Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico. Fall 2004

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1 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions: The exam consists two parts: 5 short answers (6 points each) and your pick of 2 out 3 long answer problems (35 points each). Where possible, show all work, partial credit will be given. Personal notes on two sides of a 8X11 page are allowed. Total time: 3 hours Good luck! Short Answers: S1. Graphed below are the equipotential contours associated with two point charges. (i) Sketch the electric field lines. Show arrows denoting the direction of the field (ii) Which charge distribution could create this potential? (a) (b) (c) (d) +3q 3q -3q +3q q +q q +q

2 S2. Which field lines could represent a static magnetic field. S3. A charge q is placed a distance d from a grounded infinite perfectly conducting plane. With what force is it attracted to the plane? S4. A resistor, capacitor, and inductor are connected in parallel across a battery. At t=0 the switch is closed V R C L Describe the current in the three elements as a function of time for t>0. S5 A plane wave solution to Maxwell s equations in a homogeneous, linear dielectric is given by E(z, t) = E 0 cos( 6y t) ˆ x, where t is in seconds, y is in centimeters. (a) What is the direction of propagation? (b) What is the index of refraction of the medium? (c) What would the wavelength be if this wave traveled in free space?

3 Long Answers: Pick two out of three problems below L1. Two concentric metal spherical shells of radius a and b, respectively, are separated by a weakly conducting material of conductivity σ. b a (a) If they are maintained at a potential difference V, what current flows between them? (b) What is the resistance between the shells? (c) Notice that if b>>a the outer radius (b) is irrelevant. How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea held quite far apart, if the potential between them in V. (This arrangement can be used to measure the conductivity of sea water.) L2. Consider wave incident on an nonmagnetic neutral conductor. Treat the electrons as responding according to Ohm s law with static conductivity σ 0, J = σ 0 E. (a) Using Maxwell s equations, show that inside the metal the electric field satisfies the following wave equation, E c 2 t 2 E = µ 0 σ 0 t. (We neglect here any frequency dependence associated with the conductivity). (b) Show that plane waves oscillating at frequency ω propagate inside the material with complex wave number, k = ω c 1 + i σ 0 ε 0 ω. (c) What is the physical meaning of the real and imaginary parts of k? (d) Consider a microwave at 10 GHz reflected from a silver mirror with σ 0 = (ohm-m) 1, ε 0 = Farad/m. Approximately how many meters will the microwave penetrate into the mirror (sometimes known as the skin depth ).

4 L3. An insulating circular ring (radius b) lies in the x-y plane, centered at the origin. It carries a linear charge density λ = λ 0 sinφ, where λ 0 is constant and φ is the usual azimuthal angle. The ring is now set spinning at a constant angular velocity ω about the z axis. r z x ω y (a) Calculate the total power radiated into the far field (r>>b) as electric dipole radiation? Hint: Recall the Larmor formula for the instantaneous radiated power, P(t) = 1 2 d 2, where d is the electric dipole moment. 4πε 0 3 c 3 (c) What is the polarization of this radiated field on the z-axis and on y-axis? (d) What power is radiated as magnetic dipole radiation?

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14 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Fall 2006 Instructions: The exam consists of 10 problems, 10 points each; Partial credit will be given if merited; Personal notes on two sides of an 8 11 page are allowed; Total time is 3 hours. Problem 1: A charge q is uniformly distributed on the surface of a sphere of radius R. What is the potential energy stored in this charge configuration? Problem 2: A charge q is placed a distance d above an infinite, grounded, conducting plane. Find the induced surface charge density as a function of coordinates on the plane. q d Problem 3: Hall effect: A uniform magnetic field B 0 in the z-direction is applied to a semiconducting material carrying current in the y-direction. In steady state, a voltage develops across d and the charge velocity v y does not vary (electric and magnetic forces balance). - What is the Hall voltage V h in terms of v y, B 0 and d? - Can this measurement determine the polarity of the charge carriers (n-type vs. p-type semiconductor)? Explain. Problem 4: Consider a transmission line consists of two parallel conducting strips of width w, length l, separated by distance d<<w<<l. Ignoring fringing fields, what is the inductance per unit length? d w l

