Coulomb s Law and Coulomb s Constant

Transcription

1 Pre-Lab Quiz / PHYS 224 Coulomb s Law and Coulomb s Constant Your Name: Lab Section: 1. What will you investigate in this lab? 2. Consider a capacitor created when two identical conducting plates are placed parallel and close to each in other in vacuum. The surface area for each plate is m 2. The two plates are separated by a distance of m. When an electric potential difference of 50.0 V is established between the two plates, what is the magnitude of the electrostatic force between the charged plates? Is the force attractive or repulsive? 1

2 3. What mass on the Earth surface will experience a gravitational force with the same magnitude as the above electrostatic force? 2

3 Lab Manual / PHYS 224 Coulomb s Law and Coulomb s Constant Your Name: Lab Section: Objectives In this lab you will investigate Coulomb s law and measure Coulomb s constant or the permittivity of free space (actually of air). Background Coulomb s law: For two point electric charges, respectively of qq 1 and qq 2, placed in free space and separated by distance rr, the magnitude of the electrostatic force (FF) between them is: FF = kk ee qq 1 qq 2 rr 2, (1) where the Coulomb constant kk ee is given by kk ee = NN mm 2 /CC 2. The permittivity in free space, εε 0 = CC2 (NN mm 2 ), is related to the Coulomb constant by εε 0 = 1 4 ππ kk ee. Electrostatic force between two charged parallel plates: Based on the Principle of Superposition, one can use Coulomb s law to calculate the electrostatic force between any two charged 3

4 objects. For example, two identical conducting plates parallel to each other, separated by distance dd, and each plate with a surface area of AA, form a parallel-plate capacitor with capacitance CC = εε 0 AA dd. When an electric potential difference of VV is established between them, the two plates become oppositely charged with the same magnitude of charge QQ given by: QQ = CC VV = εε 0 AA dd VV. (2) If the lateral dimensions of the plates are much larger than dd, one can neglect the edge effect and assume that on each plate the electric charge is uniformly distributed with a surface charge density given by: QQ AA = εε 0 dd VV. And a uniform charge density on an infinitely large plane produces a uniform electric field EE perpendicular to the plane, which is given by the magnitude of the surface charge density divided by 2 εε 0, EE = 1 2 εε 0 QQ AA = 1 2 VV dd. (3) Thus, the attractive electrostatic force on one charged plate exerted by the other has a magnitude FF = QQ EE = εε 0 AA 2 VV dd 2. (4) 4

5 EXPERIMENT Apparatus The main part of the set-up used in this lab is shown in Figure 1. The bottom plate is attached to the rod marked as AB, which is fixed on the board of the Coulomb balance (not shown). The height of bottom plate is thus fixed but it can rotate around rod AB. As shown in Figure 1, the top plate is fixed on the shortest rod of a trapezoidal metal-rod frame. Rod KK has two supporting knife edges (not shown): one under K and another under K. The frame is supported only by the two knife edges which are to be placed on the two short parallel brass supporting beams on the board. It can thus rotate freely around the line connecting the two knife edges, allowing the top plate to move away from or towards the bottom plate. First, adjust the knife edges until the two plates directly face each other. The frame is balanced mainly by the top plate and the weight P, and secondarily by weight P (located under Rod KK ). By rotating it, weight P can move (by rotating) along the attached rod to change its lever arm. Weight P, which can slide up and down by rotating to change its lever arm, is used to change the time period of oscillation of the frame and equilibrate the frame. Metallic pane Q is placed between two 5

