Problem 18 - University Physics with Modern Physics

Question

Coherent sources \(A\) and \(B\) emit electromagnetic waves with wavelength 2.00 cm. Point \(P\) is 4.86 m from \(A\) and 5.24 m from \(B\). What is the phase difference at \(P\) between these two waves?

Step-by-Step Solution

Answer

The phase difference is \(38\pi\) radians.

Step 1: Determine Path Length Difference

Calculate the difference in the path lengths from point P to both sources A and B.\[ \Delta L = L_B - L_A = 5.24 \, \text{m} - 4.86 \, \text{m} = 0.38 \, \text{m} \]

Step 2: Convert Path Length Difference to Wavelengths

Convert the path length difference into wavelengths. Since the wavelength \( \lambda = 2.00 \, \text{cm} = 0.020 \, \text{m} \), divide the path length difference by the wavelength.\[ \Delta N = \frac{\Delta L}{\lambda} = \frac{0.38 \, \text{m}}{0.020 \, \text{m}} = 19 \]

Step 3: Calculate Phase Difference

The phase difference in terms of wavelengths is \( 2\pi \) radians per wavelength. So multiply the number of wavelengths by \( 2\pi \) to convert this into a phase difference in radians.\[ \Delta \phi = 2\pi \times 19 = 38\pi \, \text{radians} \]

Key Concepts

Coherent Sources

Coherent sources refer to wave sources that maintain a constant phase difference and have the same frequency. This concept is crucial in observing interference patterns where the wave fronts from each source overlap. The term 'coherent' implies synchronization between the sources, much like two violinists playing the same note in perfect harmony. Coherent sources are essential for creating clear interference patterns as their waves combine predictably. In the given exercise, the coherent sources A and B ensure that the electromagnetic waves they emit have a stable phase relationship, enabling us to determine the phase difference at point P effectively.

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that travel through space. Unlike sound waves, they don't need a medium and can even travel through a vacuum. These waves are all around us, forming the light we see, radio waves, and even microwaves. In our problem, the electromagnetic waves emitted by sources A and B allow us to explore the concept of wave interference, crucial for understanding phenomena such as color patterns from thin films, like soap bubbles. Each wave travels a certain distance to reach point P, which affects how they sum up, depending on their phase difference.

Path Length Difference

Path length difference is a key concept to understand how waves interfere with each other at a point. Simply put, it's the difference in the distances that waves from two sources travel to reach a certain point. In the exercise, point P is 4.86 meters from source A and 5.24 meters from source B, which gives us a path length difference of 0.38 meters. This difference directly influences whether the waves will interfere constructively or destructively. If the path length difference equates to an integer number of wavelengths, constructive interference occurs, while a half-integer results in destructive interference.

Wavelength

Wavelength is the distance between consecutive crests (or troughs) of a wave. It's a fundamental property that defines the wave and its interactions. In simple terms, it's the 'size' of a wave. For the problem at hand, the wavelength is given as 2.00 cm, or 0.020 meters. Understanding the wavelength is crucial because it helps convert the path length difference into a phase difference, determining how the waves add up at point P.

• Smaller wavelengths mean more oscillations over a given distance.
• Calculating how many wavelengths fit into the path length difference gives insight into interference patterns.