241_hw10 - Physics 241 - Electromagnetic Waves 1: Determine...

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Unformatted text preview: Physics 241 - Electromagnetic Waves 1: Determine and explain the direction of the induced magnetic field in each region of space. You may assume that each region has cylindrical symmetry so that E will be CW or CCW around the electric field direction. a: The electric field increases with time. b: The electric field decreases with time. 0 O O O —)> —é~ O O O 0 —fi 0 O O O a O O C C —> 2a: The displacement current also guarantees that Maxwell’s Equations conserve charge. Imagine that you have a solid conducting sphere of radius R which is initially neutral. You place a large quantity of positive charge at the center of the sphere. This produces a radially outward current which varies with time. Eventually, all the charge will be on the outside of the sphere and the current will be zero. By symmetry there can be NO magnetic field anywhere in space. Imagine that you have a closed Amperian loop of area A at a constant radius r < R (so the loop is part of a spherical surface). That loop definitely encloses some current. Hence, the first term on the right hand side of Ampere’s Law (below) is nonzero. Since we know that B = 0 by symmetry, the second term on the right hand side of Ampere’s Law must be equal and opposite the first term. d<I>E dt jgé‘ffs=#ol+flofo Now charge conservation requires that dQ(T, t) dt = —j[r,t)(47rrz) In this expression, Q(r, t) is the charge INSIDE a sphere of radius 7‘ at time t and j('r,t) is the current density at radius T at time t. Hence, the right side is the amount of current moving away from the spherical surface at radius r at time t and that is equivalent to the rate at which charge is carried away from the spherical volume of radius 7‘. The minus sign must be there since the charge contained within radius 7" decreases with an outward current [so ‘31—? should be negative). Use this charge conservation equation and Ampere’s Law. Verify that the two terms on the right hand side of Ampere’s Law do cancel each other in this case. b: N ow make your Amperian loop on a spherical surface with r > R. Verify that Ampere’s Law is consistent with B = 0. 3a: A thick wire of radius R carries a current Ht) = at where a is a constant. There is a gap in the Wire so the two faces of the wire next to the gap effectively represent capacitor 1 plates. Determine the charge on each plate as a function of time. Assume that each plate is uncharged at t z 0. b: Determine the electric field between the plates as a function of time. Ignore any fringing of the electric field lines. c: Determine the magnetic field in the WIRE as a function of time for p < R. Assume that the current density in the wire is constant. Then determine the magnetic field between the plates as a function of time for p < R. Then compare the results. _1, _1, 2R 4: Verify that E(a:, t) = Emma sinUcm — wt) satisfies the wave equation We i 62E 3'52 = [14060 8152 n:_ _ 1 when k —c— #060. 5a: The magnetic field of an electromagnetic wave is described by _. B(z, t) : IO’SSinUOsz — wt)(—§:)(T) Determine the electric field completely. b: What is the distance from one wave peak to another wave peak? c: Determine the Poynting vector. d: Determine the spatial gradient of the magnetic field (63—13) at z = 10’4m and t = 10—133. e: Determine the maximum time rate of change of the electric field (which would be the same at any point in space). f: A proton with velocity 27 = 1073“; + 2 X 107?] + 3 X 1072 (In/s) is at a point in space where E and E are both at maximum strength and B" points in the i direction. Determine the magnitude and direction of the net force on the proton. 6a: A parallel plate capacitor (see diagram on next page) has a plate radius R and a plate separation d. Show that the Poynting vector points radially into the capacitor. Note that I? must be the induced magnetic field due to the displacement current. b: Show that the rate at which energy flows into the capacitor (which is the integral of g over the cylindrical surface of the capacitor) equals the rate at which energy is stored in the capacitor (which is %(uV) where u is the energy density and V is the volume of the capacitor): a —‘ . d 1 Ignore the fringing of the electric field lines. From this viewpoint, the energy “enters” the capacitor from the space around the wires and the plates rather than through the wires. 7: A source of electromagnetic radiation radiates equally in all directions. How does the magnitude of the electric field associated with the wave depend upon the distance from the source? Explain. 8a: Determine the maximum values of E and 1;" 5m from a P = 100W light bulb. b: Determine the radiation force exerted on a 0.01m2 object that is oriented perpendicular to the light from the bulb. Assume that the object perfectly reflects the light. 9: Calculate the radiation force on the Earth due to sunlight. Assume that the Earth absorbs 70% of the light and reflects the other 30%. Compare this force with the force of gravity between the Earth and the Sun. Use Emu = 1.5 x 1011777., REarm = 6.37 x 106m, MEN = 5.98 x 1024kg, MSW = 1.99 X 103%9 and PM = 3.83 x 1026W. 10: A rocket (ll/{total = 5000199) is propelled by light reflected off a large solar sail of area A = 60007712. The rocket is 5 X 1011m from a massive bright star with an average power of 5 X 1030W'. The mass of the star is 3 X 1031kg. Assume that the sail is perpendicular to the radiation. Determine the rocket’s speed 1055 after it starts from rest. You may assume that the rocket will not move far enough for any of the forces acting on it to change significantly. 11: A 2000kg spacecraft is stationary in space. The astronaut turns on a 100kW laser to propel the ship. If the laser runs continuously for one year, how fast will the spacecraft be going? 12: A vertical electric dipole is used to generate an electromagnetic wave. The signal is supposed to be received by a cylindrical solenoid. How should the solenoid be oriented so that it can best receive the signal? Be completely specific. Explain. ...
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This note was uploaded on 01/19/2011 for the course PHYS 241 taught by Professor Milsom during the Fall '08 term at University of Arizona- Tucson.

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241_hw10 - Physics 241 - Electromagnetic Waves 1: Determine...

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