Chapter 10: Light and optics (C1201327)

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1 Light

Remember from  a changing magnetic field creates an electric field (Faraday's law of induction). Maxwell's correction to Ampere's law states that the vice versa is also true, a changing electric field creates a magnetic field. An example of these two principles is the electromagnetic wave, which have a magnetic field and electric field both perpendicular to each other, and the direction of propagation. The electric and magnetic fields of electromagnetic waves have a fixed ratio, equal to the speed of light (in a vacuum), or alternatively, $\dfrac{E}{B}=c$, where $c=3\times 10^8 m/s$. Note that in a medium, light usually does not propagate at a speed equal to $c$. The refractive index describes the speed of light in a vacuum, relative to the speed of light in a medium, such that $n=\dfrac{c}{v}$. Generally, light will travel slower than in a vacuum, and hence have a refractive index greater than 1. Note that as refractive index is inversely proportional to speed of light (in a medium), as refractive index increases, speed of light (in a medium) decreases. Electromagnetic waves are created when charged particles are accelerated.

Electromagnetic spectrum describes the frequency range [and therefore, since $c=f\lambda$ and $c$ is constant, related wavelength] in which electromagnetic radiation exists. Visible light is the portion of the spectrum that is visible to the human eye, between a wavelength of 390 (violet) and 750nm (red). The spectrum is divided into seven named colors, arranged in increasing order of frequency [or decreasing order of wavelength]: red, orange, yellow, green, blue, indigo and violet; which can be memorized with the mnemonic Roy G. Biv. Frequencies just above violet is ultraviolet, and frequencies just below red is infrared. Remember that since $E=hf$, frequency is directly proportional with energy, meaning that higher frequencies have greater energy.

Light has wave-particle duality, meaning it exhibits both wave and particle (known as photons, discussed ) properties, but not both simultaneously.

“That’s like Miley and Hannah,” Mandy commented.

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“I guess it’s more like the Trinity,” one of the girls in the bible study remarked.

“Well,” Mandy replied, “that’s a bit eisegetical, because the Trinity are three persons who are God, not three expressions of God. It’s more like three angles of a triangle, than water/ice/steam.”

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“What’s this?” Emily asked walking in, “The Salem Witch trials? Crusades? The Spanish inquisition? No one expects the Spanish inquisition!”

Visible light is the portion of the electromagnetic spectrum visible to the human eye, between 380-740nm. An optic ray is an idealized narrow beam of light. When the ray (known as the incident ray) enters a new medium, some of it is reflected by the surface back into the original medium; and some of it is refracted into the new medium. The angle of incidence is the angle between the incident ray and a line normal [or perpendicular] to the surface. Note that the angle of reflection is equal to the angle of incidence. The angle of refraction can be calculated by Snell’s law, which states that $n_{i}sin(\theta_{i})=n_{R}sin(\theta_{R})$, where $n$ is the refractive index of the respective media. This law conveys several important ideas:

When light enters a new medium, based on $c=f\lambda$, $c$ is evidently constant, but also, the frequency remains constant, but wavelength changes. This is because light changes speed entering a new medium, which therefore alters wavelength. However, the same number of waves are passing a fixed point per second, meaning frequency is not altered.

The energy of a photon is $E=hf$, where $h$ is Planck’s constant ($6.63\times 10^-34$) and $f$ is frequency. Note that this also explains why frequency is not changed; because this would mean changing energy, meaning energy is not conserved.

(Chromatic) dispersion is the phenomenon of breaking up light on a frequency. This is possible because refractive index is dependent on frequency, $n=n(f)$, such that higher frequencies have higher refractive indices ($n$). Given it was noted  that higher refractive indices cause a greater angle of incidence, higher frequencies will therefore cause this too. Because white light is a combination of all frequencies, a prism will separate it into the entire color spectrum. Since violet has the greatest frequency, it will bend the most; and since red has the least frequency, it will bend the least.

Diffraction is the phenomenon of light bending through an aperture (small opening), which is on the same order of, or much smaller than, the wavelength. The diffraction angle is dependent of the ratio of the wavelength and the width of the slit, in accordance to $sin(\theta_{min})=\dfrac{\lambda}{d}$, meaning that longer waves will bend more when moving through the hole. Note therefore, that whereas dispersion bends short wavelengths (high frequencies) the most; diffraction bends long wavelengths the most.

 Formative learning activity Maps to RK10.A What is light?

2 Optics

There are two types of images:

• Virtual images, which per the name, are only imagined, appearing to exist. An example is a (flat) mirror, where although one’s image is perceived behind the mirror, if someone went behind the mirror, there’d be nothing there. It can also be produced by convex mirrors, which has a reflecting surface which is convex (bulges outward). It can also be produced by diverging lens, which have a center of lens thinner than either side, and are therefore convex. Convex therefore reflects light, diverging (spreading) it
• Real images, which actually exists. It can be produced by concave mirrors, which has a reflecting surface which is concave (bulges inward). It can also be produced by converging lens, which have a center of lens thicker than either side, and are therefore concave. Concave therefore reflects light toward a focal point, converging (concentrating) it, such as a projector sending an image onto a cinema screen

Radius of curvature is based upon a principle that any curve, can be extrapolated to a circle. The radius of that circle, is the radius of curvature therefore, of that curve. For this reason, sharper curves have a smaller radius of curvature; and vice versa.

