OBJECTIVE: To understand the relation between electric fields and electric potential, and how conducting objects can influence electric fields.

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1 Name Section Question Sheet for Laboratory 4: EC-2: Electric Fields and Potentials OBJECTIVE: To understand the relation between electric fields and electric potential, and how conducting objects can influence electric fields. APPARATUS: 1. Numerical EM simulation applet, and voltage source, electrometer, graphite paper, and potential probes. INTRODUCTION: You use two different ways to investigate electric fields and potential. The first is a real-time digital simulation of electric fields and potentials in vacuum ( It lets you position various charges and conducting objects by dragging and clicking, then calculates the electric field direction and equipotentials for you. It divides space up into discrete blocks to do the calculation, and so is not particularly accurate. The second is a realtime analog simulation of electric potential using a high-resistance (but still conducting) medium to simulate vacuum. Small currents flow through this medium along electric field lines, which generate electric potential variations that you measure with the electrometer. This lets you quantitatively measure electric potentials and calculate electric fields. A. Numerical Simulation (Don t spend more than 1 hour on part A) Point Firefox at and start up the applet. It starts up showing the electric field directions and equipotentials from a point charge. Enlarge the window to take up most of the screen, but be careful to preserve the aspect ratio (e.g. so the circular charge is still circular). You should be able to click and drag the positive charge around on the screen. A1. The white line contours are equipotentials, connecting points in space that have the same electric potential. Each contour is a different electric potential, and the electric potential difference between adjacent contours is a constant value ΔV. Why do the equipotentials get farther apart as you move away from the charge?

2 A2. Under the setup menu choose Double Charge. What is the direction of the electric field midway between the two charges? Where do you think the electric potential is equal to zero? Is the electric potential a local minimum, local maximum, or neither, midway between the two charges? (Hint think about the equipotential countours) 2

3 A5. Under the setup menu choose Dipole. Click and drag the charges so that the dipole plane is horizontal near the bottom of the screen, and takes up most of the screen. Select Mouse=Add Conductor (Gnd) and draw a filled rectangular object near the top of the screen. Select Mouse=Make Floater and convert the grounded conductor to a floating conductor by clicking on it. It is now an isolated conductor with approximately zero charge on it. Select Mouse=Move Object and drag the conductor around on the screen. The local charge density on the conductor is color coded, blue for negative and yellow for positive. i) Drag the conductor down between the dipole charges. Describe what happens to the charge distribution on the conductor, and how the electric fields change. Explain what is going on. ii) Drag the conductor vertically upward, away from the dipole. Describe what is happening to the charge density on the conductor. Explain. iii) Suppose the dipole is an electrogenic fish, i.e. a fish that can cause a charge separation in its own body between its head and tail. Suppose that the conducting object is its (conducting) prey. The electrogenic fish senses its prey by detecting changes in electric fields on its skin caused by the conducting prey. Move the prey around and watch the electric fields in the region of the dipole. Approximately how close must the conductor be to the dipole before it noticeably affects the electric field? What are your thoughts about this? There is no simple answer to this. It depends on several quantities, including the conductor size, and the point at which the electric field is being observed. 3

4 B: Analog simulation Here you use a piece of carbonized paper in which currents flow to simulate electric fields and equipotential surfaces in vacuum. Field plotting board: Get a piece of graphite paper with two silver dots (representing conducting spheres), one on each end. On the field plotting board, first put down a sheet of printer paper, then a sheet of carbon paper, and finally the graphite paper on top. Power supply: Attach the +30V output (red) of the DC power supply to one connector on the field plotting board, and the ground output (black) to the other. This maintains a constant potential difference between the two painted conductors on the graphite sheet. Digital multimeter: Attach the red and black voltage probes to the Keithley digital multimeter (DMM) by attaching a BNC to banana-plug adaptor to each probe. Then connect a banana plug cable from the red terminal of red probe to the DMM red connnection, and from the red terminal of the black probe to the DMM black connection. Do not connect anything to the black terminal of either probe. Sit one probe in each of the field plotting board electrical connections. Turn the multimeter on. The multimeter can measure multiple quantities (voltage, resistance, or current), so you have to tell it to measure voltage by pushing the V button. Put it on the 20V scale of the DC voltage measuring function by pushing the 20 button. Adjustments: Turn on the DC power supply and adjust the voltage until the multimeter reads just less than 20V (the switch just above the connections should be on 30V and not 100V ). If the voltage is too high and the multimeter reads OL (for overload ), you should reduce the power supply voltage until you get a reading. Leave everything on and connected. The display on the multimeter is the electric potential difference across its inputs, V red "V black. DMM Volts 4

