Shortcut Methods

Shortcut Methods and Tricks to Solve JEE Main Numericals

1. A glass prism of refractive index 1.52 is immersed in water of refractive index 1.33. A light ray is incident on the prism at an angle of 45 degrees. Calculate the angle of emergence from the prism.

Shortcut Method: Use Snell’s law at each surface of the prism. At the first surface: $$\sin i_1 = n_2 \sin r_1 \Rightarrow \sin 45^\circ = 1.33 \sin r_1 \Rightarrow r_1 = 32.9^\circ$$ At the second surface: $$\sin r_2 = n_1 \sin i_2 \Rightarrow \sin r_2 = 1.52 \sin 32.9^\circ \Rightarrow r_2 = 41.1^\circ$$
Trick: The sum of the incident angle and the emergent angle is equal to the angle of deviation, which is given by: $$\delta = i_1 + e - A \Rightarrow \delta = 45^\circ + 41.1^\circ - 60^\circ = 26.1^\circ$$ Therefore, the angle of emergence is $$e = 41.1^\circ$$

2. A light ray passes from air into a glass block of refractive index 1.52 at an angle of incidence of 42 degrees. Calculate the angle of refraction.

Shortcut Method: Use Snell’s law: $$\sin i_1 = n_2 \sin r_2 \Rightarrow \sin 45^\circ = 1.52 \sin r_1 \Rightarrow r_1 = 26.5^\circ$$
Trick: The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the refractive index of the second medium relative to the first. $$\frac{\sin i_1}{\sin r_1} = n_2 \Rightarrow \frac{\sin 45^\circ}{\sin 26.5^\circ} =1.52. $$

3. A parallel beam of light is incident on a convex lens of focal length 10 cm. Calculate the distance between the lens and the screen when a sharp image is formed on the screen.

Shortcut Method: Use the lens formula: $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \Rightarrow \frac{1}{10} = \frac{1}{\infty} + \frac{1}{v}$$ Solving for v, we get $$v = 10 cm.$$
Trick: For a parallel beam of light, the object is at infinity, so the distance between the lens and the screen is equal to the focal length of the lens.

4. A point source of light is placed at a distance of 20 cm in front of a concave mirror of focal length 15 cm. Calculate the position and nature of the image formed.

Shortcut Method: Use the mirror formula: $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \Rightarrow \frac{1}{-15} = \frac{1}{-20} + \frac{1}{v}$$ Solving for v, we get $$v = -60 cm.$$
Trick: The negative sign of the image distance indicates that the image is formed behind the mirror and it’s virtual.

5. A telescope has an objective lens of focal length 120 cm and an eyepiece of focal length 5 cm. Calculate the magnifying power of the telescope.

Shortcut Method: The magnifying power of a telescope is given by: $$M = \frac{f_o}{f_e} = \frac{120 \text{ cm}}{5 \text{ cm}} = 24$$
Trick: Telescopes have large objective lenses to gather more light and small eyepiece lenses to magnify the image.

6. A compound microscope has an objective lens of focal length 4 mm and an eyepiece of focal length 2 cm. The distance between the objective lens and the eyepiece is 16 cm. Calculate the magnifying power of the microscope.

Shortcut Method: The magnifying power of a compound microscope is given by: $$M = \frac{L (f_o + f_e)}{f_o f_e} = \frac{16 \text{ cm} (4 \text{ mm} + 2 \text{ mm})}{4 \text{ mm} \times 2 \text{ mm}} = 300$$
Trick: Compound microscopes have multiple lenses to achieve high magnification.

7. A diffraction grating has 1000 lines/mm. Calculate the angle of diffraction for a light of wavelength 600 nm.

Shortcut Method: The angle of diffraction is given by: $$\sin \theta = \frac{n \lambda}{d}$$ where n is the number of lines per mm, λ is the wavelength of light, and d is the distance between the slits. Substituting the given values, we get: $$\sin \theta = \frac{1000 \times 600 \times 10^{-9} \text{ m}}{1 \times 10^{-3} \text{ m}} = 0.6$$ $$\Rightarrow \theta = 36.9^\circ$$
Trick: Diffraction gratings are used to separate light into its component wavelengths.

8. A double-slit experiment is performed using light of wavelength 500 nm. The distance between the slits is 0.1 mm and the distance between the slits and the screen is 1 m. Calculate the distance between the central bright fringe and the first dark fringe.

Shortcut Method: The distance between the central bright fringe and the first dark fringe is given by: $$x = \frac{\lambda L}{d}$$ where λ is the wavelength of light, L is the distance between the slits and the screen, and d is the distance between the slits. Substituting the given values, we get: $$x = \frac{500 \times 10^{-9} \text{ m} \times 1 \text{ m}}{0.1 \times 10^{-3} \text{ m}} = 5 \text{ mm}$$

CBSE Board Exams

1. Define total internal reflection.

Shortcut Method: Total internal reflection occurs when light traveling from a denser medium to a rarer medium is reflected back into the denser medium because the angle of incidence is greater than the critical angle.
Trick:TIR occurs when the angle of incidence is greater than the critical angle.

2. State the conditions for total internal reflection.

Shortcut Method:
The light must travel from a denser medium to a rarer medium.
The angle of incidence must be greater than the critical angle.

3. Derive the formula for the critical angle of a medium.

Shortcut Method: The critical angle is the angle of incidence at which the refracted angle becomes 90°. Using Snell’s law: $$n_1 \sin i = n_2 \sin 90^\circ$$ $$n_1 \sin i = n_2$$ $$\sin i = \frac{n_2}{n_1}$$ $$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$
Trick: Remember, the critical angle is always less than 90°.

4. Explain the phenomenon of total internal reflection and its applications.

Shortcut Method:TIR occurs when light undergoes total internal reflection due to the differences in refractive indices of the media. It has applications such as:
- Prisms and mirrors
- Optical fibers
- Periscopes
- Fiber optics communication
Trick: TIR finds practical application in fiber optic communication due to minimum signal loss.

5. Describe the working of an optical fiber based on the principle of total internal reflection.

Shortcut Method:
- An optical fiber consists of a core and a cladding with different refractive indices.
- Light signals are transmitted through the core by multiple internal reflections.
- The critical angle is utilized to ensure total internal reflection and efficient transmission of light.
Trick: The numerical aperture of an optical fiber is a measure of its light-gathering capacity.

6. Explain the construction and working of a prism.

Shortcut Method:
- A prism is a transparent optical device made of materials like glass or plastic.
- It consists of two plane faces inclined at a specific angle.
- When light strikes the prism, it undergoes refraction (bending) as it enters and exits the prism.
- The angle of deviation is the angle between the incident and emergent rays, allowing for dispersion (separation) of light into different colors or wavelengths.
Trick: Refractive index is directly proportional to the deviation of light through a prism.

**7. Discuss the various types