need help on the ones with red X's thank you ### Electromagnetic Wave PropertiesA plane electromagnetic wave varies sinusoidally at 81.4 MHz as it travels through a vacuum along the positive x-direction. The peak value of the electric field is 2.12 mV/m, and it is directed along the positive y-direction.#### (a) Calculating the Wavelength- **Wavelength**: 3.69 m ✔️#### (b) Finding the Period- **Period**: 12.3 ns ✔️#### (c) Maximum Magnetic Field- **Maximum Magnetic Field**: 7.07 pT ✔️#### (d) Expressions for Space and Time Variations- Electric Field: \[ E = (2.12 \times 10^{-3}) \cos 2\pi( \text{______} x - 81.4 \times 10^6 t) \] - **Direction**: +y ✔️- Magnetic Field: \[ B = (7.07 \times 10^{-12}) \cos 2\pi( \text{______} x - 81.4 \times 10^6 t) \] - **Direction**: +z ✔️*Note: Check if divisions or inverses were applied correctly.*#### (e) Average Power per Unit Area- **Average Power per Unit Area**: 5.97E-9 W/m² ✔️#### (f) Average Energy Density in Radiation- **Average Energy Density**: 2E-17 J/m³ ✔️#### (g) Radiation Pressure on a Reflective Surface- **Radiation Pressure**: Incorrect; rework the calculation carefully.*Ensure to revisit all steps if discrepancies appear in your solutions.*This educational content outlines steps and calculations related to plane electromagnetic waves, helping students understand concepts of wave behavior in a vacuum, field variations, and energy dynamics.

The electric field () and magnetic field () vary sinusoidally with space (x) and time (t). In SI…

A plane electromagnetic wave varies sinusoidally at 81.4 MHz as it travels through vacuum along the positive x direction. The peak value of the electric field is 2.12 mV/m, and it is directed along the positive y direction. (a) Find the wavelength. 3.69 ✔m (b) Find the period. 12.3 ✔ns (c) Find the maximum value of the magnetic field. 7.07 ✓ PT pT (d) Write expressions in SI units for the space and time variations of the electric field. (x is in meters, and t in seconds.) E = (2.12E-3 ✓) cos 2x( x magnitude Did you accidentally divide or take the inverse in your calculation?x - 81.4E6 direction +y Write expressions in SI units for the space and time variations of the magnetic field. (x is in meters, and t in seconds.) B = (7.07E-12✔ ) cos 2л( [ x Did you accidentally divide or take the inverse in your calculation?x - 81.4E6 ✔ t) magnitude direction +z (e) Find the average power per unit area this wave carries through space. 5.97E-9 W/m² t) (f) Find the average energy density in the radiation. 2E-17 J/m3 (g) What radiation pressure would this wave exert upon a perfectly reflecting surface at normal incidence? x Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. Pa

College Physics

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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need help on the ones with red X's thank you

### Electromagnetic Wave Properties

A plane electromagnetic wave varies sinusoidally at 81.4 MHz as it travels through a vacuum along the positive x-direction. The peak value of the electric field is 2.12 mV/m, and it is directed along the positive y-direction.

#### (a) Calculating the Wavelength
- **Wavelength**: 3.69 m ✔️

#### (b) Finding the Period
- **Period**: 12.3 ns ✔️

#### (c) Maximum Magnetic Field
- **Maximum Magnetic Field**: 7.07 pT ✔️

#### (d) Expressions for Space and Time Variations

- Electric Field:
\[
E = (2.12 \times 10^{-3}) \cos 2\pi( \text{______} x - 81.4 \times 10^6 t)
\]
- **Direction**: +y ✔️

- Magnetic Field:
\[
B = (7.07 \times 10^{-12}) \cos 2\pi( \text{______} x - 81.4 \times 10^6 t)
\]
- **Direction**: +z ✔️

*Note: Check if divisions or inverses were applied correctly.*

#### (e) Average Power per Unit Area
- **Average Power per Unit Area**: 5.97E-9 W/m² ✔️

#### (f) Average Energy Density in Radiation
- **Average Energy Density**: 2E-17 J/m³ ✔️

#### (g) Radiation Pressure on a Reflective Surface
- **Radiation Pressure**: Incorrect; rework the calculation carefully.

*Ensure to revisit all steps if discrepancies appear in your solutions.*

This educational content outlines steps and calculations related to plane electromagnetic waves, helping students understand concepts of wave behavior in a vacuum, field variations, and energy dynamics.

Expert Solution

Step 1: General expressions for Electric and Magnetic Fields

The electric field ( $E$ ) and magnetic field ( $B$ ) vary sinusoidally with space (x) and time (t). In SI units, they can be written as:

$E left parenthesis x comma t right parenthesis equals E subscript 0 sin left parenthesis k x minus omega t right parenthesis$

$B left parenthesis x comma t right parenthesis equals B subscript 0 sin left parenthesis k x minus omega t right parenthesis$

Where:
$E subscript 0$ is the maximum electric field, which is $2.12 cross times 10 to the power of negative 3 end exponent$ V/m in this case.

$B subscript 0 equals E subscript 0 over c$ is the maximum electric field, which is $7.07 cross times 10 to the power of negative 12 end exponent$ T in this case.
$k equals fraction numerator 2 pi over denominator lambda end fraction$ is the wave number, with $lambda$ being the wavelength.
$omega equals 2 pi f$ is the angular frequency, with $f$ being the frequency.
$c$ is the speed of light in a vacuum, $3 cross times 10 to the power of 8$ m/s.

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