XPHA 4279 SECOND PUBIC EXAMINATION Honour School of Physics and Philosophy Part A A2P: EECTROMAGNETISM Tuesday June 2012, 2.30 pm 4.10 pm Time allowed: 1 hour 40 minutes Answer all of Section A and two questions from Section B. Start the answer to each question on a fresh page. A list of physical constants and conversion factors accompanies this paper. The numbers in the margin indicate the weight that the Examiners expect to assign to each part of the question. Do NOT turn over until told that you may do so. 1
Section A 1. State Gauss s law in electrostatics, defining all symbols used, and show how it can be combined with the concept of an electrostatic potential V to produce aplace s equation. [5] Three identical charges +q are arranged in an equilateral triangle of side a. Find the energy required to assemble this configuration, given the convention that the energy is zero when the charges are all separated at infinity. [3] 2. A solenoid of radius a and infinite length has n turns of wire per unit length carrying an electrical current I. (a) Use Ampère s law to show that the magnetic flux density within the solenoid is uniform with magnitude B = µ 0 ni. [4] (b) Calculate the inductance of a solenoid of length l. [3] 3. Two infinitely long parallel conducting wires, each of radius a have centres separated by a distance d. Given that one wire has a charge +λ per unit length, and the other λ, show that the potential difference between the two wires is given by λ πε 0 ln d a a. Hence calculate the capacitance per unit length of a twin cable, given that a d. [7] 4. Explain why Ampère s law in integral form needs to be modified to include the concept of a displacement current, using the parallel plate capacitor as an example. Show how this modified equation leads to one of the four Maxwell equations. [5] XPHA 4279 2
Section B 5. In a variant of Millikan s famous oil drop experiment, Professor Oxonious attempts to measure the charge of an electron using the magnetic field associated with a straight horizontal wire carrying a current I. Drops of oil of mass m are inserted at rest at the left end of the apparatus, and become charged due to ionising radiation. et the charge of a drop be Ne where e is the magnitude of the charge on a single electron. Once charged, the drops are accelerated by a uniform horizontal electric field E for a distance d to a non-relativistic velocity v. Their initial velocity on passing from the electric field region into the remainder of the apparatus, shown as the large rectangle in the diagram below, is purely horizontal. Thereafter, their motion is confined to the vertical half-plane bounded on the bottom by the wire and they experience the combination of gravitational and magnetic forces. This configuration is shown in the following sketch. E d Straight horizontal wire z N (i) Show that on exiting the electric field region, an oil drop has a speed v = 2N eed/m [3] (ii) Show that in the vertical plane containing the wire, the magnetic field is perpendicular to this plane and varies inversely with distance from the wire. [4] (iii) Write down the equation of motion for the drop and show that if the vertical component of velocity is to remain zero the drop must be a distance z N = Nevµ 0I 2πmg above the position of the wire. Indicate clearly on a sketch the direction of I needed to satisfy this condition and the resulting B. [5] (iv) Hence by combining your results from (i) and (iii) show that there are discrete vertical positions z N where drops with N electronic charges will pass through horizontally given by z N = (Ne) 3/2 /α where α = πg 2m 3 /(µ 0 I Ed). [4] (v) For N = 2 calculate the expected vertical position z 2 when m = 10 19 kg, I = 1000 A, and Ed = 1000 V. [4] XPHA 4279 3 [Turn over]
6. An induction device is set up as shown below. It has a U-shaped conductor of negligible resistance along which a conducting slider of resistance R moves in the x- direction. The assembly is immersed in a uniform magnetic flux density B of magnitude B 0 that is directed out of the page. The slider is moved such that its position with respect to the end of the U-shaped conductor is given by x = + x 0 cos ωt with x 0. A l I 0 B 0 V 0 x Slider B (i) Derive the voltage V appearing across AB and the current I 0 flowing around the U. [6] (ii) The U-shaped conductor now has a finite resistance per unit length α; show that the voltage V appearing across AB is now given by R V = B 0 lx 0 ω sin ωt R + α[l + 2( + x 0 cos ωt)]. [6] (iii) A resistive load R 1 is now placed across the ends of the circuit, as shown in the following figure. l I 0 I R R 1 I1 0 x (a) Show that I 0 = I R + I 1. [2] (b) By considering various closed loops within this circuit, derive the following relations I R R + I 0 α[l + 2( + x 0 cos ωt)] = B 0 lωx 0 sin ωt I R R + I 1 [R 1 + 2α( x 0 cos ωt)] = B 0 lωx 0 sin ωt. (c) Show that in the limit R 1 we must have I 1 0 and that the above relations are then consistent with the results in part (ii) of this question. [2] (d) For the case α 0, but R 1 bounded, show that I 1 0. [2] [2] XPHA 4279 4
7. A thin metal sphere of radius b has charge Q. (a) Calculate its capacitance. [3] (b) Calculate the energy density of the electric field at a distance r > b from the sphere s centre. [3] (c) Calculate the total energy in the field. [4] (d) Calculate the work expended in charging the sphere by carrying infinitesimal charges from infinity. [3] (e) A potential V is established between inner (radius a) and outer (radius b) concentric thin metal spheres. What is the radius of the inner sphere such that the electric field near its surface is a minimum? [7] XPHA 4279 5 [AST PAGE]