This lab will find the electric field of a system by measuring the electric potential.

If we were to apply feelings and desires to electric charges, like we do people, the electric potential would tell us if a charge wants to move, or if it's happy where it is. In other words, knowing the electric potential of a given system will tell us about the way charges will behave in that system. For example, will charges flow and make a current? Or will they just stand still? If they do flow, which way will they flow? These are all questions that can be answered be understanding the electric potential and its distribution as a function of position. It is also related to the electric field by the following relationship:
$$
\begin{equation}
E_x = - \frac{\Delta V}{\Delta x}
\label{eq:fieldgradpot}
\end{equation}
$$
This says the rate of change of the potential as a function of position (otherwise known as the gradient) is equal to the negative of the electric field in that direction. What does *that* mean? That's the project of this lab.

Please make sure your station has all of the following items. If not, check again, then talk to your lab instructor.

- One set of concentric rings mounted on panel
- Wire assembly
- Yellow Multimeter and Probes
- Ruler

Before taking an actual measurement with real electricity and currents and wires, let's practice with a virtual measurement. Below is a parallel plate capacitor set up. The left hand plate is set at 0 volts and the right plate is at 7 Volts. You can move the cursor in between the plates and see the measured electric potential at that position. Go ahead and play with it for a moment.

In general, it's easy to measure potential, but harder to measure electric fields. However, we can use the relationship mentioned above to figure out the field based on a several potential measurements at different locations. That's the goal here: measure the potential at several (at least 5) locations in the capacitor. Record the horizontal position as well as the reading on the virtual volt-meter to the right in a table of data.

We are assuming that the plate separation is small compared to the plate area. This enables us to neglect any y- components to the electric field. In reality, this arrangement shown above would not have uniform and isotropic field lines.

After you have the table, complete, you can open excel and enter the data into a new spreadsheet. One column is position, the other will be voltage.

Since we are only working in the x dimension (i.e. 1 dimension), the gradient will be effectively just the derivative with respect to position, or just the slope of the $V(x)$ graph (you should know how to get the slope by now). Use this information to find a value for the electric field inside the parallel plate setup. Also remember the negative sign in the equation relating field and potential. Enter your field measurement in the box below. (Positive will be for a field pointing to the right, negative for a field pointing to the left.):

$\overrightarrow{E}_x$ = =

Now that you know the field strength, you could also figure out the force acting on a charge. What would the electric force acting on a proton be if the proton were located at x = 5 cm?

$\overrightarrow{F}_E$ = =

Now we'll use a physically simpler, more mathematically more complex arrangement and measure potentials again. Below is the potential distribution around a point charge. Use the virtual voltmeter and radial position grid to measure the potential as a function of position for a few locations and create a plot of the potential vs. r. Use this to establish the electric field function for a point charge. Compare it the expected result from Coulomb's law.

The setup consists of two concentric conducting rings. We will apply a potential difference between the rings, and measure the potential **drop** in the space between them, as a function of position.

First, we'll need to know the geometry of the system. Namely, the outer radius of the small ring, $r_a$, and the inner radius of the large ring, $r_b$. Find a way to measure these two values.

Make sure the rings are not connected to the power while doing this!

Enter your values for $r_a$ and $r_b$ here, in millimeters:

$r_a$ = =

$r_b$ = =

Be sure the power plug is not connected until your wiring has been approved by the instructor.

The two rings will be connected to a 5V power supply via the blue and brown cable assembly. The blue wire will be connected to the 0V, and the brown will have +5V volts. Connect the 0V (blue) wire to the inner ring using the electrode. Connect the +5V (brown) wire to the outer ring via the electrode. If you've got the wiring approved by the lab instructor, then go ahead an plug the cable into the wall socket.

This experiment does not use normal household power to create the potential difference. The plug you are using is connected to a special, low voltage power supply.

On the desk is a yellow instrument called a multimeter. It's probably one of the best things ever invented. It can measure many different electrical properties quickly. We'll use it to measure DC Volts. Since it is a *multi*meter, there are different modes of operation and you'll need to make sure it's setup to measure DC Volts.

- First the rotary switch should be directed to the DC Volts symbol which looks like this: . (The DC stands for direct current, which is in contrast to AC, alternating current, which has the squiggly line above the V).
- Next make sure the brown wire is connected to the plug labeled COM. This is effectively the
*ground*, or the zero potential. The probe wire should be connected to the plug to right of the COM labeled with a V.

Now you should be ready to measure the electric potential in the area between the rings. The conductive paper under the rings allows some charges to flow.

**Gently** touch the voltmeter probe to the paper at a point about 5 mm from the edge of the inner electrode.**Do not press so hard you poke through the paper.**

You'll want to make a table of data that has columns for $r$, and $V_1$, $V_2$, $V_3$, $V_{avg}$ where the three voltage measurements are taken at the same $r$ values, but along different radii, and the $V_{avg}$ is the average of the three measurements at each $r$ position. Try to take voltage measurements every 5 mm to get enough to make a nice plot.

## Sample Data Table |
||||
---|---|---|---|---|

$r$ [mm] | $V_1$ | $V_2$ | $V_3$ | $V_\textrm{avg}$ |

5 | ||||

10 | ||||

15 | ||||

$\vdots$ |

(The analysis section can be done on your own later, after you have obtained the table of data above. It will make up the bulk of the report for this lab.)

It can be shown that the analytical prediction for the potential in between the two conducting rings will be described by the expression: $$ \begin{equation} V(r) = A \ln \left( \frac{r}{r_a} \right) \label{eq:potentialfunction} \end{equation} $$ where $A$ is a constant determined by the actual potential difference between the two rings.

1. Prepare a plot of your Potential vs. Position data using the averages for the voltage measurements at various positions. It should resemble a log function.

2. Also plot $V$ vs. $\ln(r)$. This should be a straight line.

3. Use the above to establish a value for the constant $A$. What units should it have?

3. Use equation \eqref{eq:fieldgradpot} and \eqref{eq:potentialfunction} to obtain an expression for the electric field.

4. Make an analytical plot of the electric field as a function of position between the rings, $E(r)$

5. On the sheet provided here, draw the equipotential lines corresponding to 1, 2, 3, and 4 V at approximately the correct locations based on your data and understanding of equation \eqref{eq:potentialfunction}.

6. Also draw several field lines indicating the direction of the electric field between the rings.

Submit this visualization along with the rest of the lab write-up.