Optics (Topic 5B)
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Refraction occurs when light passes from one medium to another with a different refractive index, such as from air to glass or from glass to water. This change in medium causes a change in the speed of light, which in turn alters its direction.
The direction of refracted light is measured relative to an imaginary line known as the normal, which is drawn perpendicular to the surface of the boundary where the light ray enters or exits.
The angle formed between the light ray and the normal before refraction is called the angle of incidence, while the angle after refraction is called the angle of refraction.
Refraction at a boundary depends on the refractive indices on either side:
- When light moves from a lower refractive index medium to a higher one, it slows down and bends towards the normal.
- When light moves from a higher refractive index medium to a lower one, it speeds up and bends away from the normal.
- If light travels directly along the normal line, no bending occurs because the light remains perpendicular to the boundary.
Only the speed and wavelength of light change during refraction. The frequency remains constant. This is a consequence of energy conservation.
Refractive index is a measure of how much a material slows down light compared to its speed in a vacuum.
Refractive index is defined mathematically as:
where:
- is the speed of light in a vacuum
- is the speed of light in the material in .
Since light travels more slowly in every medium compared to in a vacuum, the refractive index is always greater than one.
For example, glass has a refractive index of approximately 1.5, which means light travels 1.5 times slower in glass than in a vacuum.
Air has a refractive index that is very close to one. In most calculations, the refractive index of air can be approximated as 1.
Snell’s law is a mathematical relationship that describes how light refracts when it crosses the boundary between two media. It relates the angles of incidence and refraction to the refractive indices of the two media:
where:
- and are the refractive indices of the first and second media, respectively
- and are the angles of incidence and refraction, measured from the normal.
The first medium, where the light originates, is referred to as material 1, while the second medium, where the light enters, is referred to as material 2.
The angles in Snell’s law must always be measured relative to the normal.
Note that if a question gives an angle relative to the surface of the boundary, you will need to subtract it from to calculate the correct angle relative to the normal.
From measurements of the angle of incidence and angle of refraction , the refractive index can be calculated using Snell’s law, which can be written as:
If we are dealing with a material surrounded by air, then represents the refractive index of air, which is approximately 1, and the equation simplifies to:
To accurately determine the refractive index, plot a graph of (Y axis) against (X axis). The gradient of this graph is equal to the refractive index of the block.
Several errors and uncertainties may affect the accuracy of the results.
Systematic errors:
- Ensure that the block of material that is being measured is positioned correctly. Misalignment can lead to consistent errors in the measurements of angles.
Random errors:
- Use a sharp pencil to draw lines accurately, reducing uncertainty in the measured angles.
- The light from the lightbox may appear slightly blurry or dispersed, making it difficult to trace the ray accurately. A concentrated laser beam produces sharper rays, reducing this uncertainty.
Question walkthrough
Refraction Through a Parallel-Sided Diamond Slab
Apply Snell's law twice to find the angles of refraction as light passes from water into a diamond slab and back out into water, using the refractive indices of water and diamond.
When light travels from a medium with a high refractive index to one with a lower refractive index, such as from glass to air, it refracts away from the normal.
As the angle of incidence increases, the angle of refraction also increases.
Eventually, as the angle of refraction increases, the refracted ray travels exactly along the boundary between the two media. This specific angle of incidence is known as the critical angle, denoted by
At the critical angle, the angle of refraction is
If the angle of incidence is greater than the critical angle, light no longer refracts into the second medium. Instead, it is completely reflected back into the denser medium. This phenomenon is called total internal reflection.
The critical angle can be calculated using the formula:
where:
- is the critical angle in degrees,
- is the refractive index of the denser medium,
- is the refractive index of the less dense medium.
This formula is derived from Snell’s Law:
At the critical angle: so Substituting this into Snell’s Law gives:
For total internal reflection to occur, both of the following conditions must be satisfied:
- The refractive index of the denser medium must be greater than the refractive index of the less dense medium
- The angle of incidence, , must be greater than the critical angle
Total internal reflection has numerous practical applications:
Question walkthrough
Total Internal Reflection at a Glass Boundary
Determine whether total internal reflection occurs for light travelling through a glass block, calculate the critical angle at the glass-air boundary, and find the path of the reflected ray.
