Topic 4: Waves and particle Nature of light (Part 2)
1. Intensity of Light:
- Intensity of light is a measure of the amount of light energy per unit area per unit time. It is typically denoted by the symbol “I” and is measured in units of watts per square meter (W/m²).
- When we talk about the radiation from the Sun falling on the Earth, we sometimes use the phrase radiation flux density instead of intensity. You need to be familiar with both terms.
- If we are a distance r from a point source of radiation, the radiation will have spread out in all directions over a sphere of surface area [mat] 4πr^2/math] by the time it reaches us. The intensity will therefore be:
- [math] \text{Intensity} = \frac{\text{Power}}{4\pi r^2} [/math]
-
⇒Examples:
- (1)
- A solar heating panel has an area of 2.8m2. How much energy falls on the panel per second if the solar radiation flux density is [math] 300Wm^{-2}[/math] when:
- the panel is normal to the Sun’s radiation
- the radiation makes an angle of 55° with the panel, as in Figure 1
Figure 1 Measure the power of light
- Solution:
- [math] \text{Intensity of light } (I) = \frac{\text{Power}}{\text{Area}} \\
\text{Power } (P) = \text{Intensity of light } (I) \times \text{Area } (A) \\
\text{Power } (P) = 300 \, \text{W/m}^2 \times 2.8 \, \text{m}^2 \\
\text{Power } = 840 \, \text{J/s} [/math] - If the angle between the radiation and the panel is 55°, we need to consider the component of the radiation that falls at right angles (that is, ‘normal’) to the panel, so the energy reduces to:
- [math] \text{Intensity of light } (I) = \frac{\text{Power}}{\text{Area}} \\
- (2)
- The solar constant (the power of the radiation from the Sun falling normally on [math]1m^2[/math] of the outer surface of the Earth’s atmosphere) is approximately [math]1.4 kWm^{-2} [/math]. Estimate the power radiated by the Sun, assuming that it is 150 million kilometres away.Solution:
- [math] \text{Intensity of light } (I) = \frac{\text{Power}}{4\pi r^2} \\
\text{Power } (P) = \text{Intensity of light } (I) \times 4\pi r^2 \\
\text{Power} = 1.4 \times 4 \times 3.14 \times (150 \times 10^6 \times 10^3)^2 \\
\text{Power} = 4 \times 10^{26} \, \text{W} [/math]
2. Refraction of light:
- For all refracted waves the path is deviated towards the normal when the wave is slowed down and away from the normal when the speed increases.
- In Figure 1, the light entering the glass block bends towards the normal on entering the block and away from the normal on leaving.
- The light travels more slowly in the glass than in the air.
- The size of the deviation of the wave path depends on the relative speeds in the two media.
Figure 2 Refracted ray one medium to another medium- The ratio of the speeds is called the refractive index between the media:
- [math] \text{Refractive index from medium 1 to medium 2} = \frac{\text{speed in medium 1}}{\text{speed in medium 2}} \\ {}_{1}n_{2} = \frac{v_1}{v_2} [/math]
- Analysis of the wavefront progression shows that the ratio of the speeds is equal to the ratio of the incident angle and the refracted angle:
“Snell’s law states that the refractive index for a wave travelling from one medium to another is given by the expression”: - [math]{}_{1}n_{2} = \frac{\sin{\theta_1}}{\sin{\theta_2}} [/math]
- This is known as Snell’s law.
- In most cases, we observe the effects of light passing across an interface between air and the refracting medium.
- It is convenient to ignore any reference to the air and state the value as the refractive index of the material (sometimes called the absolute refractive index).
- The angle of incidence is usually represented by and the angle of refraction by r, so Snell’s law gives the refractive index of a medium by the expression:
- [math] n = \frac{\sin{\theta_i}}{\sin{\theta_r}} \\
n = \frac{c}{v} [/math] - where c is the speed of light in a vacuum (or air) and v is the speed in the medium.
