Light – Reflection and Refraction

Solution

The laws of reflection state that (i) the angle of incidence is equal to the angle of reflection, and (ii) the incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane. These laws are applicable to all types of reflecting surfaces, including spherical mirrors. For spherical mirrors, the reflection occurs at the curved surface, and the laws help in determining the path of light rays after reflection. The first law ensures that the direction of the reflected ray can be predicted if the incident ray's direction is known. The second law ensures that all rays lie in a plane perpendicular to the mirror's surface at the point of incidence. These principles are used in designing mirrors for various applications, such as in telescopes and headlights, where precise control over light direction is necessary. For example, concave mirrors are used in headlights to focus light into a parallel beam, enhancing visibility. Understanding these laws is crucial for solving problems related to image formation by spherical mirrors.

Solution

The nature, position, and size of the image formed by a concave mirror depend on the object's position relative to the mirror's focal point (F) and center of curvature (C). (1) When the object is at infinity, the image is formed at F, highly diminished, real, and inverted. (2) When the object is beyond C, the image is between F and C, diminished, real, and inverted. (3) At C, the image is at C, same size, real, and inverted. (4) Between C and F, the image is beyond C, enlarged, real, and inverted. (5) At F, the image is at infinity, highly enlarged, real, and inverted. (6) Between F and the mirror, the image is behind the mirror, enlarged, virtual, and erect. These variations occur due to the mirror's ability to converge light rays. For instance, in solar furnaces, concave mirrors focus sunlight to a point to generate high temperatures. Understanding these positions helps in applications like shaving mirrors, where a virtual and erect image is desired.

Solution

A convex lens converges light rays, whereas a concave lens diverges them. Convex lenses are thicker at the center and bend light rays inward, meeting at a focal point. They form real or virtual images depending on the object's position. For example, when the object is beyond the focal length, a real, inverted image is formed. Concave lenses are thinner at the center and spread out light rays, making them appear to diverge from a focal point. They always form virtual, erect, and diminished images. Convex lenses are used in magnifying glasses and cameras, while concave lenses are used in correcting myopia. The refractive index and lens thickness determine the degree of convergence or divergence. Understanding these differences is essential for designing optical instruments and correcting vision defects.

Solution

The refractive index of a medium measures how much it slows down light compared to vacuum. It's defined as the ratio of the speed of light in vacuum to its speed in the medium (n = c/v). A higher refractive index means greater light bending. For example, diamond's high refractive index (2.42) causes significant light dispersion, creating sparkle. The refractive index determines the angle of refraction when light passes between media, governed by Snell's Law (n1 sinθ1 = n2 sinθ2). This principle is crucial in lens design, fiber optics, and understanding phenomena like mirages. Different wavelengths have slightly different refractive indices in the same medium, causing dispersion (rainbows). Knowing refractive indices helps in selecting materials for specific optical applications, like eyeglasses or camera lenses.

Solution

When an object is placed at twice the focal length (2F) of a convex lens, the image is formed at 2F on the opposite side. The image is real, inverted, and the same size as the object. This occurs because light rays from the object converge symmetrically after refraction. A ray parallel to the principal axis passes through the focal point, and a ray through the optical center continues straight. Their intersection determines the image position. This setup is used in photocopiers to produce life-size copies. The lens formula (1/f = 1/v - 1/u) confirms this when u = 2f, leading to v = 2f. Magnification (m = v/u) is -1, indicating same size but inverted. Understanding this helps in designing optical systems requiring 1:1 imaging.

Solution

Concave mirrors are used where light focusing or magnification is needed. In torches and headlights, they reflect light into a strong parallel beam. As shaving mirrors, they provide magnified, erect virtual images when the object is within the focal length. Dentists use them to see enlarged tooth images. Solar furnaces employ large concave mirrors to concentrate sunlight for high heat. Convex mirrors, offering a wider field of view, are ideal as rear-view mirrors in vehicles, showing traffic behind. They're also used at road bends and driveways for safety. Shop security often uses convex mirrors for broad surveillance. The diverging nature of convex mirrors always gives diminished, erect images, ensuring more area coverage. Understanding these applications highlights the importance of mirror curvature in practical designs.

