Electricity & Magnetism Study Questions for the Spring 2018 Department Exam December 4, 2017

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1 Electricity & Magnetism Study Questions for the Spring 2018 Department Exam December 4, a. Find the capacitance of a spherical capacitor with inner radius l i and outer radius l 0 filled with dielectric of permittivity ε. b. Find the capacitance if the dielectric fills only the lower half of the capacitor. 2. A current I flows from (,0,0) to (-a,0,0) and following a quarter-circle in the x-z plane to (0,0,a), and then to (0,0, ). What is the force on a particle with charge q which is at (0,0,0) and moves,,0? with velocity x y 3. A rectangular coil of N turns and size a b is near a long straight wire carrying a steady current I 0. The coil has total resistance R, and is being pulled away from the wire at a constant speed v. a. Find the induced current in the coil as a function of the instantaneous coil-wire separation r. b. What instantaneous force F must be applied, also as a function of the coil-wire separation r? c. What total work input is required to remove the coil all the way to r? 4. a. Two spheres initially uncharged are connected by a battery of voltage V. After the switch is closed, what is the charge on the larger sphere? b. What is the capacitance of the larger sphere? 5. A linear molecule with a permanent electric dipole moment p o and moment of inertia I (for example, HCl) is placed in a uniform electric field E. a. Make a sketch showing the charges that make up the dipole, the dipole moment and the electric field when the dipole is in its equilibrium orientation. b. Derive an expression for the frequency o of small amplitude oscillations about the equilibrium orientation. c. Describe the polarization and power of the radiation produced by this oscillating dipole, assuming it has been initially displaced by angle 90 degrees from its equilibrium orientation. Make a sketch of the angular distribution of this power d. Over what time scale will it continue radiating appreciably, and therefore what range of frequencies are emitted.

2 6. A current I flows into a parallel plate capacitor with circular plates of radius R separated by d. The current was 0 before t = 0 and I 0 after. a. What is the charge on the plates as a function of time? b. What is the electric field between the plates? c. What is the displacement current density between the plates? d. What is the magnetic field between the plates at r = R/2 from the center of the plates. e. What is the Poynting vector between the plates at R/2? 7. A cylindrical capacitor consists of two long, concentric tubes of sheet metals of radii R 1 and R 2, respectively. The space between the tubes is filled with a dielectric of constant. a. Find the capacitance of this capacitor. b. Suppose the potential difference between the shells is V o, find the electrostatic energy. 8. A plane wave is propagating in a conductor which has a dielectric constant epsilon and conductivity sigma. The wave is plane polarized along the x-direction and is propagating along the z-direction. Starting with Maxwell's equations, calculate the electric and magnetic fields of the plane wave inside the conductor. In the case of a good conductor calculate the depth to which the plane wave can penetrate (skin depth). 9. A hollow dielectric sphere is centered on the origin. It has dielectric constant K 1. Its inner radius is a 1 and its outer radius is a 2. It is uncharged. Now a point charge q > 0 is placed at the origin. a. Find the electric field strength E for r < a 1, a 1 < r < a 2 and a 2 < r. b. Find the surface charge density (charge/area) on the inside (at r = a 1) and on the outside (at r = a 2). c. Now the situation is changed. A thin conducting coating is applied to the outside of the sphere, and this surface is maintained at +V 0 volts relative to infinity. Find E(r) for r > a A cylindrically shaped conductor has length L, radius a, and conductivity. It carries a uniform time-independent current density J Jeˆ z, parallel to its long axis. a. Determine the electric and magnetic fields within the conductor. b. Calculate the Poynting vector s within the conductor. c. Write out Poynting's theorem for this situation. Verify that it is satisfied for any point within the conductor.

3 11. An electromagnetic wave travelling in free space is incident normally on a perfect conductor. a. Show that the reflected component of the electric field is equal and opposite to the incident electric field. b. Show that on the surface of the conductor B total = 2B 0, where B 0 is the incident magnetic field. c. Calculate the surface current. d. Show that the electric field forms a standing wave outside the conductor. Locate the nodes and antinodes. 12. A plane wave is polarized in the +x direction and travelling in the +z direction. The wave is reflected from the xy plane at z = 0, where the index of refraction is n 1 for z < 0 and n 2 for z > 0. a. Write down the E and B fields for the incident, the reflected and the transmitted wave. b. Calculate the transmission and reflection coefficients. 13. A copper wire with a diameter of 2 mm has a resistance of 1.6 ohm for every 300 meters of length. It is carrying a current of 25 Amps. Assuming that this DC current density is uniformly distributed over the cross section, a. What is the numerical value of the resistivity of the copper from the information given? b. Determine the electric field vector E, the magnetic induction vector B, and the Poynting vector S at the surface of the wire. (Be sure to state their directions and actual numerical values. If you need it, o / , 1/ 4 o 9 10 in MKS units.) c. How much energy (in joules/meter) is stored in the electric field and in the magnetic field within the wire? d. Suppose the current is increasing at the rate of 1 Amp/s. What statement can be made about the radial dependence of the induced electric field that results within the wire? A thin non-magnetic, conducting disk of thickness h, radius a, and conductivity is placed in a region where the net, spatially uniform alternating magnetic field B zb ˆ o sin t is parallel to the z axis, as shown in the figure. a. Find the induced current density as a function of radial distance from the axis of the disk. b. What is the direction of this current at any instant in the first quarter of the period /(2 )? c. Find the total induced current at any instant in the above period. d. Calculate the average Joule heating i.e., the electromagnetic energy that is converted to heat.

