Optics and Light Lecture Review

Reflection of Light Principles and Visual Representation

Reflection of light is defined as the bouncing back of light rays from a surface to the same medium after striking it. This phenomenon involves several key components. The incident ray is the light ray that strikes the surface, while the reflected ray is the ray that bounces off. The normal line is a conceptual perpendicular line drawn to the surface at the point of incidence. The angle of incidence, denoted as $\theta_i$, is the angle between the incident ray and the normal line. Correspondingly, the angle of reflection, denoted as $\theta_r$, is the angle between the reflected ray and the normal line.

The Laws of Reflection

The behavior of light during reflection is governed by two primary laws. The first law states that the angle of incidence is always equal to the angle of reflection, expressed mathematically as $\theta_i = \theta_r$. The second law of reflection dictates that the incident ray, the reflected ray, and the normal line at the point of incidence all lie in the same plane.

Image Formation in a Plane Mirror

An image formed by a plane mirror possesses several distinct characteristics. The image is the same size as the object and is located at the same distance behind the mirror as the object is in front of it. These images are laterally inverted, meaning left and right are reversed. Furthermore, the image is virtual, meaning it is formed due to diverging rays that appear to come from behind the mirror, rather than actual light rays converging.

In scenarios where two plane mirrors are arranged at a specific angle $\theta$, multiple images of an object placed between them will be formed. The number of images, denoted as $n$, can be calculated using the formula $n = \frac{360}{\theta} - 1$. For example, if the angle $\theta = 90^{\circ}$, then $n = \frac{360}{90} - 1 = 3$ images will be formed. If the angle $\theta = 0^{\circ}$ (mirrors are parallel), an infinite number of images are formed.

Refraction of Light and Snell's Law

Refraction is the bending of light that occurs when a ray of light travels from one medium to another, such as from air to water. This bending happens because the speed of light changes depending on the medium. In a vacuum, light travels at its maximum speed, $c = 3 \times 10^8\,\text{m/s}$. When light enters a denser medium, it slows down (v_a > v_w). The refractive index, denoted by $n$, is a dimensionless number that describes how much light slows down in a material compared to a vacuum. It is calculated as $n = \frac{c}{v_{\text{medium}}}$. A higher refractive index indicates a denser medium where light bends toward the normal line.

Snell's Law describes the mathematical relationship between the angles of incidence and refraction as light passes between media. It is expressed as $n_1 \sin(\theta_i) = n_2 \sin(\theta_r)$. For light passing from air (medium 1) to water (medium 2), the formula is $n_a \sin(\theta_i) = n_w \sin(\theta_r)$. It is professionally noted that as light passes from one medium to another, the frequency ($f$) remains constant ($f_a = f_w = \text{constant}$, while the speed ($v$) and wavelength ($\lambda$) change according to the equation $v = f \times \lambda$.

Quantitative Examples in Refraction

Consider a red light with a wavelength $\lambda_a = 7 \times 10^{-7}\,\text{m}$ traveling from air ($n_a = 1$) to water ($n_w = 4/3$). To find the speed of light in water ($v_w$), we use $v_w = \frac{v_a}{n_w} = \frac{3 \times 10^8}{4/3} = 2.25 \times 10^8\,\text{m/s}$. To find the wavelength in water ($\lambda_w$), we calculate $\lambda_w = \frac{\lambda_a}{n_w} = \frac{7 \times 10^{-7}}{4/3} = 5.25 \times 10^{-7}\,\text{m}$. If the light strikes at an incident angle $\theta_i = 53^{\circ}$, the refracted angle $\theta_r$ is found via $\sin(\theta_r) = \frac{n_a \sin(\theta_i)}{n_w} = \frac{1 \times \sin(53^{\circ})}{4/3} \approx 0.6$, resulting in $\theta_r = \sin^{-1}(0.6) = 36.7^{\circ}$.

In another example, light travels from air to a diamond with an incident angle of $45^{\circ}$. Given the refractive index of diamond is $2.4$, the refracted angle is calculated: $\sin(\theta_r) = \frac{\sin(45^{\circ})}{2.4} \approx 0.294$, giving $\theta_r \approx 17.1^{\circ}$. The speed of light in the diamond is $v = \frac{3 \times 10^8}{2.4} = 1.25 \times 10^8\,\text{m/s}$.

