties? As it turns out, we add them like Pythagoras. Square each, add, then take the square root. Hopefully your difference will be smaller than this number, if not your conclusion should explain why not.

In your conclusion discuss what you have learned about the electric fields and how well your Q values agreed. If they were outside the limits suggested by their uncertainties, provide some explanations. Remember that "human error" is not an explanation.