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.

Answer 1

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  

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

                           $= - 6.14*10^{-7} T$