Solution

Total internal reflection (TIR) occurs when light travels from a denser to a rarer medium at an angle greater than the critical angle, causing complete reflection back into the denser medium. The critical angle depends on the media's refractive indices (sinθc = n2/n1). TIR is utilized in optical fibers, where light signals are transmitted over long distances with minimal loss, enabling high-speed internet and telecommunications. It's also used in binoculars, periscopes, and endoscopes for efficient light guidance. Diamond cutting exploits TIR to enhance brilliance by ensuring light entering the diamond reflects multiple times before exiting. Understanding TIR principles is essential for designing devices that require efficient light transmission without leakage.

Solution

A rainbow forms when sunlight is refracted, dispersed, and reflected inside water droplets. As white light enters a droplet, it slows down and bends (refraction), separating into different colors due to varying wavelengths (dispersion). Violet bends the most, red the least. The light then reflects off the droplet's inner surface and refracts again upon exiting, further spreading the colors. This double refraction and single reflection create the rainbow's arc. The observer sees different colors from droplets at specific angles (42° for red, 40° for violet). Rainbows appear circular from the air but are typically seen as arcs from the ground. Understanding this process explains why rainbows appear opposite the sun and require water droplets in the atmosphere.

Solution

The power (P) of a lens is the reciprocal of its focal length (f) in meters: P = 1/f. For a focal length of 25 cm (0.25 m), P = 1/0.25 = +4 D. The positive sign indicates a convex (converging) lens. Lens power measures its light-bending ability, with higher diopter values indicating stronger refraction. Convex lenses are used to correct hypermetropia (farsightedness) by converging light onto the retina. Understanding lens power is crucial for prescribing corrective glasses and designing optical instruments. For example, a +4 D lens would be prescribed if the eye's focal length is too long, causing near objects to appear blurry.

Solution

A concave lens diverges light rays, causing them to spread out as if they originated from a point on the same side as the object. Since the rays don't actually converge, no real image forms. Instead, the diverging rays' extensions meet to form a virtual, erect, and diminished image. This occurs because the lens's shape causes light to bend outward, making it impossible for rays to meet on the opposite side. For any object distance, the image appears between the lens and its focal point. This property makes concave lenses suitable for correcting myopia, where diverging light rays need to be adjusted to focus properly on the retina. Understanding this helps in applications requiring image size reduction without inversion.

Solution

The laws of reflection state that (i) the angle of incidence is equal to the angle of reflection, and (ii) the incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane. For refraction, Snell's law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media. These laws apply to spherical mirrors and lenses by governing how light rays are reflected or refracted to form images. Diagrams should show incident, reflected, and refracted rays with angles marked.

Solution

Concave mirrors can form real and inverted images when the object is placed beyond the focus, and virtual and erect images when the object is placed between the focus and the mirror. Convex mirrors always form virtual, erect, and diminished images regardless of the object's position. A table comparing image characteristics (nature, position, size) for various object positions would illustrate the differences clearly.

Solution

The mirror formula is 1/v + 1/u = 1/f, where v is the image distance, u is the object distance, and f is the focal length. The sign convention follows the New Cartesian system: (i) distances measured in the direction of incident light are positive, (ii) distances measured opposite are negative, (iii) heights above the principal axis are positive, and (iv) heights below are negative. This convention helps in determining the nature and position of the image.

Solution

Given: u = -10 cm, m = -3 (since the image is real and inverted). Using m = -v/u, we get v = 30 cm. Substituting in the mirror formula: 1/f = 1/v + 1/u = 1/30 + 1/(-10) = (1 - 3)/30 = -2/30 = -1/15. Thus, f = -15 cm. The negative sign indicates the mirror is concave.

Solution

Total internal reflection occurs when light travels from a denser to a rarer medium at an angle greater than the critical angle, causing the light to be completely reflected back into the denser medium. Examples include mirages and sparkling of diamonds. In optical fibers, this principle is used to transmit light signals over long distances with minimal loss, as the light reflects repeatedly inside the fiber.

Solution

For a real and inverted image of the same size, the object must be at 2F. Thus, u = 50 cm. Using the lens formula: 1/f = 1/v - 1/u = 1/50 - 1/(-50) = 2/50 = 1/25. So, f = 25 cm = 0.25 m. Power P = 1/f = 1/0.25 = +4 D.