15 Problem 5: Betatron: An electron with speed v, undergoing cyclotron motion in a magnetic field B(r) at the cyclotron radius r 0 = mv/qb(r 0 ) can be accelerated by ramping B-field in time. - Since magnetic fields do no work, what is increasing the kinetic energy of the electron? - Show that if the field at r 0 is half of the average across the orbit, B(r 0,t) = 1 2 " B(r,t)! da 2, #r o then the radius of the orbit is constant in time. Assume nonrelativistic speeds. Problem 6: Consider a parallel RLC circuit driven by an ac voltage source at frequency ω. In steady state, what is the current drawn from the source as a function of time (Hint: this is easiest if you use complex impedance). I(t) V 0 cos!t R L C Problem 7: Starting with Maxwell s equations, show that the electric and magnetic fields are derivable from scalar and vector potentials, E =!A!t " #$, B =! " A. How can we change the vector and scalar potentials without changing the electric and magnetic fields? Problem 8: A transverse electromagnetic wave travels inside a neutral plasma, inducing a current density J =!en e v, where n e is the density of electrons and with instantaneous velocity v driven by the electric field. Use Maxwell s equations to show that these waves satisfy the equation $ 1 '! 2 p = & ) 4"n ee 2 % 4"# 0 ( m Ignore any electron damping. $! 2 " 1 # 2 ' & % c 2 #t 2 ( ) E = * p 2 E, where c 2 is the square of the plasma frequency.

16 Problem 9: Consider a plane wave of amplitude E 0, normally incident on a dielectric with permittivity ε. Use the boundary conditions on E and B at the interface to show the amplitude of the transmitted wave is E r = 1! " /" " /" 0 E 0. Problem 10: A +q is set in circular orbit above a charge q as shown with angular velocity ω. ω -q ρ +q r What is instantaneous the rate at which the charge loses energy by electromagnetic radiation?

17 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Fall, 2007 Instructions: The exam consists of 10 problems, 10 points each. Partial credit will be given if merited. Personal notes on two sides of an 8 11 page are allowed. Total time is 3 hours. 1. An ideal electric dipole of moment p = pẑ is situated at the origin. What is the force, caused by the dipole, on each of two separate point charges, of amount q. The first is located at a distance a from the origin along the ˆx-axis, i.e., so that the charge has the Cartesian coordinates (a, 0, 0), and the other is also at a distance a from the origin, but along the ẑ-axis, i.e. so that the charge has the Cartesian coordinates (0, 0, a)? 2. Please find the capacitance per unit length of two coaxial, hollow, metal, cylindrical tubes, of radius a and b > a. 3. A hollow sphere carries charge density ρ = c/r 2 in the region a r b. Find the electric field in each of the three regions: within the hollow of the sphere, i.e., for r a; within the interior of the sphere, i.e., for a r b; and exterior to the sphere, i.e., for b r. Provide the result in terms of the total charge, q, of the shell. Provide a plot of the magnitude of the electric field as a function of the distance r from the center of the system.

18 4. A uniformly charged shell of surface charge density σ and radius a is rotating at a constant angular velocity ω, and we take the ẑ-axis along ω. At an arbitrary location, r, it has a magnetic vector potential given by A(r, θ, ϕ) = 1 3 µ 0Rσω r sin θ ˆϕ, r R, i.e., inside the shell, 1 3 µ 0R 4 σω sin θ r 2 ˆϕ, r R, i.e, outside the shell. Show that the magnetic field inside the rotating shell is uniform, and along the ẑ-direction. Also determine the magnetic field outside the shell. Can you describe that field in simple language? Is the field continuous at the boundary of the shell? Explain physically your answer. Note that for a vector of the form A = A ˆϕ, one has the following relation for its curl: (A ˆϕ) = ˆr (A sin θ) ˆθ (Ar) r sin θ θ r r. 5. A very long solenoid carries a current I. Coaxial with the solenoid is a large, circular ring of wire, with resistance R. When the current in the solenoid is gradually decreased, a current is induced in the ring. Take the solenoid to have n turns per unit length, and radius a, while the ring has radius b >> a. What is the current in the ring, as a function of di/dt? 2

19 6. A previously-charged capacitor, of amount C and charge separation Q, is in a simple open circuit along with a resistor, R, and an inductor, L. At time t = 0, a switch is closed so that this circuit now constitutes a single, closed, series circuit. What is the time dependence of the current through the resistor? 3

20 7. Consider a monochromatic wave moving through vacuum, of frequency ω, and with an electric field that is the sum of two separate parts, which are presented here in their complex forms: E = E1 + E2, E 1 = E 0 ẑ e i(kx ωt), E 2 = E 0 ẑ e i(kx+ωt), where E 0 is real. Determine the associated, real-valued magnetic field, and the time-averaged Poynting vector for the entire wave system. Please explain the meaning of your result for the Poynting vector. 8. Consider a circularly-polarized electromagnetic plane wave, propagating in vacuum with frequency ω. Write down the complex form for the electric field, and then, before you perform any averages over cycles, determine the (real-valued) intensity for the wave, making comments about the time dependence of the result. 9. At a certain time, which we take to be t = 0, we turn on the current, everywhere at once, in a particular infinitely-long wire, so that the current in this wire may be expressed in the following way: { 0, t < 0, I(t) = I 0, t 0. (We take the wire to lie along the ẑ-axis.) At any later time, t > 0, and at any particular measurement location, say a distance s directly away from the wire, only some finite portion of the wire can have communicated to this observation point the information that there is now a current running in that portion of the wire. For such a given positive time, t, and distance s, what total length of wire can have communicated this information? 10. An electromagnetic plane wave of frequency ω is traveling in the ˆx-direction through the vacuum. It has amplitude E 0, is polarized in the ŷ-direction, and has (time-averaged) intensity I 0. The observer, S, who made these statements is at rest. However, she sees another observer, S, coming past her, moving in the same direction as the plane wave, at half the speed of light. What are the frequency and intensity of the wave as seen by this other observer, S? 4