The refractive index of a transparent semi-circular block can be calculated by measuring the light passing through the block at various angles of incidence.
A semi-circular block is used because light passing through the curved surface always hits the boundary at normal incidence, so it does not refract. Therefore, if light is shone on the flat surface, the angle at which the light leaves the block can be measured to determine the angle of refraction from the flat surface.
Variables in this experiment:
- Independent variable: The angle of incidence . This is the angle at which the light ray strikes the flat surface of the semi-circular block.
- Dependent variable: The angle of refraction . This is the angle at which the light ray exits the block relative to the normal.
Experimental method:
- Setting up the apparatus: Place the semi-circular block on a protractor such that its flat side lies along the line.
- Defining the normal line: Direct the light ray from the lightbox or laser toward the flat edge of the block at . Using a ruler and pencil, draw a dotted line perpendicular to this edge to represent the normal.
- Tracing the light rays: Direct the light ray into the block at various angles of incidence, starting from and increasing in increments up to . For each angle, trace the path of the ray entering the block and the ray exiting the block.
- Measuring refraction: After tracing the light rays, remove the block to measure the angles of incidence and refraction relative to the normal using the protractor. Record these values in a table.
- Repeating for accuracy: Perform multiple trials for each angle to calculate averages, reducing random errors.
The human eye contains a converging lens, so it brings the light rays entering the eye closer together.
When incident light rays are parallel to the principal axis, a converging lens focuses them to a single point, known as the principal focus, also called the focal point, as illustrated in the diagram below.
The focal length is the distance between the optical centre of the lens and the principal focus.
Light rays from an object that pass through a convex lens are refracted and form an image at the point where the rays converge.
Shown below are three symbols used to denote a convex lens:
When ray diagrams are constructed using at least two of the three principal rays from a point on the object, the location of the corresponding image can be determined.
The image of the top of the object forms where the refracted rays from the top of the object intersect. For a real inverted image, this point lies below the principal axis.
If the base of the object lies on the principal axis, then the base of the image will also be on the principal axis. Therefore, only two rays from the top of the object are needed to find where the image will form. A third ray can be used as a check if necessary.
If the base of the object does not lie on the principal axis, then two additional rays must be drawn from the base of the object to determine the position of the base of the image.
To find where the image is formed by a converging lens using a ray diagram drawn to scale.
The object is drawn as an arrow above the principal axis. Two light rays are drawn from the top of the object and pass through the lens, creating an image.
For a converging lens, the light rays obey the following rules:
- Light ray 1: This ray passes straight from the tip of the object through the centre of the lens, and its path is not affected.
- Light ray 2: This ray travels from the tip of the object to the lens parallel to the principal axis, and refracts towards the axis passing through the focal point on the other side of the lens.
The top of the image is formed where the two light rays meet.
To find the location of where an image is formed by a diverging lens, it is helpful to draw a ray diagram to scale.
The object is drawn as an arrow above the principal axis. Two light rays are drawn from the top of the object and pass through the lens, creating an image.
For a diverging lens, the light rays obey the following rules:
- Light ray 1: This ray travels from the tip of the object to the lens parallel to the principal axis and refracts away from the axis, appearing to have come from the principal focal point.
- Light ray 2: This ray passes from the tip of the object straight through the centre of the lens, and its path is not affected.
The top of the image is formed where the two light rays meet. The image formed through a diverging lens is virtual (i.e. it cannot be projected onto a screen) and upright.
The power of a lens is defined as its ability to refract light; the greater the power of a lens, the more light it refracts. This can also be applied to the lens of the eye:
Where:
- is the power of the lens in dioptres , and
- is the focal length .
Power and focal length are inversely proportional to each other. The shorter the focal length, the more powerful the lens, and vice versa.
When thin lenses are placed in contact, their individual powers add algebraically to give the total power of the combination. A converging lens contributes a positive power, and a diverging lens contributes a negative power:
This additive rule assumes the lenses sit at effectively the same position, so any small separation between them is ignored. Power is measured in dioptres , where , so remember to substitute each focal length in metres, not centimetres, when calculating individual powers.
It is important to note that in optics, a thin lens is a lens with a thickness that is negligible compared to the radii of curvature of the lens surfaces.
The focal length of a lens can be determined by the thin-lens equation:
Where:
- is the focal length (m)
- is the object distance (m)
- is the image distance (m).
