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Online Homework 12 Solution Electric and Magnetic Field Vectors Conceptual Question Part A The electric and magnetic field vectors at a specific point in space and time are illustrated. Based on this information, in what direction does the electromagnetic wave propagate? Sol:) 由右手定則可判斷電磁波傳播方向為 + ẑ Part B The electric and magnetic field vectors at a specific point in space and r r time are illustrated. ( E and B are in the xy plane. Both vectors make 45° angles with the +y axis.) Based on this information, in what direction does the electromagnetic wave propagate? Sol:) 由右手定則可判斷電磁波傳播方向為 − ẑ Part C The magnetic field vector and the direction of propagation of an electromagnetic wave are illustrated. Based on this information, in what direction does the electric field vector point? Sol:) 已知磁場與電磁波傳播方向,由右手定則可判斷 電場方向為 + ŷ Part D The electric field vector and the direction of propagation of an r electromagnetic wave are illustrated. ( E is in xz plane and makes a 45° angle with the +x axis.) Based on this information, in what direction does the magnetic field vector point? Sol:) 已知電場與電磁波傳播方向,由右手定則可判斷 磁場方向為 at a -45° angle in the xy plane 1 Traveling Electromagnetic Wave Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves. Electromagnetic waves comprise combinations of electric and magnetic fields that are mutually compatible in the sense that the changes in one generate the other. The simplest form of a traveling electromagnetic wave is a plane wave. For a wave traveling in the x direction whose electric field is in the y direction, the electric and magnetic fields are given by r E = E0 sin (kx − ωt ) yˆ , r B = B0 sin (kx − ωt )zˆ . This wave is linearly polarized in the y direction. Part A In these formulas, it is useful to understand which variables are parameters that specify the nature of the wave. The variables E 0 and B0 are the __________ of the electric and magnetic fields. Choose the best answer to fill in the blank. Sol:) amplitudes Part B The variable ω is called the __________ of the wave. Choose the best answer to fill in the blank. Sol:) angular frequency Part C The variable k is called the __________ of the wave. Choose the best answer to fill in the blank. Sol:) wave number 2 Part D What is the mathematical expression for the electric field at the point x = 0, y = 0, z at time t? Sol:) r E = E 0 sin (− ωt ) yˆ Part E For a given wave, what are the physical variables to which the wave responds? Sol:) x and t Part F What is the wavelength λ of the wave described in the problem introduction? Express the wavelength in terms of the other given variables and constants like π. Sol:) sin (kx ) = sin (kx + kλ ) sin (θ ) = sin (θ + 2π ) 2π ⇒λ = k Part G What is the period T of the wave described in the problem introduction? Express the period of this wave in terms of ω and any constants. Sol:) 2π T= ω Part H What is the velocity v of the wave described in the problem introduction? Express the velocity in terms of quantities given in the introduction (such as ω and k) and any useful constants. Sol:) 2π ω v= = k = T 2π k λ ω 3 Poynting Flux and Power Dissipation in a Resistor Consider a cylindrical resistor of radius r0, length l, and resistance R with a steady current i flowing along the axis of the cylinder. Part A Which of the following is the most accurate qualitative description of r the magnetic field vector B inside the cylindrical resistor? Sol:). The magnetic field vector is everywhere tangential to circles centered on the axis of the cylinder. r Part B Find the magnitude of the magnetic field B(r ) inside the cylindrical resistor, where r is the distance from the axis of the cylinder, in terms of i, r, r0, l, and other given variables. You will also need π and µ 0 . Ignore fringing effects at the ends of the cylinder. Sol:). r r B ∫ ⋅ dl = µ 0 iencl B ⋅ (2πr ) = µ 0 iπr 2 πr02 r ir B(r ) = µ 0 2πr02 r Part C What can you say about the electric field vector E inside the resistor? Sol:). The electric field vector points along the axis of the resistor in the direction of the current. 