21 EM Prelim August 22, 2008 p.1 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Winter 2008 Instructions: the exam consists of 10 problems, 10 points each; partial credit will be given if merited; personal notes on two sides of 8 11 page are allowed; total time is 3 hours. 1. Two charges +q, q are separated by a distance d. Define coordinates as in the figure. Find the potential V (x, y) at any point in the x, y plane. 2. A point charge q is held at a distance h above an infinite conducting sheet. Determine the surface charge density on the sheet. 3. Consider two capacitors of identical construction except one is filled with dielectric having K = 2 (capacitor a) and the second is filled with air K 1 (capacitor b). Initially, switch one is closed and two is open so that capacitor a has an initial charge q 0. Subsequently, switch one is open and switch two is closed, so that the capacitors are connected in parallel What are the charges on q a, q b on the capacitors when connected in parallel?

22 EM Prelim August 22, 2008 p.2 4. An electron (mass m, charge e) moves between two parallel plates, where the plates have a potential difference V and separation d. Between the plates there is also magnetic field B (into the page as shown in the figure). The electron starts at rest and follows the trajectory indicated in the sketch after almost reaching the distance d in y, the particle will continue to move parallel to the upper plate with constant velocity in the x direction. Find the magnitude of the magnetic field B. 5. Consider the circuit in the figure with values V 0 = 12V, R 1 = 200kΩ,R 2 = 300kΩ, and C = 2µF. The switch is open after having been closed for a very long time. a) What is the voltage on the the capacitor just before the switch is opened? b) After how much time (in seconds) does the voltmeter read 3V? 6. Consider a square loop of wire (with side length d) oriented parallel to a long straight wire carrying current I. The loop is pulled with constant speed v in a direction perpendicular to the wire. a) What is the induced EMF in the loop? b)what is the direction of the induced current? Indicate on the figure with an arrow.

23 EM Prelim August 22, 2008 p.3 7. A plane EM wave of angular frequency ω propagates through a material with index of refraction n 1. The wave is normally incident upon the surface interface with another material with index of refraction n 2. What fraction of the incident wave energy is reflected from the surface? 8. A straight metal wire of conductivity σ and radius a carries a steady current I. a. Determine the Poynting vector as a function of the distance from the center of the wire. b. Integrate the normal component of the Poynting vector over the surface of the wire and compare to the power loss per unit length due to the resistance of the wire. 9. Write Maxwell s equations for a plane EM wave propagating in a neutral, conducting medium having permeability µ, permittivity ɛ and conductivity σ. Show that the wave equation for the electric field is given by 2 E 2 = µɛ E t + µσ E. 2 t 10. a) Find the dispersion relation (equation relating the wave number k to the angular frequency ω) for a plane wave of angular frequency ω propagating in a neutral, conducting medium (see previous problem). Note that in this case k is a complex number. b) For σ >> ω, find the skin depth in the medium.

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49 Department of Physics and Astronomy, University of New Mexico E&M Preliminary Examination Fall 2012 Instructions: The exam consists of 10 problems (10 pts each). Partial credit will be given if merited. Personal notes on the two sides of an 8.5 x 11 sheet are allowed. Total time: 3 hours. Possibly Useful Formulas Relation of spherical polar coordinates, (r, θ, φ), to Cartesian coordinates: Unit vectors: x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. ˆr = sin θ cos φ ˆx + sin θ sin φ ŷ + cos θ ẑ; ˆφ = sin φ ˆx + cos φ ŷ; ˆθ = ˆφ ˆr. Laplacian in spherical polar coordinates: 2 = 1 ( r 2 r 2 ) + r r 1 r 2 sin θ ( sin θ ) θ θ r 2 sin 2 θ 2 φ 2. Biot-Savart Law for the magnetic field at position r due to a steady current element I dl located at position r : B( r) = µ 0 Idl ( r r ) 4π r r 3. 1

