A. Width of central maxima increases with increase in wavelength keeping slit width constant.
B. Width of central maxima increases with decrease in wavelength keeping slit width constant.
C. Width of central maxima increases with decrease in slit width at constant wavelength.
D. Width of central maxima increases with increase in slit width at constant wavelength.
E. Brightness of central maxima increases for decrease in wavelength at constant slit width.
Single Slit Diffraction Analysis
This question asks about the factors affecting the width and brightness of the central maxima in single-slit diffraction.
Diffraction Width Dependence
The width of the central maxima ($W$) in single-slit diffraction is determined by the positions of the first minima. The angular positions of the minima are given by the equation:
$ \sin \theta_m = \frac{m \lambda}{a} $
where $m = \pm 1, \pm 2, \dots$ is the order number, $\lambda$ is the wavelength of light, and $a$ is the slit width.
For the first minima ($m = \pm 1$), the angular positions are $\pm \frac{\lambda}{a}$. The width of the central maxima on a screen at distance $L$ is approximately:
$ W \approx L \tan \theta_1 \approx \frac{2 \lambda L}{a} $
From this formula, we can analyze the statements:
- Statement A: If wavelength ($\lambda$) increases and slit width ($a$) is constant, $W$ increases. This statement is True.
- Statement B: If wavelength ($\lambda$) decreases and slit width ($a$) is constant, $W$ decreases. This statement is False.
- Statement C: If slit width ($a$) decreases and wavelength ($\lambda$) is constant, $W$ increases ($\frac{1}{a}$ term). This statement is True.
- Statement D: If slit width ($a$) increases and wavelength ($\lambda$) is constant, $W$ decreases ($\frac{1}{a}$ term). This statement is False.
Diffraction Brightness Dependence
The peak intensity ($I_{max}$) of the central maxima in single-slit diffraction is approximately proportional to the square of the slit width:
$ I_{max} \propto a^2 $
The intensity distribution also depends on the wavelength, but the peak intensity itself, at the center, primarily depends on the slit width for a given incident light intensity.
- Statement E: If wavelength ($\lambda$) decreases and slit width ($a$) is constant, the peak intensity ($I_{max} \propto a^2$) remains constant. Therefore, the brightness of the central maxima does not increase solely due to the decrease in wavelength. This statement is False.
Conclusion
Based on the analysis, the true statements are A and C.
The correct option is the one that includes only statements A and C.
Similar Questions
- The magnitudes of power of a biconvex lens (refractive index $1.5$) and that of a plano-concave lens (refractive index = $1.7$) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is ________.
- Given below are two statements:
**Statement I:** A plane wave after passing through prism remains as plane wave but passing through small pin hole may become spherical wave.
**Statement II:** The curvature of a spherical wave emerging from a slit will increase for increasing slit width.
In the light of the above statements, choose the correct answer from the options given below - A convex lens of refractive index $1.5$ and focal length $f = 18 \text{ cm}$ is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is $\alpha \times f$. The value of $\alpha$ is ________.
(refractive index of water = $4/3$) - Consider an equilateral prism (refractive index $\sqrt{2}$). A ray of light is incident on its one surface at a certain angle $i$. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to _________.
- A thin convex lens of focal length 5 cm and a thin concave lens of focal length 4 cm are combined together (without any gap) and this combination has magnification $m_1$ when an object is placed 10 cm before the convex lens. Keeping the positions of convex lens and object undisturbed a gap of 1 cm is introduced between the lenses by moving the concave lens away, which lead to a change in magnification of total lens system to $m_2$. The value of $\left| \frac{m_1}{m_2} \right|$ is _________.
- A parallel beam of light travelling in air (refractive index 1.0) is incident on a convex spherical glass surface of radius of curvature 50 cm. Refractive index of glass is 1.5. The rays converge to a point at a distance x cm from the centre of the curvature of the spherical surface. The value of x is _________ cm.
- The wavelength of light, while it is passing through water is 540 nm. The refractive index of water is $4/3$. The wavelength of the same light when it is passing through a transparent medium having refractive index of $3/2$ is _________ nm.
- In parallax method for the determination of focal length of a concave mirror, the object should always be placed :
- A laser beam has intensity of $4.0 \times 10^{14} \text{ W/m}^2$. The amplitude of magnetic field associated with beam is _________ T. (Take $\varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2\text{/Nm}^2$ and $c = 3 \times 10^8 \text{ m/s}$)
- An unpolarised light is incident at an interface of two dielectric media having refractive indices of 2 (incident medium) and $2\sqrt{3}$ (medium) respectively. To satisfy the condition that reflected and refracted rays are perpendicular to each other, the angle of incidence is ______ .
Important Questions from Optics
- The magnitudes of power of a biconvex lens (refractive index $1.5$) and that of a plano-concave lens (refractive index = $1.7$) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is ________.
- Given below are two statements:
**Statement I:** A plane wave after passing through prism remains as plane wave but passing through small pin hole may become spherical wave.
**Statement II:** The curvature of a spherical wave emerging from a slit will increase for increasing slit width.
In the light of the above statements, choose the correct answer from the options given below - A convex lens of refractive index $1.5$ and focal length $f = 18 \text{ cm}$ is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is $\alpha \times f$. The value of $\alpha$ is ________.
(refractive index of water = $4/3$) - Consider an equilateral prism (refractive index $\sqrt{2}$). A ray of light is incident on its one surface at a certain angle $i$. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to _________.
- A thin convex lens of focal length 5 cm and a thin concave lens of focal length 4 cm are combined together (without any gap) and this combination has magnification $m_1$ when an object is placed 10 cm before the convex lens. Keeping the positions of convex lens and object undisturbed a gap of 1 cm is introduced between the lenses by moving the concave lens away, which lead to a change in magnification of total lens system to $m_2$. The value of $\left| \frac{m_1}{m_2} \right|$ is _________.
