Ray Optics Chapter-Wise Test 14

When light travels from a denser to a rarer medium (e.g., glass to air) and the angle of incidence exceeds the critical angle, total internal reflection occurs. This is because the refracted ray would otherwise require a sine value greater than 1, which is physically impossible.

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

Water (\( n_1 = 1.33 \)), glass (\( n_2 = 1.62 \)), \( i = 45^\circ \).

\( 1.33 \times \sin 45^\circ = 1.62 \times \sin r \).

\( \sin 45^\circ = 0.707 \Rightarrow 1.33 \times 0.707 = 1.62 \sin r \Rightarrow 0.941 = 1.62 \sin r \).

\( \sin r = \frac{0.941}{1.62} \approx 0.581 \Rightarrow r = \sin^{-1}(0.581) \approx 35.5^\circ \).

A convex mirror always forms a virtual, erect, and diminished image. As the object moves closer, the image size increases slightly but remains diminished (less than the object size), and the image distance increases, approaching the focal length as a limit, though it never exceeds it.

In a compound microscope, the objective lens forms a real, inverted image of the object. The eyepiece then acts as a magnifying lens, forming a virtual image of this inverted intermediate image. Since the eyepiece does not reinvert the image, the final image remains inverted relative to the original object.

\( f_1 = 60 \, \text{cm} \), \( f_2 = -30 \, \text{cm} \).

\( \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} = \frac{1}{60} + \frac{1}{-30} = \frac{1}{60} - \frac{2}{60} = \frac{1 - 2}{60} = \frac{-1}{60} \).

\( f = -60 \, \text{cm} \) (diverging system).

\( f = 20 \, \text{cm} \), \( u = -12 \, \text{cm} \).

\( \frac{1}{v} + \frac{1}{-12} = \frac{1}{20} \Rightarrow \frac{1}{v} = \frac{1}{20} + \frac{1}{12} = \frac{3 + 5}{60} = \frac{8}{60} = \frac{2}{15} \).

\( v = 7.5 \, \text{cm} \).

Magnification: \( m = -\frac{v}{u} = -\frac{7.5}{-12} = 0.625 \).

A concave lens diverges light rays, making them appear to originate from a point on the same side as the object when traced backward. This results in a virtual image that cannot be projected on a screen, always forming on the object’s side regardless of its position.

Focal length: \( f = -10 \, \text{cm} \) (concave lens).

Image distance: \( v = -5 \, \text{cm} \) (virtual image).

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

\( \frac{1}{-5} - \frac{1}{u} = \frac{1}{-10} \Rightarrow \frac{1}{u} = \frac{1}{-5} - \frac{1}{-10} = \frac{-2 + 1}{10} = \frac{-1}{10} \).

\( u = -10 \, \text{cm} \).

For a convex lens, when the object is at the focal point (F), the rays after refraction become parallel and do not converge to a point on the other side. The image is formed at infinity, as the rays appear to diverge from an infinitely distant point when traced backward.

When light travels from a denser medium (e.g., water) to a rarer medium (air), it bends away from the normal. This refraction makes the rays appear to diverge from a point closer to the surface than the actual object, reducing the apparent depth compared to the real depth.