Introduction | Class 12 Physics Chapter 9 Ray Optics and Optical Instruments Notes

Welcome, students! Today we are diving into one of the most visual and interesting chapters in Physics: Ray Optics. We perceive the beautiful world around us primarily through light. But have you ever wondered how light travels? Why do stars twinkle? Or how does a tiny pair of glasses correct your vision?

In this chapter, we take a specific approach called “Ray Optics.” We assume that light travels in straight lines. You might ask, “Sir/Ma’am, doesn’t light behave like a wave?” Yes, it does! But, the wavelength of visible light is so tiny (400 nm to 750 nm) compared to the huge objects we interact with daily (like mirrors, lenses, or cars), that we can safely ignore the wave nature for now and treat light as a straight arrow or a “Ray.”

We will explore how light bounces (Reflection) and how it bends (Refraction). Understanding these two simple behaviors will help us master complex instruments like Microscopes and Telescopes. Let’s get started!

1. Reflection of Light by Spherical Mirrors

1.1 The Basics of Reflection

You recall from your earlier classes that reflection follows two golden rules:

• The angle of incidence ($i$) is always equal to the angle of reflection ($r$).
• The incident ray, the reflected ray, and the normal all lie in the same plane.

While this is easy for flat mirrors, in Class 12, we focus on Spherical Mirrors—imagine slicing a piece off a hollow glass ball and painting one side.

1.2 Understanding the Terminology

Before we solve problems, we must speak the language of optics:

• Pole (P): The center of the mirror’s surface.
• Centre of Curvature (C): The center of the sphere from which the mirror was cut.
• Radius of Curvature (R): The distance CP.
• Principal Axis: The straight line passing through P and C.
• Focus (F): The point where parallel rays meet (converge) or appear to come from (diverge).

Relation between f and R: For mirrors with small apertures (small openings), the focal length is exactly half the radius of curvature.
$$f = \frac{R}{2}$$

1.3 The Cartesian Sign Convention (The Most Important Rule)

Students, 90% of mistakes in Board Exams happen here. Please pay attention! We treat the mirror like a graph paper.

1. Origin: The Pole (P) is the origin $(0,0)$.
2. X-axis: The Principal Axis is the X-axis.
3. Direction of Incident Light: This is always the positive X-direction.

The Golden Rules for Signs:

• If you measure distance in the direction of the incident ray, it is Positive.
• If you measure distance against the incident ray (from P towards the object usually), it is Negative.
• Heights measured Upwards (above axis) are Positive.
• Heights measured Downwards (below axis) are Negative.

1.4 The Mirror Equation

This formula connects the object distance ($u$), the image distance ($v$), and the focal length ($f$).
$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$

Linear Magnification (m): This tells us how big the image is compared to the object.
$$m = \frac{\text{Height of Image }(h’)}{\text{Height of Object }(h)} = -\frac{v}{u}$$
Teacher’s Tip: If $m$ is negative, the image is Real and Inverted. If $m$ is positive, the image is Virtual and Erect.

Figure-1: Image formation by a Concave Mirror. Notice that the real image forms in front of the mirror, and we measure distances from the Pole (P).

2. Refraction of Light

When light travels from one transparent medium to another (say, from air to glass), it changes its speed. This change in speed causes the ray to bend. This phenomenon is called Refraction.

2.1 Snell’s Law

The relationship between the angle of incidence ($i$) and the angle of refraction ($r$) is governed by Snell’s Law:
$$\frac{\sin i}{\sin r} = \frac{n_2}{n_1} = n_{21}$$
Here, $n_{21}$ is the refractive index of medium 2 with respect to medium 1.

Physical Meaning: If light enters a denser medium (higher $n$), it bends towards the normal. If it enters a rarer medium (lower $n$), it bends away from the normal.

