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Ch 31: Electromagnetic Fields and Waves
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 31: Electromagnetic Fields and WavesProblem 13a
Chapter 31, Problem 13a

The magnetic field of an electromagnetic wave in a vacuum is Bz=(3.0 μT)sin[(1.00×107)x−ωt]B_{z}=(3.0\(\text{ }\]\mu\) T)\(\sin\)[(1.00\(\times\)10^7)x-\(\omega\) t], where x is in m and t is in s. What are the wave's wavelength?

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Step 1: Identify the wave equation provided in the problem. The magnetic field is given as Bz = (3.0 μT) sin[(1.00 × 10⁷)x − ωt]. This is a sinusoidal wave equation, where the term (1.00 × 10⁷) represents the wave number k, which is related to the wavelength λ.
Step 2: Recall the relationship between the wave number k and the wavelength λ. The wave number k is defined as k = 2π/λ, where λ is the wavelength. Rearrange this formula to solve for λ: λ = 2π/k.
Step 3: Substitute the value of k from the wave equation into the formula for λ. Here, k = 1.00 × 10⁷ m⁻¹. Using λ = 2π/k, plug in k = 1.00 × 10⁷ m⁻¹.
Step 4: Perform the division and simplify the expression for λ. This involves dividing 2π by the value of k (1.00 × 10⁷ m⁻¹). Ensure the units are consistent, and the result will be in meters (m), which is the standard unit for wavelength.
Step 5: Interpret the result. The calculated wavelength represents the distance between successive peaks of the electromagnetic wave in the vacuum. This is a fundamental property of the wave and is inversely proportional to the wave number k.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. They travel at the speed of light in a vacuum and include a spectrum of waves, from radio waves to gamma rays. The behavior of these waves can be described by their frequency, wavelength, and amplitude, which are fundamental properties that determine their characteristics.
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Wavelength

Wavelength is the distance between successive peaks (or troughs) of a wave, typically denoted by the Greek letter lambda (λ). It is inversely related to frequency; as frequency increases, wavelength decreases. In the context of electromagnetic waves, the wavelength can be calculated using the wave's speed and frequency, following the formula λ = c/f, where c is the speed of light.
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Wave Equation

The wave equation describes the relationship between the wave's spatial and temporal characteristics. For a sinusoidal wave, it can be expressed as B(z, t) = B₀ sin(kx - ωt), where B₀ is the amplitude, k is the wave number, and ω is the angular frequency. The wave number k is related to the wavelength by k = 2π/λ, which is essential for determining the wavelength from the given wave equation.
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