Tentamen i ELEKTROMAGNETISK FÄLTTEORI

Transcription

1 Karlstads Universitet Fysik Tentamen i ELEKTROMAGNETISK FÄLTTEORI [ VT 2017, FYGB03] Datum: Tid: Lärare: Jürgen Fuchs Tel: / Total poäng: 50 Godkänd / 3: 25 Väl godkänd: : : 42 Tentan består av 2 delar som inlämnas separat: Del 1: 10 p. Del 2: 40 p. Hjälpmedel: Del 1 & 2: Ordbok/ordlista engelska svenska Del 2 (efter del 1 har inlämnats) dessutom: Ett handskrivet A4 ark med valfritt innehåll (skrivet på ena sidan, ej maskinskriven eller maskinkopierad) inlämnas tillsammans med tentan Physics Handbook Endast en uppgift per sida. Svaren måste vara väl motiverade. FYGB03 Tentamen

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3 Problem 1 Basic electric och magnetic phenomena 10 p. a How rapidly do the electric field and electrostatic potential fall off at distances 2 p. very far away from the following charge distributions: A point charge. An electric dipole. An infinite straight line of uniform line charge density. Two parallel infinite straight lines of uniform line charge density ρ l and ρ l, respectively. b Describe the magnetic flux density for the following systems: 2 p. a long straight wire carrying a uniform steady current; a magnetic dipole. Draw pictures of the field lines. c 2 p. Which of the following statements are correct? The divergence of the electric field strength E( r,t) is zero. In time-independent situations the divergence of the electric field strength E( r) is zero. In the absence of electric charges the divergence of the electric field strength E( r, t) is zero. In the absence of macroscopic electric charges the divergence of the electric field strength E( r,t) is zero. In the absence of electric currents the divergence of the electric field strength E( r, t) is zero. The curl of the electric field strength E( r,t) is zero. In time-independent situations the curl of the electric field strength E( r) is zero. Intheabsenceofelectricchargesthecurloftheelectricfieldstrength E( r,t) is zero. FYGB03 Tentamen

4 d Describe the diffferences in the behavior of diamagnetic, paramagnetic and 2 p. ferromagnetic materials. Explain the microscopic origin of these diffferences. e 1 p. Why do transformers usually have laminated cores? f What is a gauge, or gauge condition? 1 p. Why can it be important to impose a suitable gauge condition? FYGB03 Tentamen

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6 Problem 2 Electrostatics: Electric field 6 p. A circular disk of radius d carries a uniform surface charge density ρ s. a Determine the electrostatic potential V and the electric field intensity E at 4 p. every point on the symmetry axis of the disk. b Give an approximate formula for the E-field on the axis that is valid in the 1 p. limit that the distance from the disk becomes much larger than the radius d. c 1 p. Interpret the result of part b. Problem 3 Electrostatics: Image charges 5 p. Consider two point charges q 1 =Q and q 2 = Q above a grounded conducting plane. In Cartesian coordinates (x,y,z), let the two point charges be located at r 1 =(a,0,a) and at r 2 =( a,0,a), respectively, and the grounded plane be given by z=0, i.e. the x-y-plane. a Using the method of image charges, determine the total force that is exerted 3 p. on the charge q 1. b Find the surface charge density ρ s at the point r 3 =(a,0,0) on the grounded 2 p. plane. FYGB03 Tentamen

7 Problem 4 Conductors 5 p. A cable of length L and cross section S, and consisting of a metal of conductivity σ is immersed in the ground. The cable is isolated against the ground (i.e., the resistance between cable and ground is infinite), but it turns out to have an isolation fault (i.e., finite resistance R fault against the ground) localized at some point along the cable. 5 p. Taking the ground(earth) to have the same electrostatic potential everywhere, one can describe the situation as a circuit involving three resistors. Explain how this can be done. Combine this description with the general formula for the total resistance of a long cable to determine both the location of the isolation fault and its resistance R fault in terms of L, S, σ and of the resistances R and R L between cable and ground that one measures at the two ends of the cable. Problem 5 Magnetostatics 6 p. Consider a very long cylindrical conductor carrying a uniform current density J 0, that contains a cavity having the shape of an off-center parallel cylinder. Denote the radius of the cylindrical conductor by a, the radius of the cylindrical cavity by b, and the distance between the axis of the conductor and the axis of the cavity by d. a 5 p. Compute the magnetic flux density B( r) inside the cavity. Hint: By the superposition principle, the field in the cavity is the same as the one of two cyclinders with radii a and b carrying uniform current densities J 0 and J 0. b The calculation shows that the B-field in the cavity is constant. 1 p. Why is this result of interest for applications? Discuss whether there is a preferred value for the parameter d. FYGB03 Tentamen

8 Problem 6 Induction 6 p. A very long straight wire is placed parallel to one side of a rectangular loop of wire, as shown in the figure to the right. Regard the rectangular loop as an electric circuit consisting of a resistance R 2 (shown explicitly in the figure) and a self-inductance L 2. Denote the width of the loop by a, the height of the loop by b, and the distance between the loop and the straight wire by d. R 2 a 1 p. Compute the mutual inductance L 12 between the straight wire and the loop. b Set up the differential equation that is obeyed by the electric current in the 1 p. loop. c Solve the differential equation subject to the following constraints: Initially the straight wire and the loop do not carry any current. At some time t 1 suddenly a current of constant magnitude I 0 is switched on in the straight wire, and at time t 2 it is switched off again. 2 p. Hint: The derivative of the Heavyside step function θ(t t ) is given by the delta function δ(t t ). d Compute the total energy dissipated in the resistanve R 2. 2 p. FYGB03 Tentamen

9 Problem 7 Elektromagnetic power 7 p. An very long coaxial cable consists of a solid inner conductor of radius a and a thin outer conductor of inner radius b. At one end the inner and outer conductor are connected through a resistor of resistance R and at the other end they are connected through a battery which provides a voltage V 0. a Obtain an expression for the electric and magnetic fields E and B in the region between the inner and outer conductors, in terms of the quantities a, b, V 0 and R. (Neglect the resistance of the coaxial cable itself and of the wires that connect the cable to the resistor and to the battery.) 3 p. b Determine the Poynting vector in the region between the inner and outer 2 p. conductors. c By integrating the Poynting vector over the region between the conductors, 1 p. obtain the total power flow between the conductors. d 1 p. Interpret the result. FYGB03 Tentamen

10 Problem 8 Elektromagnetic waves 5 p. A plane electromagnetic wave in vacuum with harmonic time dependence exp(iωt) has a complex electric field strength ( phasor ) E(x,y,z) = E exp( ia(x+y+z)) ( e x e y ), where (in standard units) the parameters E and a are positive real numbers. a 1 p. What is the direction of propagation of the wave? b Determine the complex magnetic flux density ( phasor ) B(x,y,z). 2 p. c 2 p. Compute the (time-dependent) Poynting vector P( r,t) of the wave and its time average value P( r). FYGB03 Tentamen