50 Time-averaged power radiated by an oscillating electric dipole: P = µ 0 p 2 ω 4. 12πc Time-averaged power radiated by an oscillating magnetic dipole: P = µ 0 m 2 ω 4 12πc 3. Instantaneous power radiated by a non-relativistically moving charge with acceleration a (Larmor formula): P = q2 a 2 6πɛ 0 c 3. Fresnel formulas for the amplitude reflection coefficient of a plane wave incident at a planar interface between two dielectrics: r = n cos θ n cos θ n cos θ + n cos θ ; r = n cos θ n cos θ n cos θ + n cos θ, where, refer, respectively, to polarizations perpendicular and parallel to the plane of incidence. The angles of incidence and refraction are θ and θ, and n, n are the refractive indices of the medium of incidence and the medium of transmission, respectively. 2

51 1. In the Cartesian coordinate system (x, y, z), the electrostatic potential has the form V = a z, where a is a constant. Does the potential obey the Laplace equation? Derive the specific charge distribution that produces such a potential. 2. A uniformly charged ring of radius R, charge Q, and centered at the origin in the xy plane rotates uniformly at an angular velocity ω about its axis. Determine the electric and magnetic fields at the center of the ring. 3. A point charge q of mass m is released from rest a distance d from an infinitely extended, uniformly charged plane of surface charge density σ. Take q and σ to have the same sign. Either using the work-energy theorem or otherwise, write down an expression for the acceleration of the charge as a function of its speed without making any non-relativistic approximations. By integrating this expression, obtain the speed of the charge as a function of time. After how long will the charge achieve a speed equal to 0.8c? Neglect any radiation from the accelerating charge for this problem. Hint: You may find useful the indefinite-integral identity, dx (1 x 2 ) = x 3/2 (1 x 2 ). 1/2 4. Consider a charge q of mass m orbiting on a circle under the action of a uniform static magnetic field B normal to the plane of the circular orbit. Using the radiative power loss formula for circular orbits, P = q2 a 2 γ 4 6πɛ 0 c 3, where a is the acceleration and γ the relativistic Lorentz factor of the orbiting charge, show that the charge loses energy at a rate proportional to γ 2 as its speed approaches c. 5. A solid sphere of radius a is uniformly polarized with a permanent polarization (density) P = ẑp along the z axis. Take the sphere to be centered at the origin. What are the bound volume and surface charge densities in the sphere as a function of the position coordinates (r, θ, φ) inside and on the sphere? Show that the potentials V < (r, θ) = P 3ɛ 0 r cos θ, 3 V > (r, θ) = P a3 3ɛ 0 r 2 cos θ

52 correctly solve the electrostatic problem inside and outside the sphere, respectively, i.e., they both solve the Laplace equation and satisfy the two required boundary conditions at the spherical surface. Using V <, calculate the value of the electric field everywhere inside the sphere. 6. A uniform, infinitely extended current plane of surface current density K = K ˆx is located in the xy plane. A small circular loop of radius a carrying current I and located above the plane is free to rotate about a diameter that is held fixed and parallel to the x axis. z x y K What is the equilibrium orientation of the loop relative to the current plane? At what frequency will the loop perform small oscillations about the equilibrium orientation if it is rotated slightly away from that orientation? Express your answer in terms of µ 0, I, a, and I m, the moment of inertia of the loop about a diameter. 7. A plane electromagnetic (EM) wave is incident on a large planar metallic foil of area A at angle θ from the normal. The foil is slightly blackened so only a fraction R of the EM energy is reflected and the rest is absorbed. 4

53 θ A What are the magnitude and direction of the radiation force generated by the wave in terms of its (time-averaged) intensity I, R, A, and θ? 8. A long skinny bar magnet of magnetization M parallel to its axis approaches a highly permeable material with a plane surface. Take the magnetization of the magnet to be perpendicular to the material surface. With what force will the magnet attach to the surface of the material, if the cross-sectional area of the magnet is A? Make any approximations that are reasonable to arrive at your answers. You may find useful the facts that the magnetic field inside a solenoid of n turns per unit length and current I is µ 0 ni parallel to the axis of the solenoid and that the force on a uniformly magnetized bar may be written, in perfect analogy with electrostatics, as the effective magnetic 5

54 charge of amount AM times the external magnetic field B ext to which the bar is exposed. 9. An unpolarized monochromatic electromagnetic plane wave of angular frequency ω is incident from vacuum on the plane surface of an ideal plasma. The refractive index of the plasma may be exressed as n(ω) = 1 ω2 P ω 2, where ω P is the plasma frequency which we assume to be a constant and smaller than ω. (a) What fraction of the power of the plane wave would be reflected back at normal incidence? (b) For what range of values of the angle of incidence would the plane wave be fully reflected? (c) For what angle of incidence would the reflected wave be perfectly plane polarized? 10. Consider a monochromatic TEM mode of a planar metallic waveguide of plate separation w that is filled with a dieletric material of refractive index n. If the rms value of the electric field of the mode is E 0 and its angular frequency is ω, then write down complete expressions for the electric and magnetic fields, including their magnitudes and directions, inside the guide. What is the speed of propagation of the TEM mode? Calculate the time-averaged Poynting vector and energy density of the guided mode. 6

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62 1 Department of Physics and Astronomy, University of New Mexico Electricity and Magnetism Preliminary Examination Fall 2014 Instructions: You should attempt all 10 problems (10 points each). Partial credit will be given if merited. NO cheat sheets are allowed. Total time: 3 hours.

