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A plane electromagnetic wave, with wavelength \(3.0 \mathrm{~m}\), travels in vacuum in the positive direction of an \(x\) axis. The electric field, of amplitude \(300 \mathrm{~V} / \mathrm{m},\) oscillates parallel to the \(y\) axis. What are the (a) frequency, (b) angular frequency, and (c) angular wave number of the wave? (d) What is the amplitude of the magnetic field component? (e) Parallel to which axis does the magnetic field oscillate? (f) What is the timeaveraged rate of energy flow in watts per square meter associated with this wave? The wave uniformly illuminates a surface of area \(2.0 \mathrm{~m}^{2} .\) If the surface totally absorbs the wave, what are \((\mathrm{g})\) the rate at which momentum is transferred to the surface and (h) the radiation pressure on the surface?

Step by step solution

01

Calculating Frequency

The frequency \( f \) of a wave is related to its wavelength \( \lambda \) and the speed of light \( c \) in a vacuum by the equation \( f = \frac{c}{\lambda} \). Given \( \lambda = 3.0 \; \text{m} \) and \( c = 3.0 \times 10^8 \; \text{m/s} \), we find: \[ f = \frac{3.0 \times 10^8 \; \text{m/s}}{3.0 \; \text{m}} = 1.0 \times 10^8 \; \text{Hz} \]

02

Calculating Angular Frequency

The angular frequency \( \omega \) is related to the frequency \( f \) by the formula \( \omega = 2\pi f \). Using \( f = 1.0 \times 10^8 \; \text{Hz} \), we get: \[ \omega = 2\pi \times 1.0 \times 10^8 = 2\pi \times 10^8 \; \text{rad/s} \]

03

Calculating Angular Wave Number

The angular wave number \( k \) is given by \( k = \frac{2\pi}{\lambda} \). With a wavelength \( \lambda = 3.0 \; \text{m} \), we have: \[ k = \frac{2\pi}{3.0} = \frac{2\pi}{3} \; \text{rad/m} \]

04

Amplitude of Magnetic Field

The amplitude of the magnetic field \( B_0 \) related to the electric field \( E_0 \) is given by \( B_0 = \frac{E_0}{c} \). With \( E_0 = 300 \; \text{V/m} \) and \( c = 3.0 \times 10^8 \; \text{m/s} \): \[ B_0 = \frac{300}{3.0 \times 10^8} = 1.0 \times 10^{-6} \; \text{T} \]

05

Direction of Magnetic Field Oscillation

In an electromagnetic wave, the electric and magnetic fields are perpendicular to each other and to the direction of wave propagation. Since the electric field oscillates parallel to the \( y \)-axis and the wave travels in the positive \( x \)-direction, the magnetic field oscillates parallel to the \( z \)-axis.

06

Calculating Time-Averaged Energy Flow

The time-averaged energy flow per unit area, or the intensity \( S \), is given by \( S = \frac{1}{2}c\varepsilon_0 E_0^2 \). Using \( \varepsilon_0 = 8.85 \times 10^{-12} \; \text{F/m} \) and \( E_0 = 300 \; \text{V/m} \): \[ S = \frac{1}{2} \times 3.0 \times 10^8 \times 8.85 \times 10^{-12} \times 300^2 \approx 119 \; \text{W/m}^2 \]

07

Rate of Momentum Transfer

The rate of momentum transfer per unit area is equal to the intensity \( S \), thus the total rate is \( S \times 2.0 \). \[ \text{Rate} = 119 \times 2.0 = 238 \; \text{W (or Nm/s)} \]

08

Calculating Radiation Pressure

The radiation pressure \( P \) for complete absorption is given by \( P = \frac{S}{c} \). Using \( S = 119 \; \text{W/m}^2 \) and \( c = 3.0 \times 10^8 \; \text{m/s} \): \[ P = \frac{119}{3.0 \times 10^8} = 3.97 \times 10^{-7} \; \text{N/m}^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Calculation

The frequency of an electromagnetic wave is a key parameter that defines how many oscillations occur in one second. It's calculated using the equation: \[ f = \frac{c}{\lambda} \]where:

• \( f \) is the frequency in Hertz (Hz),
• \( c \) is the speed of light in a vacuum, approximately \( 3.0 \times 10^8 \; \text{m/s} \), and
• \( \lambda \) is the wavelength of the wave.

To calculate the frequency for a wave with a wavelength of \( 3.0 \; \text{m} \), plug in the values:

• \( f = \frac{3.0 \times 10^8}{3.0} = 1.0 \times 10^8 \; \text{Hz} \)

This means the wave oscillates \( 100 \) million times per second as it travels through a vacuum.

Angular Frequency

Angular frequency describes how fast the wave oscillates and is expressed in radians per second. It is closely related to the frequency and is given by:\[ \omega = 2\pi f \]

• \( \omega \) is the angular frequency in radians per second,
• \( f \) is the regular frequency in Hertz.

Using the previously calculated frequency \( f = 1.0 \times 10^8 \; \text{Hz} \), we find:

• \( \omega = 2\pi \times 1.0 \times 10^8 = 2\pi \times 10^8 \; \text{rad/s} \)

Angular frequency provides a deeper understanding of the wave's oscillation speed on a circular scale.

Radiation Pressure

Radiation pressure is the pressure exerted by an electromagnetic wave when it strikes a surface. This occurs because the wave carries momentum, which can be transferred to the surface. For complete absorption, the radiation pressure \( P \) can be calculated using the formula:\[ P = \frac{S}{c} \]where:

• \( S \) is the intensity of the wave (time-averaged energy flow per unit area),
• \( c \) is the speed of light.

From the step-by-step solution, we derived that \( S = 119 \; \text{W/m}^2 \). Substituting in the values gives:

• \( P = \frac{119}{3.0 \times 10^8} \approx 3.97 \times 10^{-7} \; \text{N/m}^2 \)

Although appearing minuscule, radiation pressure plays significant roles in areas such as solar sails for spacecraft propulsion.

Magnetic Field Amplitude

In an electromagnetic wave, the electric and magnetic fields are perpendicular to each other and to the direction of wave propagation. They oscillate in harmony but with different magnitudes. The amplitude of the magnetic field \( B_0 \) is related to the electric field amplitude \( E_0 \) by the equation:\[ B_0 = \frac{E_0}{c} \]where:

• \( E_0 \) is the electric field amplitude,
• \( c \) is the speed of light.

Given \( E_0 = 300 \; \text{V/m} \) and \( c = 3.0 \times 10^8 \; \text{m/s} \), substitute these values to find:

• \( B_0 = \frac{300}{3.0 \times 10^8} = 1.0 \times 10^{-6} \; \text{T} \)

This value indicates the strength of the magnetic field component of the wave, allowing understanding of its electromagnetic structure.