∙∙A sinusoidal electromagnetic wave of frequency 6.10 ×10^14 Hz travels in vacuum in the +x -dir

step 1 In the first part of this problem, we are going to calculate the magnitude of electric field and

∙∙A sinusoidal electromagnetic wave of frequency 6.10 ×10^14...

$\bullet$$\bullet \mathrm{A}$ sinusoidal electromagnetic wave of frequency $6.10 \times 10^{14} \mathrm{Hz}$ travels in vacuum in the $+x$ -direction. The magnetic field is parallel to the $y$ -axis and has amplitude $5.80 \times 10^{-4} \mathrm{T} .$ (a) Find the magnitude and direction of the electric field. (b) Write the wave functions for the electric and magnetic fields in the form of Equations $(23.3).$

Key Concepts

Field Orientation and the Right-Hand Rule

The electric field, magnetic field, and the direction of wave propagation in an electromagnetic wave are mutually perpendicular. The orientation is commonly determined using the right-hand rule, where the thumb, forefinger, and middle finger represent the directions of propagation, E field, and B field respectively.

Wave Number and Angular Frequency

The angular frequency (?) and wave number (k) are fundamental parameters of a wave, where ? is related to the temporal oscillation of the wave and k to the spatial variation. They are connected via the speed of light in a vacuum by the relation ? = ck, with k = 2?/? and ? = 2?f.

Plane Wave Formulation

Electromagnetic waves can be described mathematically as plane waves using sinusoidal functions. The general form involves amplitudes, angular frequency (?), wave number (k), and phase factors, expressed typically as E(x,t) = E? cos(kx - ?t + ?) and B(x,t) = B? cos(kx - ?t + ?), capturing both spatial and temporal variations.

Relationship Between Electric and Magnetic Fields

In a vacuum, the magnitudes of the electric (E) and magnetic (B) fields in an electromagnetic wave are related by the speed of light (c), with the relation E = cB. This relationship ensures that the energy carried by the wave is distributed correctly between the electric and magnetic field components.

Electromagnetic Wave Propagation

Electromagnetic waves are self-sustaining oscillations of electric and magnetic fields that propagate through space. In a vacuum, the wave travels at the speed of light, and the electric and magnetic fields oscillate perpendicular to each other and to the direction of travel.

Transcript

Please give Ace some feedback