x.øi5ss="printViewProblemDescription">Learning Goal: To show how a propagating triangle electromagnetic wave can satisfy Maxwell's equations if thewave travels at speed c.Light,radiant heat (infrared radiation), X rays, and radio waves are allexamples of traveling electromagnetic waves. Electromagnetic wavesconsist of mutuallycompatible combinations of electric and magneticfields ("mutually compatible" in the sense that changes in the electricfield generate the magnetic field, and viceversa).Thesimplest form for a traveling electromagnetic wave is a plane wave. Oneparticularly simple form for a plane wave is known as a "trianglewave," in which theelectric and magnetic fields are linear in positionand time (rather than sinusoidal). In this problem we will investigatea triangle wave traveling in thex direction whose electric field is in the y direction. This wave is linearly polarized along the y axis; in other words, the electricfield is always directed along the y axis. Its electric and magnetic fields are given by the following expressions: and,where , , and are constants. The constant , which hasdimensions of length, is introduced so that the constants and have dimensions of electric and magnetic field respectively. Thiswave is pictured in the figure at time . Notethat we have only drawn the field vectorsalong the x axis. In fact,this idealized wave fills all space, but the field vectors only vary inthe 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:The amplitude of the electric field must be directly proportional to the amplitude of the magnetic field.The wave must travel at a particular velocity (namely, the speed of light).Part CConsider the loop shown in the figure. Itis a square loop with sides of length , with onecorner at the origin and the opposite corner at the coordinates , . Recall that . What is the value of the line integral of the electric field around loop at arbitrary time ?Express the line integral in terms of ,, , , and/or .ANSWER:=Part DRecall that . Find the value of the magnetic flux through the surface in the xy plane that is bounded by theloop , at arbitrary time .Express the magnetic flux in terms of ,, , , and/or .ANSWER:=Part EPart not displayedIf the electric and magnetic fields given in the introduction are to be self-consistent, they must obey allof Maxwell's equations, including theAmpère-Maxwell law. In these lastfew parts (again, most of which are hidden) we will use theAmpère-Maxwell law to show that self-consistency requirestheelectromagnetic wave described in the introduction to propagate at thespeed of light.The Ampère-Maxwell law relates the lineintegral of the magnetic field around a closed loop to the rate ofchange in electric flux through this loop:.In this problem, the current is zero. (For to be nonzero, we would need charged particles moving around. Inthisproblem, there are no charged particles present. We assume that theelectromagnetic wave is propagating through a vacuum.)Part HFinally we are ready to show that the electric and magnetic fields given in the introduction describe an electromagnetic wave propagating at thespeed of light. If the electric and magnetic fields are to be self-consistent, they must obey all of Maxwell's equations. Using one Å;x.øi5Maxwell's equations, Faraday's law, we found a certain relationship between