E1 Report

E1 REPORT - SPRING 2023 DATA (2.5 points) Part I: Charging by conduction Initial Voltage (V) Voltage after Discharging Pail (V) White and Proof Plane (3 rubs) 5 0.33 Blue and Proof Plane (3 rubs) 12 0.6 White and Proof Plane (6 rubs) 28 0.8 Blue and Proof Plane (6 rubs) 25 0.6 Part II: Charging by induction Depth down Pail Charge of proof Plane (V) ¼ of pail length 13 ½ of pail length 16 ¾ of pail length 16 Bottom of pail 16 Using two charge producers: Charge of Instrument (V) Blue producer deep into pail -8 White producer deep into pail 8 Both producers deep into pail 0 Induction by proof plane: Initial Charge (V) Grounded Charge (V) Proof Plane removed Charge (V)

White Charge Producer 12 0 -11 Blue Charge Producer -4 0 4 Part IV: Charge on a sphere Sphere I Charges after Charging Sphere II: Point Φ ( ° ) Θ ( ° ) Charge (V) A 0 0 -8.5 B 90 0 1.6 C 180 0 3 D 270 0 -0.2 E N/A 90 2 Part V: Charge on Asymmetrical Conductors Point Trial 1 Charge (V) Trial 2 Charge (V) Avg Charge (V) A 15 14 14.5 B 10 10 10 C 10 10 10 D 10 9 9.5 E 10 9 9.5 Average Charge for all Points: 10.7 Error in Charge: +/- 3.8 Part VI: Gauss’s Law

Using your data from procedure step 4, Part II, plot the electrometer reading vs. fractional depth to which the proof plane is inserted into the pail. ( 0.5 points) Plot the charge distribution on Sphere I (Part IV) as a function of (for ). Is this plot ϕ θ = 0 𝑜 linear? Would you expect it to be linear? (1 point) The plot is not linear. It is not expected for the charge distribution to be linear because of the fact that charge distribution is dependent on the inverse of distance squared not on the angle of rotation around a sphere. This means that the graph would not be linear.

Include sketches of the apparatus showing the distribution of charges on spheres I and II in Part IV and also, separately, on the asymmetric conductor in Part V. Indicate high surface charge density by closely spaced “+” or “-” signs, as appropriate). (1 point) CALCULATIONS AND ERROR ANALYSIS (0.5 points) (General instructions:show all steps for each type of calculation or error analysis only once; must be typed) Calculate the error in repeated measurements of voltage for all your data in Parts IV, V, and VI and report the average and error values. Show this calculation of the average and error for any one of your data sets here. *Part IV did not require repeated measurements. Calculating Average Charge (ex. Part V, Point A) In order to calculate the average charge for a point, it is necessary to add all relevant measurements and divide by the number of data points as shown below: (Measurement 1+ Measurement 2)/2 = (15V + 14V)/2 = 14.5 V

4. Comment on the way charges distribute themselves on a conducting sphere depending on the external conditions. Can you explain this charge distribution by referring to Coulomb’s Law? Is it possible to have a positive net charge on the sphere, and still have some parts of its surface charged negatively? How would you design an experiment to test for this possibility? (1 point) In Part IV, we placed a neutral conducting sphere next to a positively-charged conducting sphere. By Coulomb’s law of attracting and repelling electrostatic forces, The positively-charged conducting sphere attracted the electrons on the surface of the neutral conducting sphere towards it, causing point A on the neutral conducting sphere to have a negative charge distribution of -8.5V (see Data Part IV). Since the electrons clustered at point A, the other points, which were farther away from the positively-charged sphere, were less negatively charged and some became positively charged. Point C, the furthest away from point A, had the highest positive charge of 3.0V, which makes sense as it should have the least number of electrons. Point B and E had a positive charge of 1.6V and 2V respectively. Point D had a negative charge of -0.2V. This can be explained by the fact that it was probably the closest out of all the other points to point A, so there is still a significant number of electrons accumulating at that point. It should be possible to have a positively-charged conducting sphere have some parts of its surface charged negatively. We can design an experiment where a positively-charged and negatively-charged conducting sphere are placed next to each other. By Coulomb’s law, the positive charges on the positively-charged sphere will cluster at the point closest to the negatively-charged sphere, and the negative charges on the negatively-charged sphere will cluster at the point closest to the positively-charged sphere. As a result, both spheres will have a more positive and negative side. 5. Can you conjecture what would happen to the charge distribution if the radius of Sphere I in Part IV decreased (or increased) while radius of Sphere II, the separation between the surfaces of the two spheres, and the net charge on each sphere remained the same? (0.5 points) The charge distribution at each point would decrease (become more negative) if the radius of Sphere I decreased and would increase (become more positive) if the radius of Sphere I increased. If the radius of Sphere I decreased, the surface area of Sphere I would decrease as surface area is . Since the net 4π𝑟 2 charge of both spheres remain the same, the charge density of the Sphere I would increase as . This means that the concentration of electrons per unit of surface area would increase, σ = 𝑄 𝐴 = 𝑄 4π𝑟 2 causing the charge distribution at each point to become more negative. The inverse happens when the radius increases. 6. What were the main difficulties of this experiment, and the dominant sources of systematic error? If you were approaching this experiment once again from the beginning, or giving advice to a student who has not yet performed it, what improvements or refinements of the procedure would you suggest? (0.5 points) We had a large systematic error in Part I of our experiment. Rubbing the proof plane on the white producer vs the blue producer should have produced oppositely-charged voltages for the proof plane. However, in both cases, our experiment showed that the charge of the proof plane was positive. At first we believed that we performed the procedure incorrectly, so we repeated Part I multiple times and had our

TA double check our results. However, despite that, we still had the same error. We even switched voltmeters, but the error persisted. There must have been some systematic error at play, but unfortunately both we and TA were unable to figure it out. Some of the voltmeters also had positive-charge bias. Whenever we would zero the voltmeter, it will still be slightly positive by 0.2V-0.3V. Therefore, the experiment might benefit from having more accurate and newer voltmeters. TIME STAMPED NOTES ( -2 points, if missing)