2.2 Real and Apparent Depth

Have you noticed that a swimming pool looks shallower than it actually is? This is due to refraction. Light coming from the bottom of the pool bends away from the normal as it exits the water, making the bottom appear raised.
$$\text{Apparent Depth} = \frac{\text{Real Depth}}{\text{Refractive Index (n)}}$$

2.3 Total Internal Reflection (TIR)

This is a magical phenomenon. Imagine light going from Water (Denser) to Air (Rarer). It bends away from the normal. As you increase the angle of incidence, the refracted ray bends more and more until it grazes the surface (angle of refraction becomes $90^\circ$).

The angle of incidence for which the angle of refraction is $90^\circ$ is called the Critical Angle ($i_c$).
If you increase the angle even further ($i > i_c$), the light cannot escape! It reflects back entirely into the water. This is Total Internal Reflection.
$$\sin i_c = \frac{1}{n}$$

Figure-2: Mechanism of Total Internal Reflection. Light is trapped inside the medium, which is the working principle of Optical Fibers used in the internet.

Applications of TIR:

• Mirage: An optical illusion in hot deserts where layers of air have different densities.
• Diamonds: The brilliance of a diamond is due to multiple internal reflections (cut faces and high refractive index).
• Prisms: $90^\circ$ and $180^\circ$ reflecting prisms used in binoculars.
• Optical Fibers: Carrying data signals or light without loss over long distances.

3. Refraction at Spherical Surfaces and Lenses

Now we combine two spherical surfaces to create a Lens. Lenses are the heart of all optical instruments.

3.1 Lens Maker’s Formula

This is a 5-star derivation for your Board Exams. It relates the focal length of a lens to the radii of curvature of its two surfaces ($R_1, R_2$) and the refractive index ($n$).
$$\frac{1}{f} = (n_{21} – 1) \left( \frac{1}{R_1} – \frac{1}{R_2} \right)$$
Teacher’s Note: Be very careful with signs here. For a double convex lens, $R_1$ is usually positive and $R_2$ is negative.

3.2 Thin Lens Formula

Similar to the mirror formula, but with a minus sign:
$$\frac{1}{v} – \frac{1}{u} = \frac{1}{f}$$

3.3 Power of a Lens

Power measures the ability of a lens to bend light. A thick lens bends light more and has a shorter focal length.
$$P = \frac{1}{f(\text{in meters})}$$
Unit: Dioptre (D).

• Convex lens (Converging) has Positive Power.
• Concave lens (Diverging) has Negative Power.

3.4 Combination of Thin Lenses

If two lenses are placed in contact, their powers simply add up. This is useful in designing camera lenses to correct defects.
$$\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$$
$$P_{eq} = P_1 + P_2$$

4. Refraction through a Prism

A prism is a transparent medium bounded by plane surfaces inclined at an angle $A$ (Angle of Prism). When white light passes through it, it bends and often splits into colors (Dispersion).

Angle of Deviation ($\delta$): The angle between the incident ray and the emergent ray.
It depends on the angle of incidence. The graph of $\delta$ vs $i$ is a U-shaped curve.

Minimum Deviation ($D_m$):
At a specific angle, the deviation is minimum. At this point, the light travels symmetrically through the prism (parallel to the base).
The refractive index of the prism is given by the famous Prism Formula:
$$n = \frac{\sin \frac{A + D_m}{2}}{\sin \frac{A}{2}}$$

Figure-3: (Left) Path of a ray through a glass prism. (Right) The graph showing the variation of the angle of deviation with the angle of incidence.

5. Optical Instruments

Nature has given us eyes, but we always want to see more—smaller things (bacteria) or distant things (galaxies). This is where optical instruments come in.

5.1 The Simple Microscope

It’s just a convex lens of short focal length (a magnifying glass).
When the object is placed near the focus, a virtual, erect, and magnified image is formed.
Magnification (m):
$$m = 1 + \frac{D}{f} \quad (\text{Image at Near Point } D=25cm)$$
$$m = \frac{D}{f} \quad (\text{Image at Infinity})$$

5.2 The Compound Microscope

To get higher magnification, we use two lenses:

1. Objective: Small focal length, small aperture. Facing the object.
2. Eyepiece: Moderate focal length, large aperture. Facing the eye.

