A spherical wave with a wavelength of 2.0 m is emitted from the origin. At one instant of time, the phase at r = 4.0 m is π rad. At that instant, what is the phase at r = 3.5 m and at r = 4.5 m?

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Step 1: Understand the relationship between the phase of a wave and its position. The phase of a wave changes as you move away from the source, and this change is proportional to the distance traveled divided by the wavelength. The phase difference can be calculated using the formula Δφ = (2π/λ)Δr, where λ is the wavelength and Δr is the change in distance.

Step 2: Identify the given values. The wavelength λ is 2.0 m, the phase at r = 4.0 m is π rad, and we need to find the phase at r = 3.5 m and r = 4.5 m.

Step 3: Calculate the phase difference between r = 4.0 m and r = 3.5 m using the formula Δφ = (2π/λ)Δr. Here, Δr = r_final - r_initial = 3.5 m - 4.0 m = -0.5 m. Substitute λ = 2.0 m into the formula to find Δφ.

Step 4: Calculate the phase difference between r = 4.0 m and r = 4.5 m using the same formula Δφ = (2π/λ)Δr. Here, Δr = r_final - r_initial = 4.5 m - 4.0 m = 0.5 m. Substitute λ = 2.0 m into the formula to find Δφ.

Step 5: Add the calculated phase differences to the initial phase at r = 4.0 m (π rad) to determine the phases at r = 3.5 m and r = 4.5 m. For r = 3.5 m, subtract the phase difference (since Δr is negative), and for r = 4.5 m, add the phase difference (since Δr is positive).

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Key Concepts

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

Wavelength

Wavelength is the distance between successive crests of a wave, typically denoted by the symbol λ. In this case, the wavelength of 2.0 m indicates that the wave repeats every 2 meters. Understanding wavelength is crucial for determining how the phase of the wave changes with distance.

Phase of a Wave

The phase of a wave describes the position of a point in time on a waveform cycle, usually measured in radians. A phase of π rad indicates that the wave is at its maximum negative displacement. The phase can be calculated based on the distance from the source and the wavelength, allowing us to find the phase at different points.

Spherical Waves

Spherical waves are waves that propagate outward in all directions from a point source, forming a spherical front. The phase of a spherical wave varies with distance from the source, and the relationship can be expressed as φ = (2π/λ) * r, where φ is the phase, λ is the wavelength, and r is the distance from the source. This concept is essential for calculating the phase at different radial distances.

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