4 r Part D What is the magnitude of the electric field vector E ? Sol:). V = iR r V iR E = = l l r Part E Sol:). In what direction does the Poynting vector S point? r 1 r r S= E×B µ0 In cylindrical coordinate rˆ × θˆ = kˆ kˆ × θˆ = −rˆ So the answer is ” − r̂ ”. r Part F Calculate S , the magnitude of the Poynting vector at the surface of the resistor (not at the circular ends of the cylinder). To answer this you need to take r = r0 . Sol:). r r From part B and part D we know E and B (r ) r 1 r r i2R S= E×B = µ0 2πlr0 5 Triangle Electromagnetic Wave In this problem we will investigate a triangle wave traveling in the x direction whose electric field is in the y direction. This wave is linearly polarized along the y axis; in other words, the electric field is always directed along the y axis. Its electric and magnetic fields are given by the following expressions: E y ( x, t ) = E 0 ( x − vt ) / a and BZ (x, t ) = B0 (x − vt ) / a , where E0 , B0 , and a are constants. The constant a, which has dimensions of length, is introduced so that the constants E0 and B0 have dimensions of electric and magnetic field respectively. This wave is pictured in the figure at time t = 0 . Note that we have only drawn the field vectors along the x axis. In fact, this idealized wave fills all space, but the field vectors only vary in the x direction. We expect this wave to satisfy Maxwell's equations. For it to do so, we will find that the following must be true: 1. The amplitude of the electric field must be directly proportional to the amplitude of the magnetic field. 2. The wave must travel at a particular velocity (namely, the speed of light). r Part A What is the propagation velocity v of the electromagnetic wave whose electric and magnetic fields are given by the expressions in the r introduction? Express v in terms of v and the unit vectors x̂ , ŷ , and ẑ . The answer will not involve c; we have not yet shown that this wave travels at the speed of light. Sol:) r v = vxˆ 6 In the next few parts, we will use Faraday's law of induction to find a relationship between E0 and B0 . Faraday's law relates the line integral of the electric field around a closed loop to the rate of change in magnetic flux through this loop: r r r ∫ E (r , t ) ⋅ dl C =− r d r r B (r , t ) ⋅ dA . ∫ dt S Part B To use Faraday's law for this problem, you will need to constuct a suitable loop, around which you will integrate the electric field. In which plane should the loop lie to get a nonzero electric field line integral and a nonzero magnetic flux? Sol:) The xy plane Part C Consider the loop C1 shown in the figure. It is a square loop with sides of length L, with one corner at the origin and the opposite corner at the coordinates x = L , y = L . Recall that r r E (r , t ) = [E0 ( x − vt ) / a ]yˆ . What is the value of the line integral of the electric field around loop C1 at arbitrary time t? Express the line integral in terms of E0 , L, a, v, and/or t. Sol:) Since Ex=0, the line integral alone segment 1 and 3 are zero. r r r Consider segment 2, dl = dyyˆ , E ⋅ dl = E y dy E 0 (L − vt ) E (L − vt ) L E (L − vt )L dy = 0 dy = 0 ∫ 0 0 0 a a a r r r Consider segment 3, dl = −dyyˆ , E ⋅ dl = − E y dy ∫ L E y ( x = L, t )dy = ∫ ∫ L − E y ( x = 0, t )dy = − ∫ 0 0+ L L 0 E 0 (− vt ) E vt L E vtL dy = 0 ∫ dy = 0 a a 0 a E 0 (L − vt )L E vtL E 0 L2 +0+ 0 = a a a 7 Part D r r Recall that B(r , t ) = [B0 (x − vt ) / a ]zˆ . Find the value of the magnetic flux through the surface S1 in the xy plane that is