63 2 Possibly Useful Formulas Divergence of a vector A = A ρ ˆρ + A φ ˆφ + Az ẑ in cylindrical coordinates: Laplacian in cylindrical coordinates: Vector identity: A = 1 ρ ρ (ρa ρ) + 1 ρ 2 = 1 ρ ρ A φ φ + A z z. ( ρ ) + 1ρ 2 ρ 2 φ z 2. ( A) = (. A) 2 A. Maxwell s equations:. E = ρ ɛ 0,. B = 0, E = B t, ( B = µ 0 J + ɛ E ). t Biot-Savart law for the magnetic field at position r due to a steady current element Id l at position r : B(r ) = µ 0 Id l ( r r ) 4π r r 3. Time-averaged power radiated by an oscillating electric dipole: P = µ 0 p 2 ω 4 12πc.

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82 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Spring 2006 Instructions: The exam consists of 10 problems, 10 points each; Partial credit will be given if merited; Personal notes on two sides of an 8 11 page are allowed; Total time is 3 hours. Problem 1. Consider a ring of radius R with a charge q uniformly distributed on the circumference. R z Show that the electric field a distance z from the origin on the axis of the ring is, E(z) = q z 4!" 0 ( z 2 + R 2 ) z ˆ. 3 / 2 Show that the limit z>>r (keeping first nonvashing term) is what you expect. Problem 2. Now let the ring rotate at angular speed ω. What is its magnetic moment? ω Problem 3. A dielectric medium of permittivity ε fills half of a parallel plate capacitor. Each plate has area A. The distance between the plates is d. What is its capacitance? (Ignore fringing fields) ε d Problem 4. A model of an atom consists of a point charge e is imbedded inside a uniform cloud of charge -e distributed throughout a spherical volume of radius a. -e e e b -e E An external electric field is applied which polarizes the atom, displacing the cloud (without distorting it) a distance b from equilibrium. Calculate b.

83 Problem 5. A square ring of iron, with very large magnetic permeability µ, is wrapped with a coil of wire with n 1 turns per unit length on one side and another coil with n 2 turns per unit length on the opposite side. What is the mutual inductance between the coils? µ n 1 n 2 Problem 6. Consider a circuit with resistor and capacitor, driven by an ac-voltage source V 0 cos(ωt). In steady state, show the voltage across the resistor as a function of time is: R Comment on the limits! << 1/RC,! >> 1/RC. Problem 7. The electric field associated with a plane wave traveling in a nonmagnetic dielectric medium is given by, E(r,t) = (10Volts/cm) cos(2cm!1 z! 30ns!1 t) x ˆ + cos(2cm!1 z! 30ns!1 t) y ˆ ( ) (i) To what part of the electromagnetic spectrum does this wave correspond? (ii) In what direction is the wave propagating? (iii) What is the polarization of the field (linear, circular, elliptical)? (iv) What is the dielectric constant of the medium? (v) What is the intensity of the radiation? C Problem 8. A electromagnetic wave travels through a neutral plasma generating a current density J. Use Maxwell s equations to show that the electric field satisfies the wave equation, $! 2 " 1 # 2 ' #J & % c 2 #t 2 ) E = µ. 0 ( #t Problem 9. According to classical physics, an electron orbiting the nucleus will decay due to electromagnetic radiation. For a hydrogen atom, modeled as an electron in a circular orbit initially with radius a 0 (Bohr radius), what is the instantaneous rate of decay of energy at time t=0? Problem 10. An infinite line charge, with charge λ per unit length in the lab frame moves at a speed β=v/c=.8, where c is the speed of light. Using relativistic length contraction, find the electric and magnetic fields (magnitude and direction) in the lab frame and rest frame of the rod. Show that these satisfy the Lorentz transformation of the electromagnetic field. E! = E, E! " = # E " + r $ % cb B! = B, c! # R V R (t) = Re V 0 e!i"t % $ R + i /"C& ( ) ( $ % E) B " = # cb " & r ( ) 2 cos"t! ( ) 2 # "RC = V 0 ' $ 1 + "RC where and denote the vector components parallel and perpendicular to the direction of relative motion between reference frames, r! = v c, and! = 1/ 1" # 2. "RC ( ) 2 sin"t 1 + "RC % ( &

