Electric fields 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. An example of a sketch of a solution is provided below. EXAMPLE: Parallel charged plates A point particle of mass m travels freely in the x-direction with uniform velocity v 0. At x = 0, it enters a region between two plates oriented perpendicular to the y-axis; the plate spacing is w, and then plate length in the x-direction is L. The particle enters on the mid-plane y = 0. While between the plates, it experiences a constant, spatially uniform force F in the +y-direction. After exiting the plates the particle again moves freely. Obtain an equation for the y-coordinate of the point at which the particle exits the plate. List the steps taken for finding the solution. You answer here the questions WHAT? and HOW? 1. Draw diagram, annotate it accordingly. v 0 m +w/2 w/2 y L F x 2. Apply kinematic equations for each of the directions x and y. According to Newton s 2 nd, there is no acceleration along the x-axis. 3. Calculate the flight time from the equation along x-axis, because L is given. 4. Plug in the time found at step #3 into the equation for y-position. 5. Check the units: y is a distance and the result must be in units of length. 6. Check limiting behavior: if L, then y=w/2. Explain why you chose this approach. You answer here the questions WHY? and WHEN? Initial position, velocity, and acceleration are known. Final position is asked. This is a case of kinematic equations. 1
1. Fixed beads: Three point charges are fixed on an insulating horizontal table as shown in the diagram below. The system is in vacuum. a) Draw a diagram showing the direction and magnitude of the net electric field at the point P. b) A small bead of charge Q is placed on the table at the point labeled P. Use your results from part a) to calculate the net (vector) force on the bead. c) Where should an additional fixed charge of magnitude +q be placed in order to make the net force on the bead of charge Q zero? In other words, to have the bead of charge Q at equilibrium. 2. Flying electron: The electric field inside a large plane parallel-plate capacitor is E everywhere (see diagram). The distance between the plates is d. Due to the strong field inside, the negative plate loses an electron. a) What is the force on the electron immediately after being liberated from the plate? b) What is the speed of the electron when it arrives at the positive plate (assume that the electron had zero velocity upon liberation from the plate)? c) How long does it take for the electron to travel from the to + plate? 2
3. Circle: What is the electric field at the center of the insulating-wire circle of radius R in the diagram? The top half of the circle has a charge +Q spread uniformly over its length and the bottom half of the circle has a charge -Q spread uniformly over its length. The sign of the electric charge changes abruptly between the two regions (halves). 3
4. Antennas: From atoms to antennas, many real-life systems can accurately be modeled by a system composed of two electric charges of equal magnitude q, but of opposite sign, placed at a fixed distance L from each other. This system is called an electric dipole. Calculate the electric field of an electric dipole at the following points. All distances are given with respect to the center of the electric dipole system. a) At a distance L along the axis of symmetry of the dipole. b) At a distance L along the axis of the dipole itself, on the positive charge side. 5. Torques: Three point charges are fixed at the three vertices of an equilateral triangle of side a as shown in the diagram. The system is in vacuum. Calculate the net torque on the dipole consisting of the charges +q and -q. 4
6. Rod: A thin rod of length L carries a total charge Q uniformly distributed along its length. Find an expression for the electric field along the axis of the rod starting at one end. I.e., find E(x) for x 0. 7. Large sphere: A large non-conducting sphere of radius R is non-uniformly charged. The density of the electric charge inside this sphere changes with the distance from the center according to the equations below. Find an expression for the total electric charge contained in this sphere. 5
8. Challenge problem Mapping an electric field: All the electronics that run the present world are powered on by electric currents. Electric current is nothing else than electric charges moved around by electric fields. One thing that the electrical engineers do is to probe the electric fields inside the electronic devices. The challenge of today is to probe electric fields. 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 on the surface of an electronic device. You are connecting the common lead of the voltmeter to the left rim of that device and are moving the positive lead (the probe) in the x-y plane on the surface to probe. You find out that the reading of the voltmeter decreases with 5mV for each millimeter when moving from right to left along the x-axis. If you move the positive lead (probe) parallel to the left rim at different distances x, the reading stays the same on the meter. a) What is the electric field along the surface of that electronic device? b) What type of electric charge distribution is enclosed in that electronic device? 6