Introduction to Electromagnetism - Electricity and Magnetism - Past Exam, Exams of Electromagnetism and Electromagnetic Fields Theory

Download Introduction to Electromagnetism - Electricity and Magnetism - Past Exam and more Exams Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! Introduction to Electromagnetism 171.106 Final Exam 5/13/09 2-5 You are allowed one 8.5 inch by 11 inch page of formulae, both sides filled. Also check the formulae provided on the last page of this exam. Start each problem on a fresh page and please give detailed reasoning. If in a later question you need a result from a question that you cannot solve then write formulae with the unknown as a symbol and you will get due credit. Ask for clarification if the text is unclear. Problem 1 (8 points) Consider a region of space where the electric field is oriented along the x̂ -direction under electrostatic conditions (no time dependence). (a) Show that the field is independent of the y and z coordinates in that region. (4 points) (b) Show that if there is no charge in the region then the electric field is also independent of x. (4 points) Problem 2 (30 points) Consider a parallel plate capacitor with plate area A and plate spacing, d. (a) Under the assumption that 0E outside the capacitor write an expression for the total energy stored when the plates carry charges +Q and –Q respectively. (4 points) (b) Use the expression for the total energy derived in (a) to determine the capacitance of the device. (4 points) (c) Determine the electrical dipole moment of the charged parallel plate capacitor and use it to write an approximate expression for the electric field very far from the capacitor. (4 points) (d) Assuming the result from (c) can be used down to a distance of minr A from the capacitor show that there is an additional contribution of  2 stray 3 23 Qd U A  to the stored energy of the charged capacitor. (8 points) (e) Determine a value for the capacitance that takes into account this stray field contribution to the stored energy. Discuss your expression in appropriate limits. (6 points) (f) Is it possible to design a capacitor that has no dipole stray field? (4 points) Problem 3 (12 points) A thin circular ring with radius, a, carries a total charge Q. The ring rotates with angular velocity,  about the axis which is perpendicular to the plane of the ring and passes through its center. (a) Determine the magnetic B field at an arbitrary point along the axis of rotation. (4 points) (b) Determine the magnetic dipole moment of the ring. (4 points) (c) Show that far from the ring the result of (a) is consistent with the dipole approximation. (4 points) Problem 4 (18 points) A consumer connects an impedance 1 2Z z iz  to an AC electrical outlet as indicated in Fig. 1. Here z1 and z2 are real numbers of dimension Ohm. Denote the amplitude of the AC voltage by V and the angular frequency by  . (a) Determine the average power dissipated. (5 points) The power company complains that the load is too inductive (  2 1/z z is too large and positive) and demands that the impedance presented to the grid be purely resistive (no imaginary part seen by them). (d) Show that connecting a suitably sized capacitor in series with the load can solve this problem. Determine also the required capacitance. (5 points) In a different application  2 1/z z for the load ends up too large and negative (a capacitive load). Again the power company demands a resistive load only. (e) Show how to connect a capacitor to cancel a capacitive load and once again present a resistive load only. Determine also the magnitude of that resistive load. (8 points) Figure 1.