Gauss s Law Prob.#1 Prob.#2 Prob.#3 Prob.#4 Prob.#5 Total Your Name: Your UIN: Your section# These are the problems that you and a team of other 2-3 students will be asked to solve during the recitation session next week. We may add numerical values and more questions to these problems for the actual class team work. Print this form BEFORE coming to class. Get familiar with the problems proposed, think about a solving approach, and identify the methods and equations needed. Choose five problems from this form and sketch a symbolical solution for them. Attach pages if you need more space for writing. DO NOT actually solve the problems but only indicate in words, or in a numbered list the order in which the equations will be put at work. This is the Approach. For each of the five problems you will choose, write a very short verbal Explanation. Bring your annotated pre-ice to the recitation session and hand it to the recitation instructor for grading. 1. Captive in a cube: A point charge q is at the center of a cube with the sides of length a. a) What is the electric flux through one of the six faces of the cube? 2. Make use of Gauss s Law: Gauss s law is always true, but it isn t always useful. It happens that there are exactly three classes of charge distributions that have enough symmetry to be analyzed easily with Gauss s law. a) For each of the following uniform charge distributions, find the vector form of the E-field in the point marked with an x. Distribution #1: A solid sphere of radius R carrying a volume charge density. r x Distribution #2: An infinitely long line carrying a line charge density R r x 1
Distribution #3: A plane of infinite area carrying a surface charge density. r x 3. Conductor with cavity: A conducting spherical shell with inner radius a and outer radius b has a point charge Q located at its center. The total charge on the shell is q. The shell is insulated from its surroundings. a) Find the E-field for this system for points inside the cavity (0<r<a), in the conductor shell (a<r<b), and outside the shell (r>b). b) Find the surface charge density on the inner and on the outer surfaces of the shell. Which of them is higher? 2
4. Conducting wire: A conducting wire of length L and radius R is uniformly charged with a total positive charge Q and is placed in a concentric conducting tube of same length. The tube has a thickness t, has the external radius 3R, and it was initially uniformly charged with the same Q. The setup is shown in the figure. Neglect any edge effect at the ends of the rod and tube. a) Find the electric flux passing through an external concentric cylindrical surface T of radius 4R and height h. b) Find the electric field in the point M on the cylindrical surface T. 5. Non-uniform electric fields: A cube with sides of length a is located with one corner at the origin, like in the figure. Find the flux through the shaded area if the electric field everywhere is given by the equation: a) E = αx 2 i + βy 2 j b) E = αy 2 i + βx 2 j Note: α and β are known constants. 3
6. Another universe: Suppose that physicists have just discovered a parallel universe where the laws of physics are different from what we know. While the electric charges are present in that different world, the measure of the interaction between two electrically charged point particles q1 and q2 is given by the law F = C q 1q 2 r 6 r where, just like in the Coulomb s law, r is the distance between the charged points, r is the unit vector from source to probe (test charge), and C is a positive known constant. Just like in our universe, like charges repel and opposite sign charges attract each other. In that parallel universe, if a positive charge Q is placed at the origin of the Cartesian system of axes, what would be the electric flux through the surface of a sphere of radius R enclosing the charge? See the figure. 7. Non-uniform charge density: A spherical electronic device of radius R contains a total positive electric charge Q distributed such that the volume charge density ρ(r) is given by the piece-wise function: 4
ρ(r) = [ 3α 2R r, if 0 r R/2 α (1 ( r 2) R ), if R r R 2 0, if r R Here, α is a positive constant having units of Coulomb/cubic-meter. a) Determine α in terms of Q and R. b) Derive an expression of the electric flux through the surface of this sphere of radius R. c) Derive an expression for the electric field at the distances R/4, R/2, 3R/4, and 2R from the center of this distribution of charge. 8. Challenge problem Storing energy in electric fields & Capacitance: All the electronics that run the present world are powered on by electric currents. The electric charges and the energy needed for making these current run can be stored by devices called Capacitors. One thing that the electrical & computer engineers do is to probe the voltage inside the charge-storing devices. If that voltage is too high, then the insulator between the two conducting plates could become ionized and the charge would be lost. The better the insulator between the plates, the more electric charge can be stored for the same applied voltage. The challenge of today is to find a relationship between the geometrical parameters of the insulator between the plates and the voltage applied on the insulator s surface, if the electric charge stored does not change. You should remember from the previous lab that the difference of electric potential (voltage) is nothing else than the work done by the electric field on a +1 (unit) electric charge. You may want to first read your lab manual and do the Pre-Lab assignment of the week before solving this challenge problem. The challenge: Your team is to find out what is the E-field between the plates of a plane parallel plates capacitor. The insulator between the plates is the air of the room. You are connecting the lead of the electric supply to the two metal plates of area A each. The plate connected to the + terminal acquires the total charge Q. The plate connected to the common terminal acquires the total charge Q. 5
If the distance between the plates has a really small value d, then we can consider the size of the plates infinitely large. a) Apply Gauss s law for the two infinite plates to calculate the value of the electric field in the region between them and in the region outside them. b) How does the value of the electric field between the plates of the capacitor change if the plates are pushed to each other until the distance between them halves? c) How does the value of the electric field between the plates of the capacitor change if the plates are away from each other until the distance between them doubles? d) What would be the force that the electric field between plates would exert on a particle of dust of charge q? e) What would be the work that the electric field between the plates would do to move the -q particle of dust from the surface of the negative plate until reached the surface of the positive plate? 6