3. In this question we consider an electromagnetic wave, propagating with the phase velocity, c, along the z-axis. The electric field component of this wave is shown directly below.

Figure 3: The electric field component of an electromagnetic wave, propagating along the z-axis. You may assume that the electric field is described by the equation, \vec{E}(z, t)=E_{o} \sin (k z-\omega t) i_{x} where it is the unit vector oriented along the x-axis. (a) Calculate the magnetic field component of the electromagnetic wave. You are 1=reminded of the formula B (ik × Ē), where it is the normalised vector pointing in the direction of propagation of the electromagnetic wave. [4 marks] (b) You are given the definition of the Poynting vector, S=(1/μ) (Ex B) for=a wave propagating through the vacuum. Use this, and the relationship c²1/(Moo), to show that for the electromagnetic wave discussed above: \vec{S}=\epsilon_{o} c E_{o}^{2} \sin (k z-\omega t) (c) Consider the point on the z-axis, z = 0, where the Poynting vector as a function of time is given by S(t) = €„cЕ² sin²(wt). Calculate the time-averaged Poynting vector: \langle\vec{S}(t)\rangle=\frac{1}{T} \int_{0}^{T} \vec{S}(t) d t