The objective forms a real image, which acts as an object for the eyepiece.
Total Magnification:
$$m = m_o \times m_e \approx \frac{L}{f_o} \times \frac{D}{f_e}$$
(Where L is the tube length).

5.3 The Telescope

Used to see distant objects.

1. Refracting Telescope: Two lenses. The objective has a LARGE focal length and LARGE aperture to gather maximum light. The eyepiece is small.
   
   Magnification: $m = \frac{f_o}{f_e}$
2. Reflecting Telescope (Cassegrain): Uses a large concave mirror instead of a lens. Mirrors are lighter, cheaper, and don’t have chromatic aberration (color defects). This is what modern observatories use.

Figure-4: Ray diagram of an Astronomical Telescope adjusted for normal vision (image at infinity). Notice the large objective lens.

Important Concepts & Teacher’s Tips

• Confusing Formulae: Remember, Mirror formula has a PLUS ($+$), Lens formula has a MINUS ($-$).
  
  Mnemonic: “Mirror” has ‘r’ which looks like a plus? No, that’s confusing. Just remember: Mirrors Add, Lenses Subtract (in the formula).
• Sign Convention is King: Never plug in values without checking signs. $u$ is almost always negative. $f$ is negative for Concave, positive for Convex.
• Half-covering a lens: A common conceptual question asks, “If I cover half the lens with black paper, will I see half the image?”
  
  Answer: No! You will see the FULL image, but it will be dimmer (less intensity) because less light is forming it.
• Violet vs Red: In a prism, Violet bends the most (Violent deviation), and Red bends the least.

Solved Practice Set (CBSE Pattern)

Part A: Very Short Answer Questions (1 Mark)

Q1. A convex lens is immersed in water. What happens to its focal length?

Ans: The focal length increases. This is because the refractive index difference between the glass and water is less than that between glass and air, reducing the bending power of the lens.

Q2. Define Critical Angle.

Ans: The angle of incidence in the denser medium for which the angle of refraction in the rarer medium becomes $90^\circ$ is called the critical angle.

Part B: Short Answer Questions (2-3 Marks)

Q3. A small candle, 2.5 cm in size, is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed?

Solution:
Given: $u = -27$ cm, $h = +2.5$ cm, $R = -36$ cm (Concave).
Focal length $f = R/2 = -18$ cm.
Mirror Formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$
$\frac{1}{v} + \frac{1}{-27} = \frac{1}{-18}$
$\frac{1}{v} = -\frac{1}{18} + \frac{1}{27} = \frac{-3 + 2}{54} = -\frac{1}{54}$
$v = -54$ cm.
Ans: The screen should be placed 54 cm in front of the mirror (on the same side as the object).

Q4. Explain why the sun is visible a little before actual sunrise and a little after actual sunset.

Ans: This is due to atmospheric refraction. The layers of air near the earth are denser than those above. Light from the sun below the horizon bends towards the normal as it travels through the atmosphere, making the sun appear shifted upwards by about $0.5^\circ$. This lengthens the day by about 2 minutes at dawn and dusk.

Part C: Long Answer Questions (5 Marks)

Q5. Derive the Lens Maker’s Formula for a double convex lens.

Solution Summary (Student should write full steps):
1. Consider refraction at the first surface ($ABC$). Object $O$ forms image $I_1$.
Formula: $\frac{n_2}{v_1} – \frac{n_1}{u} = \frac{n_2 – n_1}{R_1}$ …(i)
2. $I_1$ acts as a virtual object for the second surface ($ADC$). Final image forms at $I$.
Formula: $\frac{n_1}{v} – \frac{n_2}{v_1} = \frac{n_1 – n_2}{R_2}$ …(ii)
3. Add equations (i) and (ii). The term with $v_1$ cancels out.
$\frac{n_1}{v} – \frac{n_1}{u} = (n_2 – n_1)(\frac{1}{R_1} – \frac{1}{R_2})$
4. Divide by $n_1$ and substitute $\frac{1}{v} – \frac{1}{u} = \frac{1}{f}$.
Final Result: $\frac{1}{f} = (n_{21} – 1)(\frac{1}{R_1} – \frac{1}{R_2})$.