84 Instructions:?????? Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Spring, 2007 Problem 1: A thin ring, of radius a, has a non-uniform, linear charge density on it, of amount λ = λ 0 sin φ. Please find the net charge of the ring, and its dipole moment. Along the direction normal to the ring, which we call z, at very large distances the electrostatic potential is approximately proportional to 1/z n. What is the value of n? Problem 2: Find the capacitance of a pair of concentric, spherical metal shells, which have radii a and b, with b > a. Problem 3: A particular parallel plate capacitor has an area A and a distance h separating the plates. It has been charged so that one plate has a charge Q, and the other the negative of that, but has recently been disconnected from the charging battery. Half of the area is filled, over the entire separation distance, with a dielectric material with dielectric constant ɛ. You now attempt to pull this material out from between the two plates. What is the minimum force you will have to exert in order to do this? Problem 4: An originally-uncharged, metal sphere of radius a is placed in an otherwise uniform electric field, E = E 0 ẑ, which induces a charge distribution on the sphere. As a result of this the total electrostatic potential exterior to the sphere may be written as ( ) V (r, θ) = E 0 r a3 r 2 cos θ. What is the induced charge distribution? What is the potential s dependence on r interior to the sphere? Problem 5: A rectangular loop of wire hangs vertically, and supports a mass m that hangs downward from it, under the influence of gravity. The upper end of the loop finds itself in a region where there is a uniform magnetic field B, which points into the page. For what current, I, in the loop, would the mass be suspended in mid-air? What direction must that current have? Problem 6: What current density, J, would produce the magnetic vector potential, A = k ˆφ, in cylindrical coordinates, where k is a constant? Problem 7: A square loop of wire, of side length a, lies on a table, a distance s away from a very long, straight wire which carries a current, I = I 0 sin ωt. The square loop has a total resistance R. What current flows in it?

85 Problem 8: Using Maxwell s Equations with sources, show that the total charge inside any fixed, finite volume is conserved, i.e., constant in time. Problem 9: The magnetic field associated with a plane wave travelling in a nonmagnetic, dielectric medium is given by B( r, t) = B 0 {cos[ω(2x/c t)]ẑ + sin[ω(2x/c t)]ŷ}. a. What is the direction of propagation of the wave? b. What is the direction of the polarization of the wave, and what is its nature? c. What is the dielectric constant of the material through which it is travelling? d. What is the intensity of the radiation? Problem 10: A particular observer, O, measures fields in a small region in space, where he finds approximately uniform electric and magnetic fields as follows, where A is a constant: E = Aˆx, B = 3Aŷ. a. Another observer, O, passes by at a velocity v = αẑ. What must the value of α be in order that this observer measures no electric field? b. Could there be yet another observer, O, moving at a different velocity such that she would see no magnetic field? If so, what would her velocity be? 2

86 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Spring 2008 Instructions: The exam consists of 10 problems, 10 points each; Partial credit will be given if merited; Personal notes on two sides of an 8 11 page are allowed; Total time is 3 hours. Problem 1. Consider two point charges +q and -2q located on the y-axis as shown below. -2q y +q x Sketch the electric field lines and equipotential lines in the x-y plane. Problem 2. The positive terminal of a battery (ground taken at infinity) is attached to a perfectly conducting sphere of radius R, bringing it to potential V. How much work does the battery do in bringing the sphere to the same potential? +V R Problem 3. Two spherical cavities, of radii a and b, are hollowed out from the interior of a solid (neutral) conducting sphere of radius R. At the center of each cavity a point charge is placed: q a and q b. q a a R q b b (Next page)

87 (a) Find the surface charge densities at radii a, b, and R: σ a, σ b, and σ R. (b) What is the electric field outside the conductor? (c) What is the force on q a and q b? (d) Which of these answers will change if a third charge, q c, is brought near the conductor? Problem 4. A parallel plate capacitor is filled with two equal thickness layers of linear dielectrics, of permittivity! 1 = 2! 0 and! 2 = 1.5! 0, respectively (ε 0 is the permittivity of free space). A surface charge density ±σ is placed on the two plates -σ +σ ε 1 ε 2 (a) What is the electric field inside each of the dielectrics (ignore fringing fields). (b) What is the surface charge density at the interface between the two dielectrics? Problem 5. A bar magnetic with magnetic dipole m is place on the axis, a distance d from an infinitely long wire carrying current I. With what force is the magnetic dipole attracted to the wire for the two orientations shown? I N r I r N (i) m perpendicular to I and r. (ii) m parallel to I, perpendicular to r. Problem 6. A charge/area σ is distributed on the surface of a very long cylinder of radius R. The cylinder is spun about its axis so that it s instantaneous angular velocity is ω(t). Find the electric field as a function of position and time. ω(t) R σ