6 horseshoe magnets and is used to damp oscillation of the frame. Make sure that pane Q does not touch the magnets. Note: for smallangle rotation, the position of weight P is important to efficiently stop oscillation of the frame and to increase the sensitivity of the experiment. After adjusting P and P, the frame should be able to rotate freely around the line connecting the two knife edges but its oscillation should be efficiently damped. The top and bottom plates should be aligned parallel to each other and separated by only a few millimeters. Measuring the distance (dd) between the upper surface of Bottom Plate and the lower surface of Top Plate To accurately measure dd, not by directly touching the two plates, you will use a laser, a mirror, and a scale. As shown in Figure 1, the mirror is fixed on a vertical rod attached to the horizontal Rod KK. Top Plate is perpendicular to the mirror and Rod KK is parallel with both Top Plate and the mirror. If D1 is the distance from the center of Top Plate to the mid-point between the two knife edges, when the frame rotates clockwise around the axis connecting the two knife edge by a tiny angle of αα, the center of Top Plate moves up by a distance of DD 1 ssssss(αα) DD 1 αα (Figure 2). 6

7 Meanwhile, the outer edge of Top plate moves up by (DD ww) αα and the inner edge moves up by (DD ww) αα, where ww is the width of top plate. Direct the laser beam onto the mirror. Make sure that this incident beam is in the plane formed by the normal of the mirror and the mid-point between the two knife edges. Place the scale such that the reflected beam intercepts on the scale. Following the Law of Reflection, if the angle of incidence (between the incident beam and the normal) is θθ, the angle of reflection (between the incident beam and the normal) is also θθ. When the frame rotates clockwise around the axis connecting the two knife edges by angle αα, the mirror also rotates clockwise by angle αα. So does the normal of the mirror. Because the incident laser beam remains at the same direction, the angle of incidence is increased by α becoming θθ + αα. Then, the angle of reflection also becomes θθ + αα. Thus, the reflected beam changes direction by angle 2 αα (Figure 3). In accordance, the intercept of the reflected beam on the scale moves up by a distance of h = DD 2 ssssss(2 αα) 2 DD 2 αα, where DD 2 is the distance between the mirror and the scale. To measure the distance dd between the two plates, bring the top plate down to press against the bottom plate. Measure the intercept position of the reflected beam on the scale, h 0. This serves as the reference position. When releasing the top plate, the frame rotates around the axis connecting the two knife edges by angle αα, top plate moves up by distance dd DD 1 αα and the intercept of the reflected beam on the scale also moves up to h with h h 0 = h 2 DD 2 αα. The displacement h is sufficiently large and can be easily measured, enabling accurate measurement of the distance between the two plates: 7

8 dd DD 1 αα = h DD 1 2 DD 2. (5) Measuring the electrostatic force between the two charged plates In this lab, the attractive electrostatic force (FF) between the two plates (when charged to a certain magnitude) is measured indirectly by comparing it with the gravitational force on a mass mm place on Top Plate. If under either force the distance between the two plates is the same, one can conclude FF equals mm gg. In this lab, first place a weight of mass mm at the center of Top Plate which brings it down. Record the intercept position of the reflected beam on the scale, h. Next, remove the mass from Top Plate and connect the two plates to a power supply and gradually increase the potential difference between them. When the intercept position of the reflected beam on the scale is again h, the distance between the plates is the same in both cases, inferring that the corresponding attractive electrostatic force on Top Plate equals the gravitational force on mass mm, as described by εε 0 AA 2 VV dd 2 = mmmm. (6) The SI units are to be used in Equation (6): A the surface area of each plate in mm 2 ; VV the potential difference between the plates in VV; dd the distance separating the plates in mm; mm the mass of the weight in kg; gg = 9.80 m/s 2. Therefore, measuring the mm-versus VV dd 2 relation allows you to 8

9 determine the Coulomb constant or the permittivity in free space. Procedures 1. Measure the averaged width (ww) and length (ll) of the plates, and the distance (DD 1 ) between the marked center of Top Plate and the mid-point of the two knife edges. Calculate the surface area of the plates: AA = ww ll. Record them in Table 1. TABLE 1 ww (mm) ll (mm) AA (mm 22 ) DD 11 (mm) DD 22 (mm) hh 00 (mm) 2. Level the board of the Coulomb balance Use the two vertical screws on the board to level it. This should have been done by your TA. 3. Set up the circuit (Figure 4) At this moment, the power supply should remain turned off. Connect Bottom Plate to the ground (black) of the power supply and connect Top Plate to the positive terminal (red), and be sure to connect in series between them a resistor of 1 MΩ resistance. Connect the voltmeter in parallel with the two plates and make sure the polarities of the terminals are 9

