Assertion (A): A prism of refracting angle \(60^{\circ}\) is made of a material of refractive index \(\sqrt{2}\) for a certain wavelength. As light of this wavelength passes through the prism, the prism, angle of minimum deviation is \(30^{\circ}\).
Reason (R): At minimum deviation, angle of refraction of the first face is \(r_1 = A/2 = 30^{\circ}\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
For a prism at minimum deviation, the angle of refraction at the first face is \(r_1 = A/2\), where \(A\) is the prism angle. Given \(A = 60^{\circ}\), \(r_1 = 30^{\circ}\). This makes Reason (R) true. Using the prism formula \(n = \frac{sin((A + \delta_m)/2)}{sin(A/2)}\), with \(n=\sqrt{2}\) and \(A=60^{\circ}\), we find \(\delta_m = 30^{\circ}\). Thus, Assertion (A) is also true. Reason (R) provides a key condition used in calculating the minimum deviation, hence it is the correct explanation for (A).
Assertion (A): Minimum distance between a real object and its real image formed by a convex lens is three times the focal length of lens.
Reason (R): Distance between an object and its real image formed by convex lens is minimum when magnification produced by the lens has minimum value.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
For a convex lens, the minimum distance between a real object and its real image is \(4f\), occurring when the object is placed at \(2f\). At this point, the magnification is \(|m|=1\). The minimum magnitude of magnification approaches zero as the object moves to infinity. Hence, both Assertion (A) and Reason (R) are false.
Assertion (A): A concave mirror and a concave lens have the same focal length in air. When dipped in water, the focal length of the two are equal.
Reason (R): The focal length depends only on the radii of curvature.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The focal length of a mirror \(f_m = R/2\) depends only on its radius of curvature and is independent of the surrounding medium. The focal length of a lens \(1/f_l = (n_l/n_m - 1)(1/R_1 - 1/R_2)\) depends on the refractive indices of the lens material (\(n_l\)) and the surrounding medium (\(n_m\)), as well as radii of curvature. Thus, Assertion (A) is false as their focal lengths are generally not equal and lens focal length changes with medium. Reason (R) is false as lens focal length also depends on refractive indices.
Assertion (A): When a glass prism is immersed in water, the deviation caused by prism decrease.
Reason (R): Refractive index of glass prism relative to water is less than relative to air.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Deviation is given by (delta = (mu-1)A), where (mu), is the relative refractive index. When immersed in water, (mu_{gw} = mu_g/mu_w), which is less than (mu_{ga} = mu_g), (refractive index w.r.t. air). A smaller (mu), leads to a smaller deviation. Both assertion and reason are true, and R correctly explains A.
Assertion (A): When a light wave travels from a rarer to a denser medium, it loses energy.
Reason (R): When a light wave travels from a rarer to a denser medium, it loses speed and energy carried by the wave is proportional to its speed.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The energy of a light wave depends on its frequency \(E=h\nu\), which remains constant when light travels between different media. While the speed changes, the energy does not. Therefore, both assertion and reason are false.
Assertion (A): When a monochromatic beam of light is incident from one medium to another, the frequency of incident, reflected and refracted rays are equal.
Reason (R): The incident, reflected and refracted rays are co-planar.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The frequency of light is an intrinsic property of the source and remains constant upon changing media, so A is true. According to laws of reflection and refraction, the incident, reflected, and refracted rays lie in the same plane as the normal, so R is true. However, R does not explain A.
Assertion (A): The refractive index of diamond is \(\sqrt{6}\), and that of liquid is \(\sqrt{3}\), . If the light travels from diamond to the liquid, it will totally reflected when the angle of incidence is \(30^\circ\).
Reason (R): For total internal reflection, angle of incidence should be less than critical angle \(\theta_c = sin^{-1}(\frac{1}{\mu})\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The critical angle \(\theta_c = sin^{-1}(\mu_{liquid}/\mu_{diamond}) = sin^{-1}(\sqrt{3}/\sqrt{6}) = sin^{-1}(1/\sqrt{2}) = 45^\circ\), . For TIR, the angle of incidence (i), must be greater than \(\theta_c), (i.e., (i > 45^\circ),\). Since \(i = 30^\circ\), is not greater than \(45^\circ\), A is false. Reason R is also false because TIR occurs when \(i > \theta_c\).
Assertion (A): The speed of light in an optically rarer medium is greater than that in an optically denser medium.
Reason (R): One light year equals to \(9.5 \times 10^{12}\), km.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
An optically rarer medium has a lower refractive index, meaning light travels faster in it, so A is true. A light-year is the distance light travels in one year, approximately \(9.46 \times 10^{12}\), km, so R is true. However, the definition of a light-year does not explain the speed of light in different media.
Assertion (A): Consider a prism A of refracting angle \(5^0\), and another prism B of refracting angle \(10^0\), . Both prisms are made of crown glass. If white light is incident on each prism, angular dispersion caused by prism B will be more.
Reason (R): Dispersive power depends on the nature material.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Angular dispersion is given by \(\delta = (\mu_v - \mu_r)A\), . Since both prisms are of the same material, \((\mu_v - \mu_r\),), is constant. Prism B has a larger angle (A), \((10^\circ\), vs \(5^\circ\),), so it will cause more angular dispersion, making A true. Dispersive power is indeed a property of the material, so R is true. However, R does not explain why prism B has more dispersion than A, which is due to its larger refracting angle.