Ray Optics Chapter-Wise Test 8

A convex mirror always produces a virtual, erect, and diminished image, regardless of the object’s position. This provides a wide field of view, allowing drivers to see a larger area behind the vehicle, making it ideal for rear-view mirrors despite the reduced image size.

Magnifying power: \( m = \frac{f_o}{f_e} \).

\( f_o = 120 \, \text{cm} \), \( f_e = 4 \, \text{cm} \).

\( m = \frac{120}{4} = 30 \).

Apparent depth = \( \frac{\text{real depth}}{n} \).

Real depth = \( 16 \, \text{cm} \), \( n = 1.33 \).

Apparent depth = \( \frac{16}{1.33} \approx 12.03 \, \text{cm} \).

Optical fibers maintain light transmission when bent because the angle of incidence at the core-cladding interface remains greater than the critical angle, ensuring total internal reflection. This allows light to follow the bend without escaping into the cladding.

Snell’s law: \( n_1 \sin i = n_2 \sin r \).

Glass (\( n_1 = 1.5 \)), air (\( n_2 = 1 \)), \( i = 40^\circ \).

\( 1.5 \times \sin 40^\circ = 1 \times \sin r \).

\( \sin 40^\circ \approx 0.643 \Rightarrow 1.5 \times 0.643 \approx 0.964 \Rightarrow \sin r = 0.964 \).

\( r = \sin^{-1}(0.964) \approx 74.6^\circ \).

Critical angle: \( \sin i_c = \frac{1}{1.5} \approx 0.667 \), \( \sin 40^\circ < 0.667 \), so refraction occurs.

Inside the focal point, a convex lens diverges rays, forming a virtual image on the same side. Beyond the focal point, the lens converges rays to a point on the opposite side, forming a real image. This transition occurs as the object crosses the focal point, changing the ray behavior.

In a prism, minimum deviation occurs when the angle of incidence equals the angle of emergence. This symmetry ensures the refracted ray inside the prism is parallel to the base, minimizing the deviation angle.

Magnifying power: \( m = \frac{f_o}{f_e} \).

\( f_o = 200 \, \text{cm} \), \( f_e = 10 \, \text{cm} \).

\( m = \frac{200}{10} = 20 \).

Critical angle: \( \sin i_c = \frac{n_2}{n_1} = \frac{1}{1.62} \approx 0.617 \).

\( i_c = \sin^{-1}(0.617) \approx 38.1^\circ \).

At critical angle, angle of refraction = \( 90^\circ \).

In a reflecting telescope (e.g., Cassegrain design), a secondary mirror redirects the light focused by the primary concave mirror through a hole in the primary mirror to the eyepiece. This allows a compact design with a long focal length, improving magnification and convenience of observation.

Focal length: \( f = -25 \, \text{cm} \) (concave lens).

Object distance: \( u = -50 \, \text{cm} \).

Lens formula: \( \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \).

\( \frac{1}{v} - \frac{1}{-50} = \frac{1}{-25} \Rightarrow \frac{1}{v} + \frac{1}{50} = \frac{1}{-25} \Rightarrow \frac{1}{v} = \frac{1}{-25} - \frac{1}{50} = \frac{-2 - 1}{50} = \frac{-3}{50} \).

\( v = -\frac{50}{3} \approx -16.67 \, \text{cm} \) (virtual image).

Magnifying power: \( m = \frac{f_o}{f_e} \).

\( f_o = 100 \, \text{cm} \), \( f_e = 5 \, \text{cm} \).

\( m = \frac{100}{5} = 20 \).

Lens maker’s formula: \( \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \).

\( n = 1.5 \), \( R_1 = 15 \, \text{cm} \), \( R_2 = -15 \, \text{cm} \).

\( \frac{1}{f} = (1.5 - 1) \left( \frac{1}{15} - \frac{1}{-15} \right) = 0.5 \left( \frac{1}{15} + \frac{1}{15} \right) = 0.5 \times \frac{2}{15} = \frac{1}{15} \).

\( f = 15 \, \text{cm} \).

The focal point of a convex mirror is virtual and behind the mirror because reflected rays diverge and appear to originate from this point when traced backward. This indicates the mirror’s diverging nature, ensuring all images are virtual, erect, and diminished.

A periscope uses total internal reflection in prisms (or plane mirror reflection) to redirect light. Prisms reflect light internally at angles greater than the critical angle, allowing the viewer to see objects from a different height or direction without direct line-of-sight.