Question

like a wavelength 5000 A° cenit is diffected by slit width of 2 x 10⁻³ m for whal distance travel by the diffrated coove does the spreading defraction becomes greater than width of apertur.

Answer 1

4 m

Answer 2

The phenomenon described is single-slit diffraction. The spreading of the diffracted wave can be characterized by the width of the central maximum of the diffraction pattern. The angular position of the first minima on either side of the central maximum is given by $\sin \theta = n\lambda/a$, where n=1 for the first minimum. For small angles, $\sin \theta \approx \theta$, so the angular half-width of the central maximum is $\theta \approx \lambda/a$.

The linear half-width of the central maximum on a screen placed at a distance $D$ from the slit is $y = D \tan \theta$. For small angles, $\tan \theta \approx \theta$, so $y \approx D(\lambda/a)$. The full width of the central maximum is $W = 2y \approx 2D(\lambda/a)$.

We are looking for the distance $D$ where the spreading (width of the central maximum, $W$) becomes greater than the width of the aperture ($a$). So, we set the condition $W > a$.

$2D(\lambda/a) > a$

Now, we solve for $D$:

$2D\lambda > a^2$ $D > a^2 / (2\lambda)$

Substitute the given values:

Wavelength, $\lambda = 5000 \text{ Å} = 5000 \times 10^{-10} \text{ m} = 5 \times 10^{-7} \text{ m}$

Slit width, $a = 2 \times 10^{-3} \text{ m}$

$D > (2 \times 10^{-3} \text{ m})^2 / (2 \times 5 \times 10^{-7} \text{ m})$ $D > (4 \times 10^{-6} \text{ m}^2) / (10 \times 10^{-7} \text{ m})$ $D > (4 \times 10^{-6} \text{ m}^2) / (10^{-6} \text{ m})$ $D > 4 \text{ m}$

Thus, the spreading due to diffraction becomes greater than the width of the aperture for distances greater than 4 meters.

The distance $D = \frac{a^2}{2\lambda}$ is sometimes referred to as the Fresnel distance or Rayleigh distance, which marks the transition from Fresnel diffraction (near-field) to Fraunhofer diffraction (far-field). At distances much less than this, the wave propagates essentially without spreading significantly compared to the aperture size. At distances comparable to or greater than this, diffraction effects become prominent, causing the wave to spread.