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An electromagnetic wave of wavelength 435 nm is traveling in vacuum in the -\(z\)-direction. The electric field has amplitude 2.70 \(\times\) 10\(^{-3}\) V/m and is parallel to the \(x\)-axis. What are (a) the frequency and (b) the magnetic-field amplitude? (c) Write the vector equations for \(\vec{E} (z, t)\) and \(\vec{B} (z, t)\).

Short Answer

Expert verified
(a) Frequency: 6.90 × 10^14 Hz, (b) Magnetic field amplitude: 9.00 × 10^-12 T, (c) \( \vec{E}(z, t) = 2.70 \times 10^{-3} \cos(1.44 \times 10^7 z - 4.34 \times 10^{15} t) \hat{i} \); \( \vec{B}(z, t) = 9.00 \times 10^{-12} \cos(1.44 \times 10^7 z - 4.34 \times 10^{15} t) \hat{j} \).

Step by step solution

01

Convert Wavelength to Frequency

We start by calculating the frequency of the wave using the formula \( f = \frac{c}{\lambda} \), where \( c = 3 \times 10^8 \) m/s is the speed of light and \( \lambda = 435 \times 10^{-9} \) m is the wavelength.Calculating:\[ f = \frac{3 \times 10^8 \text{ m/s}}{435 \times 10^{-9} \text{ m}} = 6.90 \times 10^{14} \text{ Hz} \]
02

Calculate Magnetic Field Amplitude

The amplitude of the magnetic field \( B_{m} \) can be found using the relation between the electric field amplitude \( E_{m} \) and the magnetic field amplitude: \( B_{m} = \frac{E_{m}}{c} \).Given \( E_m = 2.70 \times 10^{-3} \text{ V/m} \) and \( c = 3 \times 10^8 \text{ m/s} \):\[ B_{m} = \frac{2.70 \times 10^{-3} \text{ V/m}}{3 \times 10^8 \text{ m/s}} = 9.00 \times 10^{-12} \text{ T} \]
03

Write the Electric Field Equation

The electric field \( \vec{E}(z, t) \) for a wave traveling in the -\( z \)-direction and polarized along the \( x \)-axis is given by:\[ \vec{E}(z, t) = E_{m} \cos(kz - \omega t) \hat{i} \]where \( k = \frac{2\pi}{\lambda} \) and \( \omega = 2\pi f \).Calculating\( k = \frac{2\pi}{435 \times 10^{-9}} \text{ m} = 1.44 \times 10^7 \text{ m}^{-1} \)and\( \omega = 2\pi \times 6.90 \times 10^{14} = 4.34 \times 10^{15} \text{ rad/s} \).Thus, the equation becomes:\[ \vec{E}(z, t) = 2.70 \times 10^{-3} \cos(1.44 \times 10^7 z - 4.34 \times 10^{15} t) \hat{i} \]
04

Write the Magnetic Field Equation

The magnetic field \( \vec{B}(z, t) \) is perpendicular to both \( \vec{E}(z, t) \) and the direction of propagation. Therefore, it is along the \( y \)-axis:\[ \vec{B}(z, t) = B_{m} \cos(kz - \omega t) \hat{j} \]Using the previously calculated values, the equation becomes:\[ \vec{B}(z, t) = 9.00 \times 10^{-12} \cos(1.44 \times 10^7 z - 4.34 \times 10^{15} t) \hat{j} \]

