A 550- nm harmonic EM-wave whose electric ficld is in the :...

A $550-$ nm harmonic EM-wave whose electric ficld is in the : direction is traveling in the $y$ -direction in vacuum. (a) What is the frequency of the wave? (b) Determine both ar and $k$ for this wave. (c) If the electric ficld amplitude is $600 \mathrm{~V} / \mathrm{m}$, what is the anplitude of $\mathrm{dx}$ magnetic ficld? (d) Write an expression for both $E(t)$ and $B(r)$ given that each is zero at $x=0$ and $t=0$. Put in all the appropriate units.

A $550-$ nm harmonic EM-wave whose electric ficld is in the :
direction is traveling in the $y$ -direction in vacuum. (a) What is the frequency of the wave? (b) Determine both ar and $k$ for this wave. (c) If the electric ficld amplitude is $600 \mathrm{~V} / \mathrm{m}$, what is the anplitude of $\mathrm{dx}$ magnetic ficld? (d) Write an expression for both $E(t)$ and $B(r)$ given that each is zero at $x=0$ and $t=0$. Put in all the appropriate units.

Key Concepts

Electromagnetic Waves in Vacuum

Electromagnetic waves in vacuum propagate with no medium and are characterized by mutually perpendicular electric and magnetic fields that oscillate in phase. Their behavior is governed by Maxwell's equations, and their speed in vacuum is a universal constant, c, which links many of their fundamental properties such as frequency, wavelength, and wave number.

Frequency and Wavelength Relationship

The frequency of an electromagnetic wave is directly related to its wavelength by the equation f = c/?, where c is the speed of light in vacuum and ? is the wavelength. This relationship enables the determination of one property given the other and is crucial in understanding the energy and oscillatory behavior of the wave.

Wave Number and Angular Frequency

The wave number, k, is defined as 2? divided by the wavelength (k = 2?/?) and represents the spatial frequency of the wave. Similarly, the angular frequency, ?, is defined as 2? times the frequency (? = 2?f) and describes the temporal oscillation rate. These two parameters are fundamental in the mathematical description of wave propagation and appear in the phase factors of sinusoidal wave expressions.

Electric and Magnetic Field Amplitude Relationship

In an electromagnetic wave traveling in vacuum, the amplitudes of the electric field (E?) and the magnetic field (B?) are directly related by the speed of light through the equation B? = E?/c. This relationship highlights the intrinsic connection between the fields and underscores the energy transport mechanism inherent in electromagnetic waves.

Sinusoidal Wave Expressions

Sinusoidal expressions for the electric and magnetic fields are used to describe harmonic electromagnetic waves. These expressions typically take the form E(r, t) = E? sin(k · r ? ?t + ?) and B(r, t) = B? sin(k · r ? ?t + ?), where the phase constant is chosen to satisfy initial conditions such as the fields being zero at a particular position and time. This form efficiently characterizes both the spatial and temporal variations of the fields in a propagating wave.

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