5 B1. Before starting the measurements, you determine the relation between voltage difference and electric field by doing a little calculation here while the power supply stabilizes. In the space below, you the relation between electric field and electric force on a charge to write down the electric potential difference between two points a small distance d r s = ("x,"y) apart in terms of the x- and y-components of the electric field ( E x and E y ), and the x- and y-components of the distance ("x and "y ). a) First, remember that the electric potential energy difference between points A and B separated by a small distance d r s is the external work F r ext d r s required to move a charge (for instance, for you to push a charge) at constant speed from A to B. Constant speed means that the force you are applying exactly cancels the electric force that is trying to keep you from pushing the charge. r Write down the work in terms of the electric force F electric and the distance d r s. (this is not complicated!) b) Remember that the electric potential difference between two points A and B is the electric potential energy difference per Coulomb of charge moved. Use this relation to write down the electric potential difference between two points separated by d r s in terms of the electric field E r. c) Now write down the electric potential difference between points A and B separated by a small distance d r s in terms of the x- and y-components E x and E y of the electric field, and the x- and y-components of the separation dx and dy. 5

6 B2. You now use the voltage probes to measure the potential difference between any two points on the graphite sheet, and determine electric fields. Put the black voltage probe in the grounded electrode connector on the silver dot. Use the other to measure the potentials on the grid points 1 cm apart. Choose a location about 2 cm over and 2 cm up from the grounded electrode. Fill in the potential on the grid of 9 points below (1 cm separation on your graphite sheet). Pick the lab partner with the steadiest hands to hold the probe the values are quite sensitive to the exact position of the probe. Search path for B4 1 cm Silver dot Find electric field here in B3 B3. From these data (electric potential at the crosses), and your answer to B1, calculate the electric field vector in volts/centimeter at the central point (not all of the data may be necessary to determine the electric field). Draw it in above. It is easiest to find the x- component and y-component of the field separately. Hint: to decide which potentials to use, think about centering them about the point of interest. 6

7 B4. Put the black probe on the cross where you just found the electric field, and search around in a circle with the red probe to find the direction in which the electric potential difference is the largest. Indicate that point on the drawing in B2. How is this direction related to the electric field you found in B2? Explain using ideas of forces, work, and potential energy. 7

8 B5. Now pick up both the black and red probes. You will use them to determine the electric field x and y components along the vertical line of crosses indicated below. Do this by measuring potential differences between grid points 2 cm apart (centered on bold crosses below). This goes quickly if one person measures and another writes down. Let the one with steady hands do the measuring: "V is quite sensitive to exact positioning of the probes. Position "V x (volts) "V y (volts) E x (volts/cm) E y (volts/cm) +8 cm +6 cm +4 cm +2 cm 0 cm -2 cm -4 cm -6 cm -8 cm In the space below, sketch your measured electric fields at these points from your data above. Probe locations to get ΔV x at +6cm Silver dot 8

9 B6. Now follow the instructions in your lab manual to trace out several equipotential curves through the carbon paper and onto the white paper on the bottom. Sketch your equipotential curves on the figure on the previous page (w/ your previously measured electric fields). What is the direction of your electric field vectors relative to the equipotential that goes through (or close to) those points? Explain why using ideas of forces, work, and potential energy. 9

10 B7. Electric flux: In this section you determine the electric flux going through the dotted boundary in B5 from the left to the right. The electric flux is the amount of electric field going through the surface. The component parallel to the surface does not go through, so only the component perpendicular to the surface counts. Since the electric field is different at every point, the component perpendicular to the surface varies continually along the dotted line above. In B5 you measured the components of the E-field at 2 cm intervals from the bottom of the graphite sheet to the top. To exactly calculate the electric flux across the dotted line in B5, you need to know the electric field vector everywhere along that line, not just at 2 cm intervals. Since the E-field you determined is changing only gradually along the dotted line, you could probably guess what it is at points where you have not measured it, then use those guesses to calculate the electric flux. In physics we usually try the very simplest approximation first. Here that approximation is to use just the measured values, suppose that the electric field vector has a constant value 1cm on either side of the bold crosses, then jumps to the next measured value. In the graph below, plot the component of the electric field perpendicular to the dotted boundary as a function of every point along the boundary, in the physics approximation, and also by visually interpolating between the measured points to more accurately estimate the actual position dependence. Electric field component POSITION ALONG BOUNDARY (CM) Using the physics approximation, calculate the electric flux through the dotted boundary. Since the paper is two-dimensional rather than three, the units of flux are not the usual ones. Discuss how the actual flux might be different from that obtained from the approximation. 10

11 B8. Gauss law. Here you measure the flux through the square Gaussian surface shown below as a dashed line using an approximation similar to that used in B6, but at 1 cm intervals. Positioning the surface near the measurement you made in section B2 works well. Remember that the electric flux through a closed suface is the electric flux outward. i) Draw the surface normal vector n ˆ on the four different sides of the surface ( ) ˆ ii) Label each side with r E x, y n expressed in terms of the components E x and E y. iii) Measure the total electric flux through this surface (we are in two-dimensions, so the units are [Field][length] rather than [Field][Area]. Remember that E r ( x, y) will vary along the surface it is not constant. iv) Based on your approximation, and Gauss law, is any charge enclosed? Flux out thru top side Flux out thru left side Flux out thru right side Total flux Flux out thru bottom side 11

12 B9. Get rid of your boring old dipole and find a carbonized sheet with a more interesting design. There should be a stack of several to look through. Or draw one yourself with the silver pen. Several suggestions are given in Figure 4 of the 202/208 lab manual. Map out several equipotential lines. Make a quick sketch below, and write something below explaining why you think the equipotential lines are arranged as you measured them. 12

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