Part D: Case Study Based Question (4 Marks)

Q6. Optical Fibers and Communication
Optical fibers are the backbone of modern communication. They work on the principle of Total Internal Reflection (TIR). They consist of a core and a cladding. The refractive index of the core ($n_1$) must be higher than that of the cladding ($n_2$). A signal in the form of light enters one end at a suitable angle, undergoes repeated TIR, and exits the other end without significant loss of energy.
(i) What is the condition for TIR to take place?
(ii) Why is the core refractive index higher than cladding?
(iii) If $n_{core} = 1.68$ and $n_{cladding} = 1.44$, calculate the critical angle.

Answers:
(i) Light must travel from denser to rarer medium, and the angle of incidence must be greater than the critical angle.
(ii) To ensure the light travels from denser (core) to rarer (cladding) medium, which is a prerequisite for TIR.
(iii) $\sin i_c = \frac{n_2}{n_1} = \frac{1.44}{1.68} \approx 0.857$. So, $i_c = \sin^{-1}(0.857) \approx 59^\circ$.

Multiple Choice Questions (CBSE Pattern)

1. Q1. A converging lens is used to form an image on a screen. When the upper half of the lens is covered by an opaque screen:
   A) Half the image will disappear.
   B) Complete image will be formed with decreased intensity.
   C) Half image will be formed with same intensity.
   D) Complete image will be formed with increased intensity.
2. Q2. The refractive index of the material of a prism is $\sqrt{2}$ and the angle of the prism is $60^\circ$. The angle of minimum deviation is:
   A) $30^\circ$
   B) $45^\circ$
   C) $60^\circ$
   D) $90^\circ$
3. Q3. For a telescope, the objective lens has a large focal length and large aperture to:
   A) Reduce chromatic aberration.
   B) Increase magnification and resolving power.
   C) Make the instrument lighter.
   D) Reduce spherical aberration.
4. Q4. Which of the following colors of white light deviates most when passing through a prism?
   A) Red
   B) Yellow
   C) Violet
   D) Green
5. Q5. An air bubble in a glass slab ($n=1.5$) appears to be at 6 cm depth when viewed from one side and 4 cm when viewed from the other side. The thickness of the slab is:
   A) 10 cm
   B) 6.67 cm
   C) 15 cm
   D) None of these
6. Q6. The power of a lens is -4.0 D. The lens is:
   A) Convex, f = 0.25 m
   B) Concave, f = 0.25 m
   C) Convex, f = 4 m
   D) Concave, f = 4 m
7. Q7. Critical angle for light passing from glass to air is minimum for:
   A) Red light
   B) Green light
   C) Yellow light
   D) Violet light

MCQ Answers & Explanations:

• 1. (B) Every part of the lens forms the full image. Covering part only blocks some rays, making it dimmer.
• 2. (A) Use $n = \sin((A+D_m)/2) / \sin(A/2)$. Here $\sqrt{2} = \sin(30 + D_m/2) / \sin(30)$. Solving gives $D_m = 30^\circ$.
• 3. (B) Large focal length increases magnification ($f_o/f_e$); large aperture gathers more light (resolution).
• 4. (C) Violet has the shortest wavelength and slows down the most, bending the most.
• 5. (C) Real Depth = Apparent Depth $\times$ Refractive Index. $d_1 = 6 \times 1.5 = 9$, $d_2 = 4 \times 1.5 = 6$. Total thickness = $9+6=15$ cm.
• 6. (B) Negative power means Concave. $f = 1/P = 1/4 = 0.25$ m.
• 7. (D) $\sin i_c = 1/n$. Since Violet has the highest refractive index ($n$), it has the smallest critical angle.