I have part 1 solved but lost with the others, I'll fail if I don't hand this in
Sussex County Community College PHYS121-41: Physics II with Calculus Term Project (Fall, 2021) Electric Field of a Uniformly Charged Circular Ring In this project, you will investigate the electric field produced by a uniformly charged circular ring. While the calculations for the electric field along the symmetry axis of a charged ring is a standard textbook problem, the general question of the electric field of a charged ring at any arbitrary point in space is highly non-trivial. Although exact analytic solutions for the field exist, the calculations involve advanced integration techniques and the mathematics of Bessel functions, which are usually outside the scope of an introductory physics course. However, on the other hand, this non-trivial problem can be studied by using simple computer simulations that can even be done with a common spreadsheet program. This is what you will do in this project. In addition to the simulations, you will develop analytic approximation formulas for the field and you will compare the approximations to your simulation results. Part I 1. Consider a uniformly charged circular ring of radius R placed on the xy-plane of a Cartesian coordinate system with its center located at the origin, and let the total charge on the ring be Q. Prove that the electric field along the z-axis is given by Here k is the Coulomb’s constant. (10 pts.) Part II In this part, you will investigate the electric field (including the off-axis E-field) of a uniformly charged circular ring by computer simulations using a discrete point charge model. At a point P with the position vector r = ( , , z), the electric field due to a single point charge q located at r’ = ( , , 0) is given by the Coulomb’s law where d = r – r’ = d ed is the displacement vector of the point P relative to the point charge. Here k is the Coulomb’s constant again, and ed is the unit vector in the direction from the point charge to P, i.e. By using the distance formula, the distance between the point P and the charge can be easily found to be Hence, the electric field at P due to the point charge can be written as E = Ex i + Ey j + Ez k with the three components given by In terms of the cylindrical coordinate system, , and the three components in cylindrical coordinates can be written as by using the relationships i = and j = , and is the same as given above. Consider a collection of N identical point charged particles, each one carries a charge q, which are placed on the circumference of a standard circle of radius R on the xy-plane with the center located at the origin. The total charge of the whole collection is Q so q = Q/N. The points are evenly spaced along the circumference. Label the N point charges by n = 0, 1, 2, … , N – 1 such that the azimuthal angle of the n-th charge is where . An example of N = 10 is shown in the figure on the right. 1. Electric field along the z-axis: For N = 40, write a spreadsheet program to calculate the three components , , and of the electric field along the z-axis for using the interval . Plot a graph of (in unit of kQ/R 2) versus z (in unit of R). On the same graph, also plot the theoretical result as obtained in Part I. Verify both and are zero along the z-axis in this case. (20pts.) (Note: The discrete point charge model constructed in this way is actually exact along the z-axis. This gives you a simple, but not complete, check of the program that you wrote.) 2. Off-axis electric field and the convergence test: You will investigate the off-axis electric field of a uniformly charged circular ring using the discrete point charge model in this part. The point charge model constructed in this way can be regarded as an approximation to a continuous charge distribution along a circle and the approximation is expected to become better and better if more and more discrete points (with a fixed total charge) are used. But exactly how many points are needed in order to get a good enough approximation? You will address this question in this part. a) For N = 5, consider a point P on the xz-plane (i.e. the azimuthal angle of P is = 0) and set z/R = 1. Compute and at P for , with equal to the distance of P from the z-axis, using the interval . (5 pts.) b) Repeat the same calculations in (a) for N = 10, 20, 40, and 80. (5 pts.) c) Plot (in unit of kQ/R 2) versus r (in unit of R) for different values of N on a single graph. On a separate graph, plot (in unit of kQ/R 2) versus r (in unit of R) for different values of N. Observe that the electric field converges to a well-defined limit as the number of charges N increases. (10 pts.) 3. Artifact periodicity in the electric field: For a uniformly charged circular ring, clearly the azimuthal component of the electric field is zero and both and are independent of the azimuthal angle because of the rotational symmetry about the z-axis. However, due to the periodic structure of our point charge model, the electric field computed would be periodic in and the period is . In this part, you will investigate how this artifact periodicity diminishes as the number of charges N increases. a) Fix r/R = 1 and z/R = 1 . Calculate , and for for N = 5 using the interval . Repeat the same calculations for N = 10, 20, 40, and 80. (5 pts.) b) Plot (in unit of kQ/R 2) versus for different values of N on a single graph. Also make two more separate similar graphs for and . Observe how the amplitudes of the periodic oscillations decrease as N increases. (10 pts.) (Note: Since this artificial periodic oscillation diminishes quickly as the number of points N increases, you can actually use the average value of the electric field over one period, which is independent of , as an approximation of the field.) 4. Visualizing the electric field: a) Fix = 0 and N = 40, calculate the components and in the range and using the intervals . Clearly, you should skip the point where the charged loop is supposed to be located because the field exactly at that point is undefined. (5 pts.) b) Plot the electric field as a 2D vector field plot. You should adjust the graph’s aspect ratio to 1:1. (10 pts.) (Note: Useful information for how to plot vector fields in Excel can be found at https://engineerexcel.com/create-vector-plot-excel/ . You will have to modify the way to specify the two components because we do not calculate and using explicit formulas. This is all you need if you add the arrows one by one manually. There are not too many to draw. It is manageable. If you prefer to automate the process of drawing arrows using a VBA subroutine, you are on your own. If you are not familiar with using VBA subroutines in Excel, here is a start: https://www.guru99.com/vba-functions-subroutine.html . You may have to do your own research if needed.) Part III In this part, you will develop analytic approximation formulas for the electric field based on power series expansions. Finally, you will compare the approximations to the simulation results in Part II. Consider a uniformly charged circular ring of radius R and total charge Q placed on the xy-plane of a Cartesian coordinate system with its center located at