4 15. Suppose that one measures the electrical conductivity of the following materials at room temperature: high purity copper n-type germanium niobium One then plunges each of these samples into helium (4K) and repeats the measurement. How much does the conductivity of each material change? (qualitative answer) In which direction? Why? 16. A circular loop of wire carrying current I is located with its center at the origin of coordinates and the normal to its plane has spherical angles θ 0 and φ 0. There is an external magnetic field with components B x = B 0 (1+ βy) and B y = B 0(1+βx). Compute the force acting on the loop. 17. A conducting sphere of radius a and total charge Q is surrounded by a spherical shell of dielectric material (with permittivity ) of inner radius a and outer radius b. Find the electrostatic energy of the system. 18. A center-fed antenna consists of two short segments, each of length d/2, each carrying a current in the same direction so that 2z i t I(z, t) I o 1 e d a. Use the continuity of charge equation to show that 2iI o i t (z) e d where is the linear charge density. b. Find expressions for the vector and scalar potentials, (Lorentz gauge) and show that only the vector potential is needed to find E and B far from the antenna. c. Find and E far B from the antenna. d. Find the total power radiated. 19. A loop of wire of radius R lies in the xy plane centered on the origin carrying a current I as drawn. a. Find the magnetic field at point P at (0, 0, z). b. Expand your result to two terms for z >> R. Identify the poles. 20. A magnetically hard material in the shape of a right circular cylinder of radius a and length L has magnetization M uniform throughout its volume and parallel to its axis. Find H Z(z) on the axis of the magnet by treating it as two sheets of magnetic charge located at z = +L/2 and z = -L/2.

5 21. You are unfortunately tied across the rails of a railroad track that are otherwise insulated from each other and from the ground. A train is rapidly approaching, making good electrical contact with the rails. The component of the earth's magnetic field normal to the surface is 2 x 10-5 Wb/m 2 at this location. Surprisingly, you are electrocuted before being hit. If you were electrocuted by a 1000 Volt potential, how fast was the train coming? 22. A particle charge of +q and mass m has a velocity of V = (0, v y, v z) when at a position of (x, 0, 0). There is a magnetic field of B = (0, 0, B). a. Find an expression for the cyclotron period, i.e. the time to go once around a circular orbit. b. Find the position of the charge after one cyclotron period. 23. The intensity of sunlight at the earth s surface is erg cm -2 s -1. a. Find the electric field of this radiation at the earth s surface in units of V m -1. b. Find the radiation pressure (in dyne cm -2 ) if the sunlight is fully reflected at normal incidence. c. Find the radiation pressure if the sunlight is absorbed without reflection. 24. A copper disk of radius 5 cm rotates at 20 revolutions per second, in a magnetic field B=0.5 T perpendicular to the disk. The rim and center are connected electrically by a fixed wire with sliding contacts. The total resistance is 10. Calculate the induced current. 25. A sphere of radius R carries charge Q distributed uniformly over its surface. The sphere is rotated at a constant angular velocity ω around the z-axis. Calculate the magnetic dipole moment. Find the magnetic field at the point (x,0,0) where x>>r. 26. A birefringent crystal has refractive indices n 1=1.42 and n 2=1.54. What thickness of this crystal would be needed to make a quarter wave plate for light of wavelength 6000Ǻ?

6 27. Most of the electromagnetic energy in the Universe is in the cosmic microwave background radiation, a remnant of the Big Bang. This radiation was discovered by A. Penzias and R. Wilson in 1965, by observations with a radio telescope. The radiation is electromagnetic waves with wavelengths around 1.1mm. The energy density is 4.0x10-14 J/m 3. (This is 2.5x10 5 ev/m 3, half the rest energy of an electron in each cubic meter of the Universe.) a. What is the RMS electric field strength of the cosmic microwave background radiation? b. How far from a 1000 W transmitter would you have to go to have the same field strength? Assume the power from the transmitter is isotropic. 28. An inverted hemispherical bowl of radius R has a uniform surface charge density. Suppose that the pole of the hemisphere lies at z=+r, and the center is at the origin of a standard coordinate system. a. Find the electrostatic potential at the pole of the hemisphere, r (0,0, R). b. Find the electric field strength at the pole. 29. In free space an electromagnetic wave has its electric field vector given by E(x, y,z, t) E xˆ cos(kz t) E ysin(kz ˆ t) o a. Describe this wave: polarization? direction of propagation? intensity? b. Write an equation for B in this electromagnetic wave. c. What is E 0 if the wavelength is 4000 Å and the intensity in the beam is 10 5 W/m 2? o 30. A point dipole sits at the origin of a spherical coordinate system. It points in the z-direction and has r,, d,90,0. It also has a dipole a dipole moment = m. A second dipole sits at a point moment = m. a. What is the vector potential due to the dipole located at the origin? b. What is the magnetic induction due to the dipole located at the origin? c. What is the value of B at the position of the second dipole? d. What is the magnitude of the magnetic dipole-dipole interaction energy for these two dipoles? e. What are the translational force and torque on the second dipole?

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Spring 2014 Final Exam Equation Sheet. B( r) = µ o 4π

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2014 Final Exam Equation Sheet Force Law: F q = q( E ext + v q B ext ) Poynting Vector: S = ( E B) / µ 0 Force on Current Carrying