The focal point is the point at which light converges at. Evidently, this will only be found in a converging/concave mirror, as diverging/convex mirrors do not focus light. All light rays parallel [to the ground] will reflect off the mirror, at $f=\dfrac{1}{2}R$, where $f$ is the focal length away from the mirror:

• In converging/concave mirrors, its distance is ½ the radius of curvature away from the mirror, from the same side of the mirror. Therefore, the focal point in the converging mirror, is in front of the mirror; and the rays therefore reflect to it

• In diverging/convex mirrors, its distance is ½ the radius of curvature away from the mirror, from the opposite side of the mirror. Therefore, the focal point in diverging mirrors, is behind the mirror; and the rays therefore reflect away from it

Figuring the focal point in lens is less intuitive, and requires the Lensmaker’s equation, which states $\dfrac{1}{f}=(\dfrac{n_{1}}{n_{2}}-1)(\dfrac{1}{r_{1}}-\dfrac{1}{r_{2}})$, and $n$ is the refractive index of the lens material, and $r$ is the radius of curvature [for either side of the lens]. Note therefore, the refractive index of the lens material and the atmosphere, will alter the focal point. Specifically, the greater the difference between these indices, the shorter the focal point, since these are inversely proportional in the formula. For example, because the combination of a magnifying glass and air, has a greater difference than a magnifying glass and water, the former will have a shorter focal point, meaning its effect will be more pronounced, and therefore will work better in air [than water]. Note also, the greater the distance between the radius of curvature [for either side of the lens], meaning the thicker the lens, the shorter the focal distance [and per , meaning its effect will be more pronounced], as these are also inversely proportional in the formula.

[Focal] power is the reciprocal of the focal point, $P=\dfrac{1}{f}$, and is the degree to which a lens bends light rays. The unit for optical power is the diopter.

Ray tracing/diagrams show the mirror or lens, focal point, an object [which emits or reflects photons], and its image [where the ray converges]. Note that these diagrams use a spherical profile, although only a parabolic reflector can only truly make a focal point. This is because spherical mirrors suffer from spherical aberration, which results in an imperfectly produced image. Each object [point] can be associated with three rays:

1. Path parallel to the ground, which as just stated , reflects to the focal point of a concave mirror [or refracts to the focal point of a converging lens], or away from the focal point of a convex mirror [or refracts away from the focal point of a diverging lens]
2. Path aimed at center of lens or mirror, reflecting off the mirror, as if it were striking a flat mirror [or refracts through the lens as if it were not present]
3. Path aimed at, or from; and focal point, or pseudo focal point. The pseudo focal point is a point of the same distance from the focal point, but on the opposite side of the lens

Using only the first two rays, you can figure where the image will meet, extrapolating those lines on the other side of the mirror, if necessary. The latter ray is selected by going through a pseudo/focal point which forms a ray parallel to the ground, and meets up at the focal point [where the other two rays meet].

A lens will often perform magnification, for example, linear [or transverse] magnification of a thin lens, $M=\dfrac{h_{i}}{h_{0}}=-\dfrac{d_{i}}{d_{0}}$, where $M$ is the linear magnification, $h_{i}$ is height of the image, $h_{0}$ is height of the object, $d_{i}$ is distance from the lens to the image, and $d_{0}$ is the distance of the lens to the object. A positive magnification indicates it is upright, and if negative indicates it is inverted. Another type of magnification is angular magnification, where $MA=\dfrac{tan(\epsilon)}{tan(\epsilon_{0})}\approx \dfrac{\epsilon}{\epsilon_{0}}$, where $\epsilon$ and $\epsilon_{0}$ are both angles between the object and the principal axis, although whereas $\epsilon$ is the angle of the height of the object, $\epsilon_{0}$ is the angle of the height of the object as seen through a lens, which is higher as a magnification makes a picture bigger. The near point is the closest focal distance at which a healthy naked eye can focus, which is approximately 25cm. Angular magnification permits placing an image closer to the eye than could normally focus, because it produces a virtual image that larger than its original [when extrapolated to the further away near point]. Note therefore, to figure out $\epsilon$ and $\epsilon_{0}$, that magnification depends not only on power of the lens, but also position of the object.

Thin lens equation is $\dfrac{1}{f}=\dfrac{1}{d_{0}}+\dfrac{1}{d_{i}}$, where $f$ is the focal length, $d_{0}$ is the distance [of the lens] from the object, and $d_{i}$ is the distance [of the lens] from the image, where per a set sign convention, left is negative, and right is positive. The thin lens equation assumes the thickness of the lens [at the thicker center], compared with the focal length, is negligible, meaning it is infinitely thin.

Double lens system is the use of two lens [rather than one lens], as in telescopes and microscopes. This means that the image of the first lens, becomes the object of the second lens. The magnification of the system is equal to the products of the magnification of each lens, $M_{t}=M_{1}.M_{2}$. The power of the system is equal to the summation of the powers of each lens, $P_{t}=P_{1}+P_{2}$.

 Formative learning activity Maps to RK10.B What is geometric optics?

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