6.1 Refraction of Light
Refractive Index and Snell's Law
Refraction of light occurs when light travels between two optical media of different optical densities, causing a change in its speed and direction of propagation.
- Snell's Law: When light passes from medium 1 to medium 2, the ratio of $\sin i$ to $\sin r$ is constant:
$$n = \frac{\sin i}{\sin r}$$
where $i$ is the angle of incidence in air (or vacuum) and $r$ is the angle of refraction in the medium.
Refractive Index ($n$) Equations
The refractive index $n$ can also be determined using the speed of light or the real and apparent depth:
$$n = \frac{c}{v} = \frac{\text{Real depth } (H)}{\text{Apparent depth } (h)}$$
- $c$: Speed of light in vacuum ($3.0 \times 10^8 \text{ m s}^{-1}$)
- $v$: Speed of light in the medium ($\text{m s}^{-1}$)
6.2 Total Internal Reflection
Critical Angle ($c$) and Total Internal Reflection
When light travels from an optically denser medium to an optically less dense medium, it bends away from the normal.
- Critical Angle ($c$): The angle of incidence in the denser medium when the angle of refraction in the less dense medium is $90^\circ$.
- Total Internal Reflection: Occurs when light travels from a denser to a less dense medium and the angle of incidence $i$ is greater than the critical angle ($c$). No refraction occurs; the light is entirely reflected back into the denser medium.
The relationship between refractive index $n$ and critical angle $c$ is:
$$n = \frac{1}{\sin c}$$
Natural Phenomena and Applications
- Natural Phenomena: Mirages, rainbow formation.
- Applications: Optical fibers in telecommunications, endoscopes in medicine, glass prisms in binoculars and periscopes.
6.3 Image Formation by Lenses
Types of Lenses
- Convex Lens (Converging Lens): Thicker at the middle than at the edges. Parallel rays converge at the principal focus ($F$).
- Concave Lens (Diverging Lens): Thinner at the middle than at the edges. Parallel rays diverge as if originating from the principal focus ($F$).
Key Lens Terms
- Optical Centre ($O$): Point at the center of the lens where rays pass through without bending.
- Principal Axis: Straight line passing through the optical centre $O$ and focus $F$.
- Focal Length ($f$): Distance between the optical centre $O$ and the principal focus $F$.
The Thin Lens Formula and Linear Magnification
The relationship between focal length ($f$), object distance ($u$), and image distance ($v$) is given by the Thin Lens Formula:
$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$
Sign Convention (Cartesian):
- Focal length $f$ is positive for convex (converging) lenses and negative for concave (diverging) lenses.
- Image distance $v$ is positive for real images and negative for virtual images.
Linear Magnification ($m$):
$$m = \frac{h_i}{h_o} = \frac{v}{u}$$
6.4 Thin Lens Formula Applications & Optical Instruments
Astronomical Telescope
- Consists of two convex lenses: an objective lens (focal length $f_o$) and an eyepiece lens (focal length $f_e$), where $f_o > f_e$.
- Normal Adjustment: Distance between the two lenses $D = f_o + f_e$.
- The first image formed by the objective lens is at its focus ($F_o$), which acts as the real object at the focus of the eyepiece ($F_e$). The final image is virtual, inverted, and magnified at infinity.
- Magnifying Power ($M$): $M = \frac{f_o}{f_e}$.
Compound Microscope
- Consists of two convex lenses: objective lens ($f_o$) and eyepiece lens ($f_e$), where $f_o < f_e$.
- Object is placed at a distance $f_o < u < 2f_o$ in front of the objective lens.
- The objective lens forms a real, inverted, and magnified first image. The eyepiece acts as a simple magnifying glass, producing a final image that is virtual, inverted, and magnified.
6.5 Image Formation by Spherical Mirrors
Types of Spherical Mirrors
- Concave Mirror (Converging Mirror): Reflecting surface curves inwards. Reflects parallel rays towards a focal point $F$.
- Convex Mirror (Diverging Mirror): Reflecting surface curves outwards. Diverges parallel rays as if coming from a focal point $F$ behind the mirror.
Key Characteristics and Mirror Formula
- Radius of Curvature ($r$): Relationship with focal length: $r = 2f$.
- The mirror formula uses the same thin lens relationship: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$.
- Applications: Concave mirrors (shaving/makeup mirrors, dentist mirrors to enlarge images); Convex mirrors (blind spot security mirrors in roads/shops to provide a wide field of view).