Question

Two travelling sinosoidal electromagnetic waves, each with an intensity 5.0 W/m2 , interfere to form a standing wave. The resulting electric field E⃗ (z,t) has nodes (i.e., is zero at all times) at z=…,−2a,−a,0,a,2a,… with a = 7.0 m , and satisifes E⃗ (z,0)=0. Furthermore, the magnetic field B⃗ (z,t) is observed to point along ±i^ everywhere. You may take c = 3.0×108 m/s and 8.9×10−12 F/m .

Part A What is the wavelength λ of the two constituent travelling waves? Express your answer in units of meters.

Part B When will the electric field be zero everywhere for the first time after t=0? Express your answer in units of ns

. Part C When will the magnetic field be zero everywhere for the first time after t=0? Express your answer in units of ns.

Part D What is the amplitude (maximum magnitude) of the electric field for the standing wave? Express your answer in units of V/m.

Part E How far from the origin is the nearest point where the amplitude of the (instantaneous) Poynting vector is maximum? Express your answer in units of meters.

Answer 1

Part (A)

Distance between successive node = Half of wavelength

Since distance between successive node = 7 m , wavelength = 14 m

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Part (B)

frequency f = c / , where c = 3 108 m / s is speed of light

frequency f = ( 3 108 ) / 14 = 2.143 107 Hz

Time period = 4.666 10-8 s = 46.7 ns

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Part(C)

Electric field will be zero every where after t=0 , when ( t ) =

where is angular frequency, that is given by = 2f = 2 /T

Hence , we have , ( 2 /T ) t = , we get t = T/2

Hence after t = 0 , electric field will be zero everywhere at t= 23.35 ns

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Part (C)

Since electric field and magnetic field variation is in phase, magnetic field also will be zero at t=23.35 ns

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Intensity of electromagnetic wave , E2 = 5 W/m2

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Poynting vector is given by

Electric field and magnetic field are maximum at mid point of successive nodes .

Since distance between sucessive nodes is 7 m , poynting vector will be maximum at 3.5 m from origin