Let E=(-2jŷ) e˜³², H =- (2 jî +ŷ) e¯jz 377 (a) Find the time-averaged Poynting's vector <$(z,t)>. (b) Find the instantaneous Poynting's vector (z,t). (c) Show that the results from (a) and (b) are consistent.
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- A uniform straight cylindrical wire has length L and radius a. An electric potential difference V is maintained between the two ends of the wire such that a constant current I flows down it. (a) Find the magnitude and direction (according to the direction of current flow) of electric and magnetic field by the surface of the wire. (b) Compute the magnitude and direction of the Poynting vector by the surface of the wire. (c) Integrate the Poynting vector over the surface of the wire to find the total flow of energy/power. (d) What happens to the energy? Interpret your results.4. Derive the maxwell equations for U, H, G and A Table 6.2.1: Maxwell Relations Function U H A G Differential dU = TdS-pdV dH = TdS + Vdp dA=-pdV - SdT dG = Vdp - SdT Natural Variables S, V S, P V, T P, T Maxwell Relation (or)--( др as V (37), = (35), as (OP), - (OV), = V (3r), - - (35), = TA long solenoid of radius R and n turns per unit length carries an alternating current I =I0sin(wt). Evaluate the Poynting vector at an interior point r=R/2.
- What is the average magnitude of the Poynting vector 3.60 mi from a radio transmitter broadcasting isotropically (equally in all directions) with an average power of 241 kW?Explain the answer to the second and last equation in detailUse the commutator results, [â, ô] = iħ, [x²,p] = 2iħâ, and [ÂÂ, Ĉ] = Â[Â, Ĉ] + [‚ Ĉ]B, to find the commutators given below. (a) [x, p²] (b) [x³, p] (c) [x², p²]
- A certain guided wave travels through air with the following magnetic field H=10+ y cos(157x) sin[(67×10° rad/s)t - Bz] [A/m] (0.3) Find the associated and B. There are a couple ways to solve this problem. Note: this is NOT a plane wave, and be sure to understand why. Nevertheless it IS a valid solution of Maxwell's equations. You will see how guided waves like this come aboutProblem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the time-harmonic electric field is given by E = u(p, ø) î, where u satisfies the Helmholtz wave equation 1 ди 1 02u + k?u = 0, 0 < 0< 27 0