88 Problem 7. Consider an RLC circuit with R /L << 1/ LC. A battery charges the capacitor to voltage V 0. At time t=0, the switch is open with the battery and closed with the resistor and inductor in series. Sketch the energy stored in the capacitor and inductor as a function of time, denoting any relevant time dependencies on your graph. V 0 C L R Problem 8. A monochromatic electromagnetic wave with complex amplitude, E(r,t) = E(r)e!i"t, travels through a neutral plasma generating a current density J(r,t) =!i" Ne2 m E(r)e!i"t, with e, m the electron s charge and mass, and N the electron density. Use Maxwell s equations to show that the electric field satisfies the following wave equation, #! 2 + " 2 & % $ c 2 ( E(r) = " 2 p ' c E(r), 2 where! p = Ne 2 /m" 0 is the plasma frequency. Problem 9. Based on the wave equation in a plasma given Problem 8, consider a plane wave with electric field E(r,t) = E 0 e i(kz!"t ) e x. (a) Derive the dispersion relation!(k). (b) What is the magnetic field associated with this wave? (c) What is the intensity of the wave? Problem 10. A charge q oscillating sinusoidally with frequency ω on line segment of length d, radiates electromagnetic radiation, observed very far away r>>d. d q θ r>>d (a) Under what condition is the dominant contribution electric-dipole radiation? (b) The dipole radiation per solid angle is not isotropic; it varies as sin 2!. Explain why. (c) What is the total electric dipole power radiated into all directions, time averaged over a cycle of oscillation.

89 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Spring 2009 Instructions: The exam consists of 10 problems, 10 points each; Partial credit will be given if merited; Personal notes on two sides of an 8 11 page are allowed; Total time is 3 hours. Problem 1. Consider two point charges +q and -q fixed on the y-axis, separated by a distance a. A charge Q is placed on the x-axis a distance r from the bisector. y +q a q r Q x What is the direction and magnitude of the force on Q in the limit a<<r (lowest nonvanishing term in a/r). Problem 2. A charge of magnitude Q is uniformly distributed throughout a sphere of radius R. What is the electric field everywhere (both inside and outside the sphere, magnitude and direction)? Q R Problem 3. A battery of voltage V is used to charge of a coaxial capacitor of length L, inner radius a, outer radius b. How much energy is stored in the capacitor once it is fully charged (ignore fringing fields) in terms of the parameters given (a, b, V)? V b a

90 Problem 4. A current density J flows uniformly along the y-direction in a slab (extending to infinity in the y and z directions) of thickness d along x. A cross section of the slab in the x-y plane is sketched below. What is the direction and magnitude of the magnetic field as function of x (do this for all x, positive and negative, inside the slab and out). y J x d Problem 5. A charge q moves in uniform electric field E = Eˆ x and uniform magnetic field B = Bˆ z. The charge starts at the origin at rest. Show that the velocity of the charge obeys the following equation of motion, d 2 v dt = q2 B 2 v + q2 2 m 2 m ( E B). 2 Sketch the trajectory of the charge for E = 0 and E 0. Problem 6. A charge/area σ is uniformly distributed and fixed on the surface of a very long cylinder of radius R and length L and zero mass. The cylinder is spun slowly about its axis from rest to a final angular velocity is ω. How much energy is stored in resulting magnetic field (ignore fringing fields)? Considering the fact that the cylinder is massless, where did this energy come from and how would you calculate the work done to create it? ω R σ

91 Problem 7. Consider an RLC circuit with R /L << 1/ LC. An ac-voltage drives the circuit V (t) = V 0 cos( ωt ). In steady state, find the time averaged power dissipated in the resistor. Sketch a plot of this power as a function of ω. Comment on its form. V(t) R C L Problem 8. A monochromatic electromagnetic wave travels in a material with dielectric permittivity ε 1 and magnetic permeability µ 1. It comes to an interface with a second material at normal incidence, with dielectric permittivity ε 2 and magnetic permeability µ 2. E inc ε 1, µ 1 E trans ε 2, µ 2 B inc k inc B trans k trans Use the boundary conditions dictated by Maxwell s equations to show that the ratio of the transmitted to incident electric field amplitude is, E trans E inc = 2Z 2 Z 1 + Z 2, where Z i = µ i /ε i is the wave impedance in the material. Problem 9. Consider a electromagnetic field traveling in a nonmagnetic, nonconducting dielectric, with bound charge described by polarization density (electric dipole per unit volume) P(r,t). Use Maxwell s equations to show that the electric field satisfies the wave equation with source, 2 2 P(r,t) 2 µ 0 ε 0 E(r,t) = µ t 2 0. t 2 Problem 10. Consider now monochromatic wave solutions to Problem 9. Take an Ansatz for the fields as plane waves, E(r,t) = E 0 e i(kz ωt ), P(r,t) = P 0 e i(kz ωt ). Suppose the medium nonlinear, so that P 0 = ε 0 χ (3) E 0 2 E 0. Show that the phase velocity of the wave is, v phase = index of refraction is n(i) = 1 + χ (3) E 0 2. c, where the intensity-dependent n(i)

