Young's Double Slit: Diffraction Pattern Analysis
This problem involves analyzing the interference pattern from a Young's double slit within the diffraction pattern of a single slit.
Given Parameters
- Slit separation, $d = 1.5 \text{ mm} = 1.5 \times 10^{-3} \text{ m}$
- Screen distance, $D = 2 \text{ m}$
- Wavelength, $\lambda = 400 \text{ nm} = 400 \times 10^{-9} \text{ m} = 4 \times 10^{-7} \text{ m}$
- Number of double-slit maxima ($N$) within the single-slit central maximum is 20.
- Unknown: Width of each slit, $a$, expressed as $x \times 10^{-3} \text{ cm}$.
Relevant Physics Principles
- The position ($y_n$) of the $n^{th}$ bright fringe (maxima) in a double-slit experiment is given by: $y_n = \frac{n \lambda D}{d}$
- The position ($y_{min}$) of the first minimum (edges of the central maximum) in a single-slit diffraction pattern is given by: $y_{min} = \frac{\lambda D}{a}$
Condition for Maxima Containment
The condition states that 20 double-slit maxima are contained within the central maximum of the single-slit diffraction pattern. This implies that the maxima range from $n=-10$ to $n=+10$. The central maximum of the single slit pattern extends to its first minima. Therefore, the $10^{th}$ double-slit maximum ($n=10$) must lie within or exactly at the position of the first minimum of the single-slit pattern.
Mathematically, this condition is expressed as:
$y_{10} \leq y_{min}$ $ \frac{10 \lambda D}{d} \leq \frac{\lambda D}{a} $Calculating Slit Width (a)
Simplifying the inequality, we get:
$ \frac{10}{d} \leq \frac{1}{a} $This leads to:
$ a \leq \frac{d}{10} $Assuming the boundary case where the $10^{th}$ maximum lies exactly at the first minimum:
$ a = \frac{d}{10} $Substitute the value of $d$:
$ a = \frac{1.5 \text{ mm}}{10} = 0.15 \text{ mm} $Unit Conversion
The question asks for the slit width in the format $x \times 10^{-3}$ cm. First, convert the calculated slit width $a$ from mm to cm:
$ a = 0.15 \text{ mm} = 0.15 \times \frac{1}{10} \text{ cm} = 0.015 \text{ cm} $Now, express $0.015$ cm in the desired format $x \times 10^{-3}$ cm:
$ a = 0.015 \text{ cm} = 15 \times 10^{-3} \text{ cm} $Determining x
Comparing $15 \times 10^{-3} \text{ cm}$ with the given format $x \times 10^{-3} \text{ cm}$, we find:
$ x = 15 $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)- If the measured angular separation between the second minimum to the left of the central maximum and the third minimum to the right of the central maximum is $30^\circ$ in a single slit diffraction pattern recorded using 628 nm light, then the width of the slit is _________ $\mu$m.
- 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 :
- 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 :
