Question

In the fraunhaufer diffraction from a single slit illuminated by polychromatic light, the first minimum with wavelength \(\lambda_{1}\) is found to be coincident with the third minimum at \(\lambda_{2}\). Then the relationship between these two wavelengths is (1) \(3 \lambda_{1}=2 \lambda_{2}\) (2) \(3 \lambda_{1}=\lambda_{2}\) (3) \(\lambda_{1}=3 \lambda_{2}\) (4) \(2 \lambda_{1}=3 \lambda_{2}\)

Short answer

(3) \(\lambda_1 = 3 \lambda_2\)

Step-by-step solution

Identify the condition for diffraction minima in single-slit diffraction

The condition for the position of minima in single-slit diffraction is given by the equation: \[ a \sin \theta = m \lambda \]where \( a \) is the slit width, \( \theta \) is the angle of diffraction, \( m \) is the order of the minimum, and \( \lambda \) is the wavelength of the light.

Express the given condition of wavelengths

According to the problem, the first minimum with wavelength \( \lambda_1 \) is coincident with the third minimum at wavelength \( \lambda_2 \). This means:\[ a \sin \theta = 1 \lambda_1 \quad (for \lambda_1) \]\[ a \sin \theta = 3 \lambda_2 \quad (for \lambda_2) \]

Equate the two expressions for minima

Set the two equations equal to each other because the minima are coincident at the same angle \( \theta \):\[ 1 \lambda_1 = 3 \lambda_2 \]

Solve for the relationship between \( \lambda_1 \) and \( \lambda_2 \)

Rearrange the equation obtained in the previous step to find the relationship between the two wavelengths:\[ \lambda_1 = 3 \lambda_2 \]

Select the correct choice

The correct relationship between the wavelengths \( \lambda_1 \) and \( \lambda_2 \) is:Option (3): \( \lambda_1 = 3 \lambda_2 \)

Key concepts

single-slit diffraction

Single-slit diffraction occurs when light passes through a narrow slit and spreads out as it travels. This spreading creates a diffraction pattern on an observation screen, where light and dark regions alternate. The phenomenon is explained by the wave nature of light. When light waves pass through the slit, they interfere with each other, causing constructive interference (bright spots) and destructive interference (dark spots). The central maximum is the brightest spot and is located at the center of the pattern, while the other bright and dark spots alternate symmetrically around it.

wavelength relationship

In single-slit diffraction, the wavelength of light plays a crucial role in determining the diffraction pattern. The relationship here can be explored using the condition for diffraction minima. The condition for the position of minima is given by the equation: \[ a \sin \theta = m \lambda \] where
* \( a \): slit width
* \( \theta \): angle of diffraction
* \( m \): order of the minimum
* \( \lambda \): wavelength of the light
For instance, our exercise involves comparing different wavelengths. The first minimum of one wavelength coincides with the third minimum of another. Setting their conditions equal gives us: \( \lambda_1 = 3 \lambda_2 \). This shows a direct and simple mathematical relationship between the wavelengths, succinctly solved using the conditions governing diffraction.

diffraction minima

Diffraction minima are the dark regions in a diffraction pattern where destructive interference occurs. They are termed 'minima' because the light intensity at those points is minimal. For a single slit, the minima occur at specific angles and can be described using the formula: \[ a \sin \theta = m \lambda \] where:
* \( m \): integer (1, 2, 3, ...), representing the order of the minimum.
* At these positions, the path difference between light waves from different parts of the slit results in them being out of phase by an integral number of wavelengths, leading to cancellation of the waves. This structured formation of light and dark bands helps us understand and calculate various aspects of wave behavior such as wavelength interactions and phase differences.