For light travelling from a medium of refractive index [math] n_1[/math] to one of refractive index [math]n_2[/math] at angles [math]θ_1[/math] and [math]θ_2[/math] (Figure 2), a more general expression for Snell’s law can be derived using the speeds [math]v_1[/math] and [math]v_2[/math]: - [math] n_1 = \frac{c}{v_1} \quad \text{and} \quad n_2 = \frac{c}{v_2} \\
\text{where } c = \text{speed of light in air.} \\
\frac{n_2}{n_1} = \frac{c/v_2}{c/v_1} = \frac{c}{v_2} * \frac{v_1}{c} \\
\frac{n_2}{n_1} = \frac{v_1}{v_2} = {}_{1}n_{2} = \frac{\sin{\theta_1}}{\sin{\theta_2}} \\
\frac{n_2}{n_1} = \frac{\sin{\theta_1}}{\sin{\theta_2}} [/math]
[math] n_1 \sin{\theta_1} = n_2 \sin{\theta_2} [/math] -
This is a very useful alternative equation for snell’ s law
3. Critical angle:
- The critical angle is the angle of incidence above which total internal reflection occurs.
- Applying the general form of Snell’s law:
- [math] n_1 \sin{\theta_1} = n_2 \sin{\theta_2}[/math]
- where [math]n_1[/math] = refractive index of glass, [math]n_2[/math] = refractive index of air = 1, [math]θ_1[/math] = critical angle, C, and [math]θ_2[/math] = 90°, leads to:
- [math] \frac{n_1}{n_2} = \frac{\sin{\theta_2}}{\sin{\theta_1}} [/math]
Hence, for a glass – air interface:
[math]n_1 = \frac{1}{\sin C}[/math] - For light travelling from a medium of refractive index [math] n_1[/math] to one of lower refractive index [math]n_2[/math], the expression becomes
- [math] \frac{n_1}{n_2} = \frac{\sin C}{\sin 90^\circ} \\
\frac{n_1}{n_2} = \sin C [/math]
4. Total internal reflection:
- Total internal reflection occurs when a wave reaches a boundary between two media and is completely reflected back into the first medium.
- This happens when the angle of incidence is greater than the critical angle, and the wave cannot pass through the boundary.
- Conditions for total internal reflection:
- The wave must be in a medium with a higher refractive index than the second medium.
- The angle of incidence must be greater than the critical angle.
- Characteristics:
- No refraction occurs; the wave is completely reflected.
- The reflected wave has the same amplitude and phase as the incident wave.
- Total internal reflection can occur with any type of wave, including light, sound, and seismic waves.
- When light travels from glass to air or glass to water, it speeds up and bends away from the normal.
- The passage of light from a medium with a high refractive index to one with a lower refractive index is often referred to as a ‘dense-to-less-dense’ or ‘dense-to-rare’ transition.
Figure 3 Total internal reflection- This relates to optical density, which is not always equivalent to physical density.
- If the angle of incidence at the glass–air interface is increased, the angle of refraction will approach 90°.
- If the angle is increased further, no light can leave the glass and so it is all reflected internally according to the laws of reflection (Figure 3).
5. Focal length of converging and diverging lenses:
- Lenses focus light by refraction.
- The surfaces of lenses are shaped so that parallel rays of light passing through the lens will either converge to a single point or diverge from a single point.
- Figure 4 shows the passage of parallel rays through a converging and diverging lens.
Figure 4 Converging and diverging lens- The points where parallel rays meet or appear to diverge from are called the focal points of the lenses.
- The focal length of a lens is the distance between the optical centre of the lens and the focal point.
- Figure 4 shows how a converging lens forms a real focus where the rays actually meet.
- The focal point for the diverging lens is on the same side as the source of the rays and forms a virtual focus where no rays intersect.
6. Ray diagram:
- When some of reflecting light passes through a lens, an image of the object is formed.
- The image can be real or virtual.
- A real image is formed by the actual intersection of rays of light and so can be projected onto a screen.
- A virtual image can only be seen when looking through the lens and appears to be at the point where the rays originate.
Figure 5 Predictable rays- The positions, size and nature of images produced by lenses can be found by drawing ray diagrams.
- There are three predictable rays from a point on an object that pass through a lens.
- These are shown in Figure 5 and are used to determine the position of the image of that point.