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Key Concepts

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

Wavelength
The wavelength of an electromagnetic wave is the distance between two consecutive peaks (or troughs) of the wave. It is usually denoted as \( \lambda \). For the provided exercise, the wavelength is specified as 435 nm (nanometers).
To convert this to meters, multiply by \( 10^{-9} \), giving \( \lambda = 435 \times 10^{-9} \) meters. This is crucial because all calculations involving the speed of light, which is in meters per second (m/s), require the wavelength to be in meters.
The relationship that comes into play with wavelength is how it dictates the energy and behavior of the electromagnetic wave. Shorter wavelengths correspond to higher energy and frequency, making them crucial in applications like visible light and X-rays.
Electric Field
The electric field of an electromagnetic wave is a vector field that represents the force that would be experienced by a positive charge placed in the field. In the exercise, the electric field amplitude is given as \( 2.70 \times 10^{-3} \text{ V/m} \).
The electric field oscillates perpendicular to the direction of wave propagation. Here, it is aligned along the \( x \)-axis. This means the wave propagates in the -\( z \)-direction while the electric field oscillates in the \( x \)-direction.
Understanding the electric field allows us to predict how the wave will interact with charged particles and matter, influencing technologies like antennas and capacitors in electronics.
Magnetic Field Amplitude
The magnetic field amplitude is a crucial component of electromagnetic waves but often less intuitive than the electric field. For the given wave, the relationship between electric field amplitude \( E_m \) and magnetic field amplitude \( B_m \) is given by:
  • \( B_m = \frac{E_m}{c} \)
where \( c \) is the speed of light, \( 3 \times 10^8 \text{ m/s} \).
Substituting the provided values, \( B_m \) calculates to \( 9.00 \times 10^{-12} \text{ T} \) (teslas).
Like the electric field, the magnetic field is perpendicular to both the electric field and the direction of wave propagation. Here, it is oriented along the \( y \)-axis, perpendicular to both the \( x \)-aligned electric field and the -\( z \)-propagation direction, forming a right-hand coordinate system with the electric field and propagation direction.
Frequency
Frequency is the number of wave cycles that pass a point in one second and is a key descriptor of electromagnetic waves. In the given problem, frequency \( f \) can be found using:
  • \( f = \frac{c}{\lambda} \)
This employs the speed of light \( c = 3 \times 10^8 \text{ m/s} \) and the wavelength to calculate frequency.
The resulting frequency \( 6.90 \times 10^{14} \text{ Hz} \) (hertz) corresponds to many cycles per second. Frequencies like these fall in the visible spectrum, crucial to understanding phenomena like color perception in light.
This also lays the foundation for technologies like radio, where varying frequencies correspond to different channels and types of communication.

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Most popular questions from this chapter

There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 nm to 400 nm. It is necessary for the production of vitamin D. UVB, with a wavelength in vacuum between 280 nm and 320 nm, is more dangerous because it is much more likely to cause skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

A sinusoidal electromagnetic wave from a radio station passes perpendicularly through an open window that has area 0.500 m\(^2\). At the window, the electric field of the wave has rms value 0.0400 V/m. How much energy does this wave carry through the window during a 30.0-s commercial?

An electromagnetic wave has an electric field given by \(\vec{E} (y, t)\) = (3.10 \(\times\) 10\(^5\) V/m) \(\hat{k}\) cos [ky - (12.65 \(\times\) 10\(^{12}\) rad/s)t]. (a) In which direction is the wave traveling? (b) What is the wavelength of the wave? (c) Write the vector equation for \(\vec{B} (y, t)\).

In a certain experiment, a radio transmitter emits sinusoidal electromagnetic waves of frequency 110.0 MHz in opposite directions inside a narrow cavity with reflectors at both ends, causing a standing-wave pattern to occur. (a) How far apart are the nodal planes of the magnetic field? (b) If the standing- wave pattern is determined to be in its eighth harmonic, how long is the cavity?

A satellite 575 km above the earth's surface transmits sinusoidal electromagnetic waves of frequency 92.4 MHz uniformly in all directions, with a power of 25.0 kW. (a) What is the intensity of these waves as they reach a receiver at the surface of the earth directly below the satellite? (b) What are the amplitudes of the electric and magnetic fields at the receiver? (c) If the receiver has a totally absorbing panel measuring 15.0 cm by 40.0 cm oriented with its plane perpendicular to the direction the waves travel, what average force do these waves exert on the panel? Is this force large enough to cause significant effects?

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