JEE Mains · Physics · STD 12 - Ray optics and optical instruments
- JEE Mains 2022
- Hard
- Numerical
The \(X-Y\) plane be taken as the boundary between two transparent media \(M _{1}\) and \(M _{2} . M _{1}\) in \(Z \geq 0\) has a refractive index of \(\sqrt{2}\) and \(M _{2}\) with \(Z <0\) has a refractive index of \(\sqrt{3}\). A ray of light travelling in \(M_{1}\) along the direction given by the vector \(\overrightarrow{ A }=4 \sqrt{3 i }-3 \sqrt{3 \hat{j}}-5 \hat{ k }\), is incident on the plane of separation. The value of difference between the angle of incident in \(M _{1}\) and the angle of refraction in \(M _{2}\) will be \(....\) degree.
Pick an option to check your answer.
Answer & Solution
Correct Answer
(B) \(15\)
Step-by-step Solution
Detailed explanation
\(\overrightarrow{ A }=4 \sqrt{3 \hat{i}}-3 \sqrt{3 \hat{j}}-5 \hat{ k }\)
As incident vector \(A\) makes \(i\) angle with normal \(z-\)axis refracted vector \(R\) makes \(r\) angle with normal \(z\) - axis with help of direction cosine
\(i=\cos ^{-1}\left(\frac{A_{z}}{A}\right)=\cos ^{-1}\left(\frac{5}{\sqrt{(4 \sqrt{3})^{2}+(3 \sqrt{3})^{2}+5^{2}}}\right)\)
\(=\cos ^{-1}\left(\frac{5}{10}\right) \Rightarrow i=60^{\circ}\)
\(\sqrt{2} \sin 60=\sqrt{3} \times \sin r\)
\(r=45^{\circ}\)
Difference between \(i \) & \(r =60-45=15\)
More questions from Physics
- Consider a binary star system of star \(A\) and star \(B\) with masses \(m_{A}\) and \(m_{B}\) revolving in a circular orbit of radii \(r_{A}\) and \(r_{B}\), respectively. If \(T_{A}\) and \(T_{B}\) are the time period of star \(A\) and star \(B\), respectively, then -JEE Mains 2021 Medium
- The half-life of \(Au ^{198}\) is \(2.7\, days.\) The activity of \(1.50\, mg\) of \(Au ^{198}\) if its atomic weight is \(198\, g mol ^{-1}\) is .......\(Ci\) \(\left( N _{ A }=6 \times 10^{23}\, / mol \right)\)JEE Mains 2021 Hard
- In two separate Young's double-slit experimental set-ups, two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slit separations and that of the wavelengths of light used are \(2: 1\) and \(1: 2\) respectively. The corresponding ratio of the distances between the slits and the respective screens ( \(D_1 / D_2\) ) is ___________ .JEE Mains 2026 Easy
- The internal energy of a monoatomic gas is 3 nRT . One mole of helium is kept in a cylinder having internal cross section area of \(17 cm^2\) and fitted with a light movable frictionless piston. The gas is heated slowly by suppling 126 J heat. If the temperature rises by \(4^{\circ} C\), then the piston will move ___________ cm.
(atmospheric pressure \(=10^5 Pa\) )JEE Mains 2026 Medium - A three coulomb charge moves from the point \((0, -2, -5)\) to the point \((5, 1, 2)\) in an electric field expressed as \(\vec{E} = 2x\hat{i} + 3y^2\hat{j} + 4\hat{k}\) N/C. The work done in moving the charge is _______ J.JEE Mains 2026 Hard
- The reading of the ammeter (A) in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is _________ A.
JEE Mains 2026 Easy
More PYQs from JEE Mains
- A capacitor of capacitance \(100\ \mu \mathrm{F}\) is charged to a potential of \(12 \mathrm{~V}\) and connected to a \(6.4 \ \mathrm{mH}\) inductor to produce oscillations. The maximum current in the circuit would be :JEE Mains 2024 Hard
- The energy associated with electric field is \((U_E)\) and with magnetic field is \((U_B)\) for an electromagnetic wave in free space. ThenJEE Mains 2019 Medium
- A parallel plate capacitor with plate separation 5 mm is charged by a battery. On introducing a mica sheet of 2 mm and maintaining the connections of the plates with the terminals of the battery, it is found that it draws \(25 \%\) more charge from the battery. The dielectric constant of mica is ___________ .JEE Mains 2026 Easy
- \(2\, kg\) of a monoatomic gas is at a pressure of \(4\times10^4\, N/m^2\). The density of the gas is \(8\, kg/m^3\). What is the order of energy of the gas due to its thermal motion ?JEE Mains 2019 Medium
- A magnet hung at \(45^{\circ}\) with magnetic meridian makes an angle of \(60^{\circ}\) with the horizontal. The actual value of the angle of dip is.JEE Mains 2022 Medium