- Ray diagrams are drawn using the predictable rays from a single point on the object (usually the top) so that the position of the image will be where the rays meet, or appear to originate from, after passing through the lens.
- It is usual to represent the object as an arrow with its base on the principle axis with the head vertically above it.
- A ray of light from the foot of the arrow along the principle axis will pass through the optical centre of the lens without deviation.
- So, it follows that the image of this point will be somewhere on the principle axis.
7. The power of lens:
- The power of a lens is a measure of its ability to converge or diverge light rays.
- It is defined as “the reciprocal of the focal length” (the distance between the lens and the point where the light rays converge or diverge).
- Mathematically:
- [math] \text{Power } (P) = \frac{1}{\text{Focal Length } (f)} [/math]
- Unit: Diopters (D or [math]m^{-1}[/math])
- Types of lenses:
- Converging lenses (positive power): Bring light rays together, form a real image, and are thicker in the middle than at the edges.
- Diverging lenses (negative power): Spread light rays apart, form a virtual image, and are thinner in the middle than at the edges.
- Power of a lens determines:
- Magnification
- Image size and position
- Light-gathering ability
- Higher power lenses:
- Have a shorter focal length
- Are more convergent (or divergent)
- Can produce greater magnification
- Remember, the power of a lens determines its ability to manipulate light rays and form images!
8. Lenses in combination:
- The power of a converging lens is given a positive value while that of a diverging lens is always negative. If a combination of two or more lenses is used, the total power, Pc is the sum of the powers of the lenses:
- [math] P_c=P_1+P_2+P_3+⋯………. [/math]
- Where P1, P2, P3, etc. are the powers of each individual lens.
- Types of lens combinations:
- Thin lenses in contact: Powers add up directly ([math]P_{total} = P_1 + P_2 [/math])
- Thick lenses or separated lenses: Powers add up, but with a correction factor
- [math] P_{total} = P_1 + P_2 – (d_1 + d_2)/f_1 f_2[/math]
- Effects of lens combinations:
- Increased magnification
- Improved light-gathering ability
- Enhanced image quality
- Changed focal length
- Applications:
- Microscopes
- Telescopes
- Camera lenses
- Spectacles (eyeglasses)
9. The lens equation:
- The distance of the object from the optical centre of the lens (the object distance) is represented as u, the image distance as v and the focal length as f, as shown in Figure 6.
- Using the similar triangles in Figure 6, the following ratios can be deduced:
Figure 6 for measure lens equation- [math] \frac{h_0}{u} = \frac{h_1}{v} \\
\frac{h_0}{f} = \frac{h_1}{v – f} [/math] - The linear magnification of the image, m, is the ratio of the size of the image to the size of the object.
- [math] m = \frac{h_0}{h_1} = \frac{f}{v – f} \\
\frac{h_1}{h_0} = \frac{v – f}{f} = \frac{v}{u} \\
m = \frac{v}{u} = \frac{v}{f} – 1 [/math] - Rearranging the above gives us the lens equation:
- [math]\frac{1}{u} + \frac{1}{v} = \frac{1}{f} [/math]
- The lens equation can be used to determine the image position for any type of lens if the real is positive sign convention is used.
- All distances associated with real images and focal points are given positive values and those associated with virtual images and focal points are negative.
- It follows that converging lenses will have a positive value for focal length and for power, and the values for the focal length and power of a diverging lens will be negative.
10. Plane polarization:
- Plane polarization is a type of light polarization where the electric field vector of the light wave vibrates in a single plane.
- This means that the light wave is confined to a single plane, rather than vibrating in multiple directions.
- The waves will lie in a vertical or horizontal plane and are said to be vertically or horizontally plane polarized.
- The plane of polarization can be varied by moving the hand in different directions. If a rope is passed through a vertical slit in a wooden board (Figure 7), horizontal vibrations would be stopped and only vertical ones transmitted.
- Such an arrangement acts as a polarizing filter.