10 correct: positive to positive, negative to negative. Note: Bottom Plate is connected through either one of the two connections on Rod AB on the board; Top Plate is connected through either one of the other two connections on the board. Ask your TA to check the circuit! 4. Align Top Plate with Bottom Plate Place the two knife edges of the frame (as shown in Figure 1) on the two short parallel supporting brass beams on the board. Adjust the knife edges to make the two plates directly face each other. Note: the frame should stand only on the knife edges. Make sure Pane Q is between the two horseshoe magnets but not touch them. Adjust the position of weight P (primarily) and Weight P (if needed) to bring the distance between the two plates within a few millimeters. Make sure that the Top Plate can freely rotate around the axis connecting the knife edges while oscillation slows down in a conveniently short period. Finally, slightly rotate Bottom Plate (along axis AB) to make it parallel with Top Plate and separated by a few millimeters (measured by unaided eyes). 5. Set up the laser and the scale Caution: Do not let the laser beams (incident or reflected) enter any one s eyes. Turn on laser and open the laser head shutter. Adjust the laser set-up such that the plane formed by the laser beam, the normal to the mirror, and the vertical scale is perpendicular to the axis connecting the two knife edges, and that the incident laser beam intercepts the mirror and the reflected beam intercepts the scale. Measure the distance (D2) between the mirror and the scale. Record it in Table 1. 10

11 6. Measure the reference position hh 00 of the reflected laser beam on the scale when the plates touch each other Place the provided small coin at the marked center of Top Plate such that it moves down and presses against Bottom Plate. Read the intercept position h 0 of the reflected beam on the scale. Record it in Table Measure the position hh of the reflected laser beam on the scale when placing a mass on the Top Plate Remove the coin and place a weight with mm = 50 mmmm at the marked center of Top Plate. When Top Plate stops oscillating, read the intercept position (h) of the reflected beam on the scale. Record it in Table Determine the corresponding potential difference between the plates which brings their separation to the same distance as the mass does in Step 7 Remove the mass from Top Plate. When Top Plate stops oscillation, turn on the power supply and gradually increase the voltage until the intercept position of the reflected beam on the scale reaches the same value as measured in Step 7. Following Equation (5), the distance between the two plates is now the same as in Step 7. Following Equation (6), the attractive electrostatic force on Top Plate due to Bottom Plate is the same as the gravitational force on the mass used in Step 7. Read the corresponding voltage (V) on the voltmeter. Record it in Table 2. 11

12 Note: always start with the lowest sensitivity for the voltmeter and then increase if necessary. 9. Repeat steps 7-8 for the weights with m = 40, 30, 20, and 10 mg. For each weight, record the corresponding readings of h and V in Table 2. TABLE 2 mass (kg) h (m) hh (m) d (m) V (volt) Analysis 1. Calculate the h = h h 0 values and record them in Table Use Equation (5) to calculate the d values. Record them in Table Plot mm-versus-(vv/dd) 2 and obtain the slop with a linear fit. Record the result. Slope = 12

13 4. According to Equation (6), the slope equals εε 0 AA. With gg = 9.80 mm/ss 2, use the measured A value, and the fitted slope to calculate εε 0 and kk ee. Record the result. 2 gg εε 0 = kk ee = 13

14 Questions 1. Do the thickness of Top Plate and the thickness of Bottom Plate affect the result? Why? 2. If the two conducting plates are replaced by insulating plates such as plastic ones, can you still use it to measure the permittivity? Why? 3. If we change the resistor of 1 MΩ resistance in Figure 4 to a resistor of 2 MΩ resistance, does it change the experimental result? 14