bounded by the loop C1 , at arbitrary time t. Express the magnetic flux in terms of B0 , L, a, v, and/or t. Sol:) r r r r dA = dxdyzˆ , B (r , t ) ⋅ dA = B z ( x, t )dxdy r r r ∫ B(r , t ) ⋅ dA = ∫ B (x, t )dxdy z S1 S1 L x − vt dx ∫ dy 0 0 a L2 L − vtL B0 L2 − vt 2 L= 2 = B0 a a = B0 ∫ L Part E Now use Faraday's law to establish a relationship between E0 and B0 . Sol:) By using Faraday’s law, we know that r r r ∫ E (r , t )⋅ dl C1 =− d dt r r r ∫ B(r , t )⋅ dA S1 r E 0 L2 r r 等號左邊可由 part C 得知 ∫ E (r , t ) ⋅ dl = C1 a r r r d d 等號右邊由 part D 可得 − ∫ B(r , t ) ⋅ dA = − dt S1 dt L B0 L2 − vt 2 2 = B0 L v a a B0 L2 v E0 L2 = ⇒ E0 = B0 v a a If the electric and magnetic fields given in the introduction are to be self-consistent, they must obey all of Maxwell's equations, including the Ampère-Maxwell law. In these last few parts (again, most of which are hidden) we will use the Ampère-Maxwell law to show that self-consistency requires the electromagnetic wave described in the introduction to propagate at the speed of light. 8 The Ampère-Maxwell law relates the line integral of the magnetic field around a closed loop to the rate of change in electric flux through this loop: r r r ∫ B(r , t ) ⋅ dl C = µ 0 I + µ 0ε 0 r d r r E (r , t ) ⋅ dA . ∫ dt S In this problem, the current I is zero. (For I to be nonzero, we would need charged particles moving around. In this problem, there are no charged particles present. We assume that the electromagnetic wave is propagating through a vacuum.) Part F To use the Ampère-Maxwell law you will once again need to construct a suitable loop, but this time you will integrate the magnetic field around the loop. In which plane should the loop lie to get a nonzero magnetic field line integral and hence nonzero electric flux? Sol:) The zx plane Part G Use the Ampère-Maxwell law to find a new relationship between E 0 and B0 . Express E 0 in terms of B0 , µ 0 , ε 0 , and other quantities given in the introduction. Sol:) r r r ∫ B(r , t ) ⋅ dl C2 = µ 0ε 0 r r r d E (r , t ) ⋅ dA ∫ dt S 2 如同 part C r r r 可得 ∫ B(r , t ) ⋅ dl = C2 B0 L2 a 如同 part D 可得 ∫ S2 r r r E (r , t ) ⋅ dA = L − E0 L2 − vt 2 a L − E0 L2 − vt 2 r r r d d 2 = µ 0ε 0 E0 L v µ 0ε 0 ∫ E (r , t ) ⋅ dA = µ 0ε 0 dt S2 dt a a 2 2 B0 L B0 µε E Lv = 0 0 0 ⇒ E0 = a a µ 0ε 0 v 9 Part H Finally we are ready to show that the electric and magnetic fields given in the introduction describe an electromagnetic wave propagating at the speed of light. If the electric and magnetic fields are to be self-consistent, they must obey all of Maxwell's equations. Using one of Maxwell's equations, Faraday's law, we found a certain relationship between E 0 and B0 . You derived this in Part E. Using another of Maxwell's equations, the Ampère-Maxwell law, we found what appears to be a different relationship between E 0 and B0 . You derived this in Part I. If the results of Parts E and I are to agree, what does this imply that the speed of propagation Express v in terms of only µ 0 and ε 0 . Sol:) Part G E0 = B0 µ 0ε 0 v Part E E 0 = B0 v v= 1 µ 0ε 0 10 must be? Electromagnetic Waves Ranking Task Part A Rank these electromagnetic waves on the basis of their speed (in vacuum). Rank from fastest to slowest. To rank items as equivalent, overlap them. Sol:) The relationship among wave speed, frequency, and wavelength is c = fλ v AM = v FM = vinf rared = v yellow = v green = v x − ray Part B Rank these electromagnetic waves on the basis of their wavelength. Rank from longest to shortest. To rank items as equivalent, overlap them. Sol:) λ AM > λ FM > λinf rared > λ yellow > λ green > λ x −ray Part C Rank these electromagnetic waves on the basis of their frequency. Rank from largest to smallest. To rank items as equivalent, overlap them. Sol:) f AM < f FM < f inf rared < f yellow < f green < f x − ray 11