Figure 7 polarizing filter- Characteristics:
- Light wave vibrates in a single plane
- Electric field vector is perpendicular to the direction of propagation
- Magnetic field vector is parallel to the direction of propagation
- Types of plane polarization:
- Linear polarization: Electric field vector vibrates in a straight line
- Vertical polarization: Electric field vector vibrates up and down
- Horizontal polarization: Electric field vector vibrates left and right
- Effects of plane polarization:
- Reduced glare
- Improved contrast
- Increased visibility
- Enhanced color saturation
- Applications:
- Polarized sunglasses
- Camera polarizing filters
- Microscopes
- LCD screens
11. Diffraction:
- Diffraction is the bending of light around obstacles or the spreading of light as it passes through small openings.
- It occurs when light encounters a barrier or a slit, causing the light waves to change direction and spread out.
- Types of diffraction:
- Single-slit diffraction
- Double-slit diffraction (interference patterns)
- Diffraction gratings (multiple slits)
- Circular aperture diffraction (e.g., camera lens)
- Characteristics:
- Bending of light around obstacles
- Spreading of light through small openings
- Interference patterns (constructive and destructive)
- Dependence on wavelength and aperture size
- Effects of diffraction:
- Image distortion
- Loss of resolution
- Creation of diffraction patterns
- Limitation of optical resolution
- Applications:
- Optical instruments (e.g., microscopes, telescopes)
- Diffraction gratings (spectroscopy, laser technology)
- Optical communication systems
- Image processing techniques
- Diffraction can be explained using Huygens’ construction (shown in Figure 8) Huygens considered every point on a wavefront as the source of secondary spherical wavelets which spread out with the wave velocity.
Figure 8 Huygens’ construction- The new wavefront is the envelope of these secondary wavelets.
- When a wavefront is obstructed by a slit or an obstacle, the secondary wavelets at the edges are transmitted into the geometrical shadow causing the diffraction spreading.
- In some instances, patterns can be seen due to the interference of imagined point sources on the wavefront as it passes through the slit or around the obstruction.
12. Diffraction grating:
- When light is reflected from a surface with thousands of equally spaced, parallel grooves scored onto each centimetre, or transmitted through thousands of equally spaced, microscopic gaps, a diffraction pattern is produced.
- Such arrangements are called diffraction gratings.
- Like the single slit, the width of the spacing must be of the same order of magnitude as the wavelength of the light, but the patterns are different.
- The maxima occur at specific angles where the small, coherent waves from each groove or slit superimpose constructively producing sharply defined lines, as shown in Figure 9.
Figure 9 Diffraction grating- If the number of lines per meter on the grating is known, it is possible to determine the wavelength of light transmitted or reflected by the grating by measuring the angles between the central maximum and the diffracted maxima.
- The relationship between the angles and the wavelength can be shown to be:
- [math] nλ=d sinθ [/math]
- where d is the slit separation and n is the order of the maximum.
- It follows that for smaller values of d, the value of θ will be bigger for a particular wavelength, and that the maximum number of orders of diffraction will be when [math] n \leq \frac{d}{\lambda} [/math].
13. CORE PRACTICAL 8: Determine the wavelength of light from a laser or other light source using a diffraction grating.
- Objective:
– Determine the wavelength of light from a laser or other light source using a diffraction grating
– Understand the principle of diffraction and its application in spectroscopy - Materials:
– Diffraction grating (with known slit spacing)
– Laser or light source (with unknown wavelength)
– Screen or detector
– Ruler or calipers - Procedure:
- Set up the diffraction grating and light source
- Measure the distance between the grating and the screen (L)
- Measure the angle of diffraction (θ) for a specific order (e.g., first order)
- Use the diffraction grating equation to calculate the wavelength:
- [math] nλ = d sinθ [/math]
- where:
n = order of diffraction (e.g., 1 for first order)
λ = wavelength of light
d = slit spacing of the grating - Repeat steps 3-4 for multiple orders (if possible)
- Calculate the average wavelength value
- Tips and Variations:
– Use a spectrometer or optical bench for more precise measurements
– Experiment with different light sources and gratings
– Analyze the diffraction pattern to determine the wavelength
– Compare theoretical and experimental values - Remember to follow proper laboratory safety protocols and obtain necessary approvals before conducting the experiment.