Question

An \(\mathrm{H}\) field in free space is given as \(\mathcal{H}(x, t)=10 \cos \left(10^{8} t-\beta x\right) \mathbf{a}_{y} \mathrm{~A} / \mathrm{m}\). Find \((a) \beta ;(b) \lambda ;(c) \mathcal{E}(x, t)\) at \(P(0.1,0.2,0.3)\) at \(t=1 \mathrm{~ns}\).

Short answer

Answer: The calculated propagation constant (β) is \(\frac{1}{3} \,\mathrm{m^{-1}}\), and the wavelength (λ) is \(6\pi \,\mathrm{m}\).

Step-by-step solution

Identify the given information

We are given the H field: \(\mathcal{H}(x, t)=10 \cos \left(10^{8} t-\beta x\right) \mathbf{a}_{y} \mathrm{~A} / \mathrm{m}\). We need to find the values of β, λ, and the E field at point P(0.1, 0.2, 0.3) at t=1 ns.

Find the propagation constant (β)

We know that for a uniform plane wave, the relation between the angular frequency (ω) and the propagation constant (β) is given by: $$\omega = \beta \nu$$ where \(\nu\) is the speed of light in the medium (in this case, free space) and equals \(3 \times 10^8 \,\mathrm{m/s}\). From the given H field equation, we can see that the angular frequency (ω) is \(10^8 \,\mathrm{rad/s}\). Therefore, we can solve for β: $$\beta = \frac{\omega}{\nu} = \frac{10^8}{3\times 10^8}$$

Calculate β

Now we can calculate the value of β: $$\beta = \frac{10^8}{3\times 10^8} = \frac{1}{3} \,\mathrm{m^{-1}}$$

Find the wavelength (λ)

Using the relation between propagation constant (β) and wavelength (λ), we can determine the wavelength: $$\beta = \frac{2\pi}{\lambda}$$ Now we can solve for the wavelength (λ): $$\lambda=\frac{2\pi}{\beta}$$

Calculate λ

Now we can calculate the value of λ: $$\lambda = \frac{2\pi}{\frac{1}{3}} = 6\pi \,\mathrm{m}$$

Find the E field expression

The relationship between the E field and the H field for a uniform plane wave can be expressed as: $$\mathcal{E}(x, t) = \frac{\eta}{\beta} \mathcal{H}(x, t) \times \mathbf{u}$$ where η is the intrinsic impedance of free space, which is approximately equal to 377 Ω, and \(\mathbf{u}\) is the unit vector pointing in the direction of propagation. Given the direction of the H field (\(\mathbf{a}_{y}\)), we can assume the wave is propagating along the x-direction, so the unit vector \(\mathbf{u}\) would be \(\mathbf{a}_{x}\). Substituting the values we calculated for β and η, we can find the function for the E field: $$\mathcal{E}(x, t) = \frac{377}{\frac{1}{3}}\left( 10 \cos \left(10^{8} t-\frac{1}{3} x\right) \mathbf{a}_{y}\right) \times \mathbf{a}_{x}$$

Calculate the E field at P(0.1, 0.2, 0.3) at t=1 ns

Now we can substitute the coordinates of point P(0.1, 0.2, 0.3) and the time t=1 ns (or \(10^{-9}\) s) into the E field equation to find the value of the E field at that location and time: $$\mathcal{E}(0.1, 0.2, 0.3, 10^{-9}) = \frac{377}{\frac{1}{3}}\left( 10 \cos \left(10^{8}(10^{-9})-\frac{1}{3}(0.1)\right) \mathbf{a}_{y}\right) \times \mathbf{a}_{x}$$ By calculating this expression, we get the E field at point P(0.1, 0.2, 0.3) at t=1 ns.

Key concepts

Plane Wave Propagation

Electromagnetic waves, including light waves, radio waves and other forms of radiation, typically travel through space as plane waves. A plane wave is a wave that has constant phase fronts, or surfaces of constant phase, with amplitude that is constant over any single front. Assuming the wave is moving through a homogenous medium, each front represents an area where the wave has the same magnitude and phase.
In the exercise, the electromagnetic field, specifically the magnetic field H, is expressed with a cosine function that oscillates with both time (t) and position (x). Here, the term 'plane wave' refers to the fact that at any fixed time, the phase of the wave is constant along planes that are perpendicular to the x-axis, which is the direction of propagation.
Understanding plane wave propagation is crucial because it is the simplest form of electromagnetic wave travel and thus forms the basis for the analysis of more complex wave phenomena. Additionally, this concept is directly linked to principles such as interference and diffraction, that are fundamental to the field of wave optics and electromagnetic theory.

Propagation Constant (β)

The propagation constant, denoted as β (beta), is a fundamental parameter in the study of wave propagation. It represents how the phase of the wave changes with distance in the direction of wave travel. Mathematically, it is defined as the change in phase per unit length along the direction the wave travels.
In the provided solution, β is calculated from the given angular frequency ω and the speed of light ν in free space. This relationship is critical in understanding how the wave evolves as it moves through space. A key application of the propagation constant is in many communication systems, where β affects how the signal degrades over distance. Understanding and managing this degradation is essential for maintaining signal quality over long distances, as seen in fiber optic communication, radio broadcasts, and other similar technologies.

Wavelength (λ)

Wavelength, represented by the Greek letter λ (lambda), is another intrinsic property of a wave. It is the distance between two consecutive points that are in phase, such as the crest of the wave, and it plays a vital role in determining the wave's properties. For instance, the wavelength is inversely related to the frequency —the higher the frequency, the shorter the wavelength.
In the context of our exercise, once the propagation constant β is found, we use it to calculate the wavelength λ. This calculation is pivotal because the wavelength tells us about the scale of the wave's interaction with objects and mediums. For example, it's important for understanding how the wave can pass through an opening or around an object, a concept known as diffraction. In telecommunications, the wavelength determines the size of the antennas and the design of the circuitry used to process the signals.

Electromagnetic Field Relationship

The electromagnetic field relationship is the connection between the electric field (E) and the magnetic field (H) components of an electromagnetic wave. These fields are perpendicular to each other and to the direction of propagation, forming a right-angled coordinate system that moves with the wave.
In free space, the intrinsic impedance (η) relates the magnitudes of E and H fields. As shown in the exercise, once we ascertain the magnetic field H, we can compute the electric field E using η and β.
This relationship is at the heart of Maxwell's equations, which describe how electric and magnetic fields are generated and altered by each other and by charges and currents. It is also the basis for how electromagnetic waves carry energy and momentum, making it a cornerstone concept in the design and function of antennas, the transmission of signals, and the general behavior of light and radio waves in various applications.