A plane monochromatic radio wave (? = 0.3 m) travels in vacuum along the positive x-axis, with a time-averaged intensity I = 45.0 W/m2. Supp

Question

A plane monochromatic radio wave (? = 0.3 m) travels in vacuum along the positive x-axis, with a time-averaged intensity I = 45.0 W/m2. Suppose at time t = 0, the electric field at the origin is measured to be directed along the positive y-axis with a magnitude equal to its maximum value. What is Bz, the magnetic field at the origin, at time t = 1.5 ns? Bz = I got .04800 but that answer didnt work.

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Ben Gia 7 months 2021-07-15T19:57:58+00:00 1 Answers 50 views 0

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    2021-07-15T19:59:16+00:00

    Answer:

    The magnetic field B_Z = - 6.14*10^{-7} T

    Explanation:

    From the question we are told that

          The wavelength is \lambda = 0.3m

           The intensity is I = 45.0W/m^2

           The time is t = 1.5ns = 1.5 *10^{-9}s

    Generally radiation intensity is mathematically represented as

                  I = \frac{1}{2} c \epsilon_o E_o^2

    Where  c is the speed of light with a constant value of 3.0 *10^8 m/s

                  E_i is the electric field

                 \epsilon_o is the permittivity  of free space with a constant value of 8.85*10^{-12} C^2 /N \cdot m^2

     Making E_o the subject of the formula we have

               E_i = \sqrt{\frac{2I}{c \epsilon_0} }

          Substituting values

              E_i = \sqrt{\frac{2* 45 }{(3*10^8 * (8.85*10^{-12}) )} }

                   = 184.12 \ V/m

    Generally electric and magnetic field are related by the mathematical equation as follows

                 \frac{E_i}{B_i} = c

    Where B_O is the magnetic field

               making  B_O the subject

                     B_i = \frac{E_i}{c}

    Substituting values

                     B_i = \frac{184.12}{3*10^8}

                           = 6.14 *10^{-7}T

    Next is to obtain the wave number

      Generally  the wave number is mathematically represented as

                              n = \frac{2 \pi }{\lambda }

    Substituting values

                              n = \frac{2 \pi}{0.3}

                                  = 20.93 \ rad/m

    Next is to obtain the frequency

          Generally  the  frequency f is mathematically represented as

                        f = \frac{c}{\lambda}

    Substituting values

                       f = \frac{3 *10^8}{0.3}

                          = 1*10^{9} s^{-1}

    Next is to obtain the angular velocity

                    Generally  the  angular velocity  w is mathematically represented as

                           w = 2 \pi f

                               w = 2 \pi (1* 10^9)

                                  = 2 \pi * 10^9 rad/s

    Generally  the sinusoidal electromagnetic waves for the magnetic field B moving in the positive z direction is expressed as

                           B_z = B_i cos (nx -wt)

    Since the magnetic field is induced at the origin then the equation above is reduced to

                       B_z = B_i cos (n(0) -wt) =  B_i cos ( -wt)

    x =0 because it is the origin we are considering

     Substituting values  

                          B_z = (6.14*10^{-7}) cos (- (2 \pi * 10^{9})(1.5 *10^{-9}))

                               = - 6.14*10^{-7} T

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