Physics, 20.04.2020 19:56 bryan519688

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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Answer from: nathand200127

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 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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A plane monochromatic radio wave (? = 0.3 m) travels in vacuum along the positive x-axis, with a tim...

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