Single Slit Diffraction Minima Condition
The positions of minima (dark fringes) in a single-slit diffraction pattern are given by the formula:
$a \sin(\theta_m) = m \lambda$
Where:
- $a$ is the width of the slit.
- $\theta_m$ is the angle of the minimum from the central maximum.
- $m$ is the order of the minimum ($m = \pm 1, \pm 2, \pm 3, \dots$).
- $\lambda$ is the wavelength of the light.
Calculating Angular Positions
We are given the second minimum to the left ($m=-2$) and the third minimum to the right ($m=3$). Let the angles be $\theta_2$ and $\theta_3$ respectively (measured from the center).
- For the second minimum: $a \sin(\theta_2) = 2 \lambda$
- For the third minimum: $a \sin(\theta_3) = 3 \lambda$
The angular separation between these two minima is $30^\circ$. Since they are on opposite sides of the central maximum, the total angular separation is the sum of the magnitudes of their angles: $\theta_2 + \theta_3 = 30^\circ$.
Applying Small Angle Approximation
For small angles (commonly assumed in diffraction problems unless stated otherwise), we can approximate $\sin(\theta) \approx \theta$ when $\theta$ is in radians.
- $\theta_2 \approx \frac{2 \lambda}{a}$
- $\theta_3 \approx \frac{3 \lambda}{a}$
First, convert the total angular separation to radians:
$30^\circ = 30 \times \frac{\pi}{180} = \frac{\pi}{6} \text{ radians}$
Substitute the approximations into the total angle equation:
$\theta_2 + \theta_3 \approx \frac{2 \lambda}{a} + \frac{3 \lambda}{a} = \frac{5 \lambda}{a}$
Equating this to the total angle in radians:
$\frac{5 \lambda}{a} \approx \frac{\pi}{6}$
Solving for Slit Width
Rearrange the equation to solve for the slit width $a$:
$a \approx \frac{5 \lambda}{\pi/6} = \frac{30 \lambda}{\pi}$
Substitute the given wavelength $\lambda = 628$ nm $= 628 \times 10^{-9}$ m:
$a \approx \frac{30 \times (628 \times 10^{-9} \text{ m})}{\pi}$
$a \approx \frac{18840 \times 10^{-9} \text{ m}}{3.14159...}$
$a \approx 6000 \times 10^{-9} \text{ m}$
Convert the slit width to micrometers ($\mu$m):
$a \approx 6 \times 10^{-6} \text{ m} = 6 \ \mu\text{m}$
The width of the slit is approximately 6 $\mu$m.
Similar Questions
Consider following statements for refraction of light through prism, when angle of deviation is minimum.
A. The refracted ray inside prism becomes parallel to the base.
B. Larger angle prisms provide smaller angle of minimum deviation.
C. Angle of incidence and angle of emergence becomes equal.
D. There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.
E. Angle of refraction becomes double of prism angle. Choose the correct answer from the options given below:
The radii of curvature for a thin convex lens are $10 \ cm$ and $15 \ cm$ respectively. The focal length of the lens is $12 \ cm$. The refractive index of the lens material is
The work function of a metal is $3 \ eV$. The color of the visible light that is required to cause emission of photoelectrons is
In the figure shown below, a resistance of $150.4 \Omega$ is connected in series to an ammeter A of resistance $240 \Omega$. A shunt resistance of $10 \Omega$ is connected in parallel with the ammeter. The reading of the ammeter is __________ mA.
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is :
A spherical surface separates two media of refractive indices 1 and 1.5 as shown in figure. Distance of the image of an object 'O', is :
(C is the center of curvature of the spherical surface and R is the radius of curvature)- A finite size object is placed normal to the principal axis at a distance of 30 cm from a convex mirror of focal length 30 cm. A plane mirror is now placed in such a way that the image produced by both the mirrors coincide with each other. The distance between the two mirrors is:
- Two polarisers $P_1$ and $P_2$ are placed in such a way that the intensity of the transmitted light will be zero. A third polariser $P_3$ is inserted in between $P_1$ and $P_2$, at particular angle between $P_2$ and $P_3$. The transmitted intensity of the light passing the through all three polarisers is maximum. The angle between the polarisers $P_2$ and $P_3$ is :
- In a Young's double slit experiment, two slits are located 1.5 mm apart. The distance of screen from slits is 2 m and the wavelength of the source is 400 nm. If the 20 maxima of the double slit pattern are contained within the central maximum of the single slit diffraction pattern, then the width of each slit is $x\times10^{-3}$ cm, where x-value is ________.
- Two thin convex lenses of focal lengths $30 \text{ cm}$ and $10 \text{ cm}$ are placed coaxially, $10 \text{ cm}$ apart. Thepower of this combination is:
Important Questions from Optics
Consider following statements for refraction of light through prism, when angle of deviation is minimum.
A. The refracted ray inside prism becomes parallel to the base.
B. Larger angle prisms provide smaller angle of minimum deviation.
C. Angle of incidence and angle of emergence becomes equal.
D. There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.
E. Angle of refraction becomes double of prism angle. Choose the correct answer from the options given below:
The radii of curvature for a thin convex lens are $10 \ cm$ and $15 \ cm$ respectively. The focal length of the lens is $12 \ cm$. The refractive index of the lens material is
The work function of a metal is $3 \ eV$. The color of the visible light that is required to cause emission of photoelectrons is
In the figure shown below, a resistance of $150.4 \Omega$ is connected in series to an ammeter A of resistance $240 \Omega$. A shunt resistance of $10 \Omega$ is connected in parallel with the ammeter. The reading of the ammeter is __________ mA.
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is :