Solution

Refractive indices: kerosene (~1.44), water (~1.33), diamond (~2.42). The speed of light is inversely proportional to the refractive index (v = c/n). Thus, light travels fastest in water (lowest n), slower in kerosene, and slowest in diamond (highest n).

Solution

Given: u = -10 cm, f = +15 cm. Using the mirror formula: 1/v + 1/u = 1/f => 1/v = 1/15 - 1/(-10) = 1/15 + 1/10 = (2 + 3)/30 = 5/30 = 1/6. Thus, v = +6 cm. The positive sign indicates the image is virtual and erect, located 6 cm behind the mirror.

Solution

A compound microscope uses two convex lenses: the objective (near the object) and the eyepiece (near the eye). The objective forms a real, inverted, and enlarged image of the object, which serves as the object for the eyepiece. The eyepiece further magnifies this image, producing a virtual and enlarged final image. Higher magnification is achieved by using lenses with shorter focal lengths and adjusting the distance between the lenses.

Solution

Given: u = -4 m, f = -2 m. Using the lens formula: 1/v - 1/u = 1/f => 1/v = 1/f + 1/u = 1/(-2) + 1/(-4) = -0.5 - 0.25 = -0.75. Thus, v = -1.33 m. The negative sign indicates the image is virtual and erect, located 1.33 m on the same side as the object.

Solution

A concave mirror is used in solar furnaces because it converges sunlight to a focal point, generating high temperatures suitable for cooking or melting materials. In headlights, it diverges light from the bulb placed at its focus to produce a powerful parallel beam, enhancing visibility. The difference lies in the mirror's curvature and the position of the light source relative to the focal point.

Solution

The size of the image will be equal to the size of the object when the object is placed at twice the focal length (2F) of the convex lens. At this position, the image is also formed at 2F on the opposite side, inverted, and of the same size. This is because the magnification (m) equals 1, indicating no change in size.

Solution

When light passes from air to glass, it slows down due to the higher optical density of glass, bending towards the normal. The refractive index (n) of glass relative to air is the ratio of the speed of light in air to that in glass, indicating how much the light slows down. This change in speed and direction is crucial for designing lenses and optical instruments.

Solution

The pencil appears bent due to refraction at the air-water interface. Light from the submerged part travels from water (denser) to air (rarer), bending away from the normal. Our brain perceives light as traveling straight, making the submerged part appear displaced, hence the bent appearance.

Solution

Concave mirrors can form real or virtual images depending on the object's position relative to the focal point (F). Beyond F, images are real and inverted; between F and the mirror, images are virtual and erect. Convex mirrors always form virtual, erect, and diminished images regardless of the object's position, due to their diverging nature.

Solution

Using the mirror formula 1/f = 1/v + 1/u, with f = -10 cm (concave mirror) and u = -15 cm, solving gives v = -30 cm. The magnification m = -v/u = -2, indicating an inverted, real image twice the size of the object, located 30 cm from the mirror.

Solution

Diamond has a higher refractive index (2.42) than glass (~1.5), resulting in a smaller critical angle. This means light entering a diamond is more likely to undergo total internal reflection, enhancing its sparkle. The precise cutting of diamonds maximizes this effect by ensuring light reflects multiple times before exiting.

Solution

For a virtual image magnified twice (m = +2), using m = v/u and the lens formula 1/f = 1/v - 1/u, with f = 20 cm, we find u = -10 cm. The object must be placed within the focal length (10 cm from the lens) to produce a virtual, erect, and magnified image.

Solution

To determine the focal length, place the concave mirror facing a distant object (like the sun). Adjust a screen until a sharp image forms on it. The distance between the mirror and the screen is the focal length. Precautions include avoiding direct sunlight viewing to prevent eye damage and ensuring the mirror and screen are aligned properly.

Solution

The power (P) of a lens is the reciprocal of its focal length (f). Convex lenses converge light, have real focal lengths (positive), hence positive power. Concave lenses diverge light, have virtual focal lengths (negative), hence negative power. This reflects their ability to converge or diverge light, influencing the nature (real/virtual) of images formed.