Download Vacuum Diode - Electricity and Magnetism - Past Paper and more Exams Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! NIU Physics PhD Exam - Electricity and Magnetism - September 2O04 **:l' Do any 3 of the 4 problems. *'lc* Problem EMl. In a vacuum diode, electrons with mass m atd.charge q are uboiled" ofi a hot cathode, at potential V :0, and accelerated across a gap to the anode, which is held at positive potential V : Vo. The cloud of moving electrons within the gap (called space charge) quickly builds up to the point where it reduces the field at the surface of the cathode to zero. From then on, a steady current .I flows between the plates. Suppose the plates are large relative to the separation (arca A> & in the figure below), so that edge effects can be neglected. Then the potential I/, the charge density p, and the speed ofthe electrons o are all functions ofo on1y. (a) Assuming the electrons start from rest at the cathode, what is their speed at point o, where the potential is V(r)? (b) In the steady state, the current .I is independent of c. Slhat, then, is the relation between p and u? (c) Use the results above and Poisson's equation to obtain a differential equation for the potential 7(c), not involving p or a, and solve it. (Hint: try a power law form.) (d) Show that a non-linear relation holds between .I and Vs, of the form whereKisaconstant rational number. I:KW that you will find in terms of the given quantities, and n, is a d Cathode (V:0) Anode (Vo) Problem EM2. Consider the right circular cylindrical cavity pictured below. The inner and outer walls of the cavity, and the ends, are perfect conductors. The inner conductor has radius o, the outer conductor has radius b, and they are coaxial with the z-anis. The height of the cavity is h. Consider the TEM modes of the cavity (the modes that have an electric field with no component in the z direction), in the vacuum region a S r S b and 0 1 z ( h, with peak electric field .Eo. Find expressions for the electromagnetic fields and the resonant frequencies and wavelengths of all such modes in terms of the grven quantities. Problem EM3. A long solenoid consists of N turns of wire with. resistance R wound on a right circular cylinder of length d and radius a, with d> a. (a) Suppose that a slowly time-varying current I(t1 -.Iscos(dt) flows in the solenoid. Find the magnetic and electric fields everywhere inside and outside of the solenoid. (b) A circular ring of wire with radius b is coa:rial .with the solenoid and outside of it (b > o). Suppose now that a slowly varying current I(t) -.I6cos(cut) flows in the ring. What current is induced in the solenoid? t I I l