Real and Apparent Depth

When looking at an object underwater, such as a coin at the bottom of a pool, it appears to be at a shallower depth than it actually is. This is due to the refraction of light as it exits the water. The relationship is defined by the formula: $\text{Refractive Index} = \frac{\text{Real Depth}}{\text{Apparent Depth}}$. If a coin is at a real depth of $2\,\text{m}$ in water ($n = 1.33$), the apparent depth ($A_d$) is calculated as $A_d = \frac{2}{1.33} \approx 1.5\,\text{m}$. This also explains why a bird seen by a fish appears higher than its actual position; the light rays bend as they enter the water, making the source appear further along the line of the refracted ray.

Critical Angle and Total Internal Reflection (TIR)

The critical angle ($\theta_c$) is the specific angle of incidence in a denser medium for which the angle of refraction in the rarer medium becomes $90^{\circ}$. This is mathematically represented as $\sin(\theta_c) = \frac{n_{\text{rarer}}}{n_{\text{denser}}}$ or $\theta_c = \sin^{-1}(\frac{n_2}{n_1})$. Total Internal Reflection (TIR) is the phenomenon where light traveling from a denser medium to a rarer medium is completely reflected back into the denser medium. This occurs only when the angle of incidence $\theta_i$ is greater than the critical angle $\theta_c$. If \theta_i < \theta_c, light is refracted. If $\theta_i = \theta_c$, the light is refracted along the boundary. Applications of TIR include optical fibers, endoscopes, and periscopes.

Lenses: Types and Properties

Lenses are transparent optical devices made of glass or plastic that bend light rays to form images. There are two primary types: convex and concave. A convex lens (converging lens) is thicker at the center than at the edges; it converges parallel light rays to a focal point ($F$). These are used in magnifying glasses, microscopes, and spectacles for farsightedness. A concave lens (diverging lens) is thinner at the middle and thicker at the edges; it spreads parallel rays apart and is used for nearsightedness correction.

Key terminology for lenses includes the focal point ($F$), the focal length ($f$), which is the distance from the center of the lens to the focal point, and the radius ($R$), where $R = 2f$. The principal axis is the horizontal line passing through the center of the lens.

Image Formation by Lenses

Images formed by a concave lens are always virtual, erected (upright), and diminished. Convex lenses, however, produce different types of images depending on the object's position. If the object is at infinity, the image is real, inverted, and diminished at $f$. If the object is beyond $2f$, the image is real, inverted, and diminished, located between $f$ and $2f$. At $2f$, the image is real, inverted, and the same size as the object, located at $2f$. Between $f$ and $2f$, the image is real, inverted, and enlarged, located beyond $2f$. At $f$, no image is formed (rays are parallel). Between the lens and $f$, the image is virtual, erected, and larger, located in front of the lens.

Eyesight Defects and Corrections

Myopia, or shortsightedness, occurs when a person can see near objects clearly but distant objects are blurred. This is caused by a long eyeball or a lens that is too powerful, causing the image to form in front of the retina. It is corrected using a concave lens. Hypermetropia, or longsightedness, occurs when distant objects are seen clearly but near objects are blurred. This is caused by a short eyeball or a weak lens, resulting in the image forming behind the retina. It is corrected using a convex lens.

Presbyopia is the difficulty in seeing objects due to the loss of elasticity in the eye lens related to aging; it is corrected with convex or bifocal lenses. Astigmatism results in blurred vision at all distances due to an irregular curvature of the eye lens or cornea and is corrected using cylindrical lenses.

Dispersion and Wave Phenomena

Dispersion of light is the separation of white light into its constituent colors (VIBGYOR) due to the difference in refraction for each color when passing through a prism. Furthermore, wave behavior such as diffraction is noted; sound waves diffract (bend) around obstacles like doors because they have longer wavelengths, whereas light travels in straight lines and does not bend enough for humans to see around corners. In pipes, a longer air column vibrates with a longer wavelength, resulting in a lower frequency ($f$) and a lower pitched note, consistent with the wave equation $v = f \times \lambda$.