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142 Preliminary Examination: Electricity and Magnetism Department of Physics and Astronomy University of New Mexico Winter 2016 Instructions: The exam consists of 10 problems, 10 points each. Partial credit will be given if merited. Total time is 3 hours. Useful formulas and relations: Relation of spherical polar coordinates, (r, θ, φ), to Cartesian coordinates: Laplacian in spherical polar coordinates: Maxwell s equations: x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. (1) 2 = 1 ( r 2 r 2 ) + r r 1 r 2 sin θ E = ρ ε 0, B = 0, ( sin θ ) + θ θ E = B t, ( B = µ 0 J + ε ) E 0. t 1 r 2 sin 2 θ 2 φ 2. (2) Biot-Savart Law for the magnetic field created by a steady current element Id l: db ( r) = µ 0 Id l r 4π r 3. Time-averaged power radiated by an oscillating electric dipole: P = µ 0 p 2 ω 4. 12πc Time-averaged power radiated by an oscillating magnetic dipole: P = µ 0 m 2 ω 4 12πc 3.

143 EM Prelim Version of December 20, /5 1. (10 points) Consider three point charges +q, +q, and 2q fixed on the y axis as shown in the figure. Another charge Q is placed on the x axis a distance r from the bisector y 2a +q -2q +q r Q x What is the direction and the magnitude of the force on Q in the limit a r (lowest nonvanishing term in a/r)? 2. (10 points) Consider three different charges with values: +2q, +q, and q placed in three different configurations (a) (b) (c) +2q +q -q -q +2q +q +q -q +2q Which of the three configurations has the minimum energy? 3. (10 points) Three conducting spheres with radius a, b, and c (a < b < c) are connected as follows: the inner and outer spheres are connected to the ground, while the middle one is connected to a potential source V. Find the electric potential in the two regions: a r b, and b r c. Also, calculate the charge on each of the spheres. c b a ground V ground ground

144 EM Prelim Version of December 20, /5 4. (10 points) A charge of magnitude Q is uniformly distributed throughout a sphere of radius R. Find an expression for the electric potential φ (r) everywhere (both inside and outside the sphere)? Sketch a graph showing φ (r) as a function of r. Q R 5. (10 points) Consider two capacitors of identical construction except one is filled with a dielectric having dielectric constant K = 2 (capacitor A) and the second is filled with air, K = 1 (capacitor B). Initially, the switch S is open and capacitor A has a charge q 0. At some time, S is closed, so that the capacitors are connected. What are the final charges q A and q B on the capacitors? S A +q 0 Κ=2 Κ=1 -q 0 B 6. (10 points) A uniformly charged ring of radius R and charge Q is rotated at a uniform angular velocity ω about its axis. Calculate the magnitude and direction of the magnetic field due to the rotating disk at a distance d from it along its axis. d ω

145 EM Prelim Version of December 20, /5 7. (10 points) A very long solenoid carries a current I (t) = I 0 (1 αt), where I 0 and α are constants. Coaxial with the solenoid is a large, circular ring of wire, with resistance R. As the current in the solenoid changes, a current is induced in the ring. Take the solenoid to have n turns per unit length, and radius a, while the ring has radius b a. What is the current in the ring? 2a 2b 8. (10 points) An electron (mass m, charge e) moves between the plates of a parallel plate capacitor. The voltage across the capacitor is V, and the plate separation is d (as shown in the figure). The electron starts from rest at the cathode and for a constant value of B it follows a certain periodic trajectory, almost reaching the anode before returning to the cathode. y x V battery B d (a) Verify that the electron velocity is determined by the following equations v x (t) = V [1 cos ebm ] db t v y (t) = V db sin eb m t (b) Find the trajectory x(t), y(t) and draw it.

146 EM Prelim Version of December 20, /5 9. (10 points) Consider a plane electromagnetic wave of amplitude E I propagating in vacuum that is normally incident on a nonmagnetic, nonconducting dielectric material with permittivity ε 1. Use the boundary conditions on the electric E and magnetic B fields at the interface to find the amplitude of the reflected and transmitted waves (denoted by E R and E T respectively). Find ε 1 such that the transmitted amplitude is half of the reflected one. ε 0, μ 0 ε 1, μ 0 E I E T k I k T B I B T E R B R k R 10. (10 points) The magnetic field associated with a plane wave traveling in a nonmagnetic dielectric medium is given by B ( r, t) = ( T ) [ cos ( 20µm 1 z 3fs 1 t ) ˆx sin ( 20µm 1 z 3fs 1 t ) ŷ ] (a) To what part of the electromagnetic spectrum does this wave correspond? (b) In what direction is the wave propagating? (c) What is the polarization of the field (linear, circular, elliptical)? (d) What is the dielectric constant of the medium? (e) What is the intensity of the radiation?