Prism Parameters and Given Condition
The problem involves an equilateral prism, which means the prism angle is $A = 60^\circ$. The refractive index of the prism is given as $n = \sqrt{2}$. A ray of light is incident on one surface, and the emergent ray grazes along the other surface. This specific condition implies that the angle of emergence is $e = 90^\circ$.
Snell's Law at the Emergent Surface
Let $r_2$ be the angle of refraction inside the prism at the second surface. According to Snell's Law at the point where the ray exits the prism:
$ n_{prism} \sin(r_2) = n_{air} \sin(e) $
Substituting the known values ($n_{prism} = \sqrt{2}$ and $e = 90^\circ$):
$ \sqrt{2} \sin(r_2) = 1 \times \sin(90^\circ) $
Since $\sin(90^\circ) = 1$:
$ \sqrt{2} \sin(r_2) = 1 $
Calculating Second Refraction Angle ($r_2$)
Solving for $\sin(r_2)$:
$ \sin(r_2) = \frac{1}{\sqrt{2}} $
Therefore, the angle $r_2$ is:
$ r_2 = 45^\circ $
Prism Angle Relationship
For any prism, the prism angle $A$ is related to the angles of refraction at the first ($r_1$) and second ($r_2$) surfaces by the formula:
$ A = r_1 + r_2 $
Calculating First Refraction Angle ($r_1$)
We need to find the angle of refraction at the incident surface, which is $r_1$. Using the prism angle formula and the calculated value of $r_2$:
$ 60^\circ = r_1 + 45^\circ $
Solving for $r_1$:
$ r_1 = 60^\circ - 45^\circ $
$ r_1 = 15^\circ $
Final Answer
The angle of refraction ($r_1$) at the incident surface is $15^\circ$.
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 = $4/3$) - 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.
- Which of the following are true for a single slit diffraction ?
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. - 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 = $4/3$) - 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.
