If Ι 0 is the intensity of the principal maximum in the single slit diff action pattern, then
what will be its intensity when the slit width is doubled ?

When we talk about single slit diffraction, the intensity of the light pattern produced is influenced by the width of the slit. If we denote the intensity of the principal maximum as Ι₀, we can analyze how this intensity changes when the slit width is doubled.

Understanding the Relationship Between Slit Width and Intensity

The intensity of the principal maximum in a single slit diffraction pattern is related to the width of the slit. The formula for the intensity at the center of the diffraction pattern can be expressed as:

• Ι = Ι₀ * (sin(β)/β)²

Here, β is defined as:

• β = (π * a * sin(θ)) / λ

In this equation:

• a is the width of the slit,
• λ is the wavelength of the light, and
• θ is the angle of observation.

Effect of Doubling the Slit Width

Now, if we double the slit width (let's say from 'a' to '2a'), we need to analyze how this affects β:

• New β = (π * (2a) * sin(θ)) / λ = 2 * (π * a * sin(θ)) / λ = 2β

Substituting this new β back into the intensity formula gives us:

• Ι = Ι₀ * (sin(2β)/2β)²

Calculating the New Intensity

Using the double angle identity for sine, we know that:

• sin(2β) = 2 * sin(β) * cos(β)

Thus, the intensity becomes:

• Ι = Ι₀ * (2 * sin(β) * cos(β) / 2β)² = Ι₀ * (sin(β) * cos(β) / β)²

From this, we can see that the intensity at the principal maximum will change due to the alteration in the slit width. Specifically, the intensity will increase because the factor of cos(β) will contribute positively to the overall intensity, while the factor of β in the denominator will not increase as significantly as the numerator, leading to a higher intensity value.

Conclusion on Intensity Change

In summary, when the slit width is doubled, the intensity of the principal maximum will not simply double but will increase due to the changes in the diffraction pattern. The exact new intensity will depend on the specific values of the wavelength and the angle of observation, but qualitatively, we can conclude that the intensity will increase as a result of the increased slit width.