Question

When two light waves, both with wavelength \(\lambda\) and amplitude \(A\), interfere constructively, they produce a light wave of the same wavelength but with amplitude \(2 A .\) What will be the intensity of this light wave? a) same intensity as before b) double the intensity c) quadruple the intensity d) not enough information

Short answer

Question: When two light waves of the same wavelength and amplitude interfere constructively, the intensity of the resulting wave is: a) Half the intensity b) Double the intensity c) Quadruple the intensity d) Equal to the intensity Answer: c) Quadruple the intensity

Step-by-step solution

Constructive interference of two waves

When two light waves of the same wavelength and amplitude interfere constructively, their amplitudes add up, which results in a wave with the same wavelength but twice the amplitude. So the amplitude of the resulting wave will be \(2 A\).

Relation between amplitude and intensity

The intensity of a wave is proportional to the square of its amplitude. Mathematically, this can be expressed as: \(I \propto A^2\) where \(I\) is the intensity and \(A\) is the amplitude of the wave.

Find the intensity of the resulting wave

As per the relationship, the intensity of the resulting wave will be proportional to the square of its amplitude \((2A)^2\). \(I_{result} \propto (2A)^2 = 4A^2\)

Compare the intensity of the resulting wave with the initial intensity

Since the intensity of the original waves is proportional to \(A^2\), let's compare the resulting intensity with the initial intensity: \(\frac{I_{result}}{I_{initial}} = \frac{4A^2}{A^2} = 4\)

Choose the correct option

The intensity of the resulting wave is 4 times the intensity of the original waves. So, the correct answer is option c) Quadruple the intensity.

Key concepts

Wave Amplitude

Wave amplitude is a measure of the size or height of a wave. It defines how much the medium is displaced from its resting position when the wave passes through. In simple terms, the amplitude is the peak value of wave displacement.
In the context of light waves, amplitude affects how bright or dim the light appears. The greater the amplitude, the brighter the light. When waves interfere constructively, as mentioned in the exercise, their amplitudes add up. This results in a wave with double the amplitude compared to individual waves.

• Example: If one wave has an amplitude of 2 units and another wave with the same amplitude interferes constructively, the resulting wave has an amplitude of 4 units.

The concept of amplitude is crucial because it directly relates to the wave's energy and intensity, which gives insight into the wave's power and effect.

Wave Intensity

Wave intensity describes the power of a wave per unit area. When discussing light waves, intensity often relates to how much energy a wave carries over an area in a given time interval.

• The formula to represent intensity is:
  \(I \propto A^2\)
  This means intensity is proportional to the square of the amplitude.
  

When thinking about constructive interference, remember that if the amplitude doubles, the intensity doesn't just double but increases by four times. That's because \((2A)^2 = 4A^2\).
This squared relationship indicates that even small changes in amplitude can cause significant changes in intensity. Therefore, understanding intensity helps in predicting how waves behave when they combine or interact with each other.

Wavelength

Wavelength refers to the distance between consecutive points of a wave that are in phase, such as crest to crest or trough to trough. Represented by the Greek letter \(\lambda\) (lambda), wavelength is critical in determining the energy and type of wave.

• For light waves, it helps determine the color of the light in the visible spectrum.
• In many wave phenomena, like sound or light, wavelength impacts how waves move through different mediums.

The concept of wavelength is essential in the exercise because constructive interference happens most effectively when the interfering waves have the same wavelength.
For two waves to interfere constructively, they need to be in phase, meaning the peaks and troughs line up which typically occurs with identical wavelengths. Understanding wavelength provides insights into how waves can combine constructively or destructively.