When using the thin-lens equation for:
- converging lenses, and are always positive, and is negative for a virtual image and positive for a real image.
- diverging lenses, and are always negative, is always positive.
It is important to note that this equation can be applied to the human eye.
Lenses can form two types of images: A real image is one that can be projected onto a screen. They are formed by a convex lens where the distance from the object to the lens is greater than the focal length:
- The image formed is always real and inverted (upside down).
- Images can be magnified, diminished or remain the same size depending on the distance between the object and the lens.
Lenses can form two types of images: A virtual image is formed by a convex (converging) lens when the object is placed closer to the lens than its focal length. Virtual images cannot be projected onto a screen because the light rays only appear to diverge from a point in space rather than actually converging there.
An example of a virtual image is the reflection in a mirror. The light rays appear to be coming from behind the mirror, but the object is not actually there. A virtual image will form on the same side of the lens as the object and will be upright.
It is important to note that dashed lines represent the backward extensions of light rays and are used to locate virtual images. They do not represent real light rays.
The linear magnification defines how much larger or smaller the image is compared to the object and is calculated by:
Where:
- the image distance, is in metres, and
- the object distance is in metres.
Question walkthrough
Finding image height using a lens
Uses the lens equation 1/f=1/u+1/v to find the image distance, then linear magnification m=|-v/u| to find the height of a real image formed by a converging lens.
Polarisation is a process by which the vibrations of a transverse wave are restricted to a single direction.
The polarisation direction is always perpendicular to the direction of wave propagation.
Only transverse waves can be polarised because their vibrations naturally occur in directions perpendicular to their travel.
Longitudinal waves cannot be polarised as their vibrations occur in the same direction as the wave travels, and so the vibrations are already restricted to one direction.
Unpolarised waves:
- Vibrations occur in all possible planes perpendicular to the wave’s travel direction.
- Example: Light from natural sources such as the Sun, bulbs, and flames is unpolarised.
- Visualisation: Imagine a rope being shaken randomly in all directions – the wave has no fixed plane of oscillation.
Polarised waves:
- Vibrations are confined to a single plane perpendicular to the wave’s travel.
- Waves may be polarised in any direction, but the most commonly discussed are vertically and horizontally polarised waves:
- In vertically polarised waves, particles are displaced in the vertical direction.
- In horizontally polarised waves, particles are displaced in the horizontal direction
- Visualisation: Imagine a rope being shaken up and down in a single fixed direction.
- Even when polarised, the vibrations remain perpendicular to the wave’s travel direction.
Polarising filters are used to polarise light waves by only allowing vibrations in a specific direction to pass through.
A polarising filter has a transmission axis aligned in a specific direction. It blocks all vibrations except those parallel to its transmission axis. For example, sunglasses with polarising lenses reduce glare by filtering out certain planes of reflected light.
Parallel filters: When two filters are aligned with their transmission axes in the same direction, maximum light passes through.
Perpendicular filters: When two filters are aligned at to each other, no light passes through.
Polarisation can occur naturally when light interacts with matter.
- Scattering: As seen in the diagram above, light scattering in the atmosphere creates polarisation – the blue light from the sky is polarised more when the sun approaches the horizon.
- Reflection: Light reflecting off shiny surfaces, such as water or glass, becomes partially polarised, which causes glare.
- Refraction: Polarisation can occur when light bends while entering a medium at a specific angle.
When microwaves pass through a metal grille, the electric field component aligned with the grille is blocked due to the motion of the free electrons in the metal bars. The electric field component that is perpendicular to the grille passes through.
The direction of polarisation for microwaves is defined as the direction of the electric field perpendicular to the grille bars. For light, it is parallel to the filter’s axis.
Radio and television signals are broadcast with either horizontal or vertical polarisation.
To receive the broadcast signal effectively, the receiving aerial (antenna) must be aligned to match the polarisation of the transmitted wave.
- For horizontally polarised signals, the aerial should be mounted flat (horizontal).
- For vertically polarised signals, the aerial should be mounted upright (vertical).
Question walkthrough
Explaining Intensity Variation Through Polarising Filters
Explain why transmitted light intensity varies with the angle between two polarising filters, and describe an experimental method to investigate this relationship using Malus's law.