In a camera, the image formed on the sensor is real and inverted. What type of lens is used in a camera, and why is the image inverted?

Cameras use convex lenses. Convex lenses can form real, inverted images, which are necessary for capturing the image on the camera's sensor. The inversion is corrected electronically or during processing.

Why are multiple lenses used in complex optical instruments instead of just one?

Multiple lenses allow for the correction of aberrations, increased magnification, and greater control over the image formation process. A single lens often can't achieve the desired level of performance.

A lens is submerged in water. How does its focal length change compared to its focal length in air? Explain why.

The focal length increases in water. The refractive index of water is closer to that of the lens material (usually glass) than air is. This means the difference in refractive indices at the lens surfaces is smaller, leading to less bending of light and thus a longer focal length.

Lenses in Ray Optics: Types, Principles, and Applications

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Lens

Lenses in ray optics are clear objects that bend light to create images through refraction. There are two primary types: concave lenses (which diverge light) and convex lenses (which converge light). Concave lenses spread light, while convex lenses focus it. The focal length and power of a lens, which are crucial for creating sharp images, are determined by its shape and curvature. Lenses are commonly used in everyday items like eyeglasses, cameras, microscopes, and telescopes to help us see or capture the world more clearly.

1.0Definition And Terminology of Lens

It is a portion of a transparent material with two refracting surfaces such that at least one is curved with the refractive index of its material being different from that of the surroundings.
A thin spherical lens with a higher refractive index than its surroundings acts as a converging (convex) lens if the central part is thicker than the edges. If the center is thinner, it behaves as a diverging (concave) lens, spreading parallel rays. This is how we classify lenses as convergent or divergent.

Basic Terminologies of Thin Lens

1.Optical Centre: It is a point O for a given thin lens, through which any ray passes undeviated.

2.Principal Axis: $C_{1} C_{2}$ is a line passing through the optical center and at a right angle to the lens.

Principal Focus: A Lens has two focal points.

1.(i) First Focus: First focal point is an object point on the principal axis corresponding to which the image is formed at infinity.

2. Second Focus: Second focal point is an image point on the principal axis corresponding to which object lies at infinity

2.0Rules For Image Formation

A ray passing through the optical centre proceeds without deviation through the lens.
A ray passing through the first focal point or aimed towards it becomes parallel to the principal axis after refracting through the lens.
A ray travelling equidistant to the principal axis passes or appears to pass through F after refraction through the lens.

3.0Image Formation in Convex and Concave Lens

Image formation in Concave Lens:

Object: Object is placed in between -\infty and optical centre.
Image: in between –f and optical centre, virtual, erect, diminished

4.0 Lens Maker’s formula & Lens Equations

$\frac{1}{f} = (\frac{μ _{l}}{μ _{s}} - 1) [\frac{1}{R _{1}} - \frac{1}{R _{2}}]$

Lens Formula

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f} \Rightarrow v = \frac{u f}{u + f}$

Important points regarding Lens formula

1. Rays should be paraxial.

2. Lenses should be thin.

3. Medium on both sides of the Lens should be the same.

$R_{1}$ is the radius of curvature of that surface on which light rays incident initially and $R_{2}$ is the radius of curvature of that surface on which light rays incident after refraction from the first surface.
$v, u, R_{1}, R_{2}, f$ (whatever is given in question) should be put along with a sign.

5.0Sign Convention

Note:

(1) For converging Lens (convex Lens in air), focal length is positive.

(2) For diverging Lens (concave Lens in air), focal length is negative.

Special Cases:

1.Sun-glasses or goggles: radii of curvature of two surfaces are equal with centres of curvatures on the same side of the lens.

$R_{1} = R_{2} = + R$

$\frac{1}{f} = (μ - 1) [\frac{1}{R} - \frac{1}{R}] \Rightarrow \frac{1}{f} = 0 \Rightarrow f = \infty an d P = 0 (∴ P = \frac{1}{f})$

2.If refractive index of medium < refractive index of lens

If $μ s < μ_{L} t h e n \frac{μ _{L}}{μ _{S}} > 1 or (\frac{μ _{L}}{μ _{S}} - 1) > 0$

A convex lens behaves as a convergent lens. While concave lens behaves as a divergent lens.

3.Refractive index of medium = Refractive index of lens $(μ_{s} = μ_{L})$

$\frac{1}{f} = (\frac{μ _{L}}{μ _{S}} - 1) [\frac{1}{R _{1}} - \frac{1}{R _{2}}]; \frac{1}{f} = 0 \Rightarrow f = \infty an d P = 0$

Lens will behave as a plane transparent plate

4.Refractive index of surrounding medium > Refractive index of lens

$μ_{s} > μ_{L} \Rightarrow \frac{μ _{L}}{μ _{S}} < 1 an d (\frac{μ _{L}}{μ _{S}} - 1) < 0$

Convex lens will behave as a divergent lens and concave lens will behave as a convergent lens. An air bubble in water behaves as a concave lens.

6.0Transverse or Lateral Magnification

$m_{t} = \frac{h e i g h t o f ima g e}{h e i g h t o f o bj ec t} = \frac{h _{i}}{h _{o}}$

$T an α = \frac{h _{_{o}}}{u} = \frac{h _{i}}{v}$

$\frac{h _{i}}{h _{0}} = \frac{v}{u}$

$m_{t} = \frac{h _{i}}{h _{0}} = \frac{v}{u}$

7.0Cutting of Lens

Case (1): When we cut the lens perpendicular to the principal axis : If the equiconvex lens is cut into equal parts by a vertical plane, the focal length of each part will be double the initial value but intensity of image will remain unchanged.

$P = \frac{1}{f} = \frac{2 ( μ - 1 )}{R}$

$P^{'} = \frac{P}{2} = \frac{( μ - 1 )}{R}$

$f^{'} = 2 f = \frac{R}{2 ( μ - 1 )}$

$f = \frac{R}{2 ( μ - 1 )}$

Case (2): When we cut the lens parallel to the principal axis: If an equiconvex lens having focal length f is cut into two identical parts by a horizontal plane AB then the focal length of each part will be equal to that of the initial lens; because , $R_{1}$ and $R_{2}$ will remain unchanged. Only intensity of image will be reduced.

$(∴ I n t e n s i t y (I) \propto (A p er t u re)^{2})$

8.0Velocity of Image in Lens

Case (1): When the object is moving along the principal axis of the Lens.

$V_{I L}^{\to} = m_{t}^{2} V_{o L}^{\to}$

$V_{I L}$ = velocity of image with respect to lens.

$V_{O L}$ = velocity of object with respect to lens.

Note:All velocities are instantaneous.

Case (2) : When the object is moving perpendicular to the principal axis of the Lens

$V_{I L}^{\to} = m_{t} V_{o L}^{\to}$ All velocities are instantaneous.

9.0Displacement Method

It is used for determination of focal length of convex lenses in the laboratory. A thin converging lens of focal length f is placed between an object and a screen fixed at a distance D apart. If D>4f. There are two positions of lens corresponding to which a sharp image of the object is formed on the screen.

Calculation of focal length $f = \frac{D ^{2} - d ^{2}}{4 D}$

Calculation of height of object $h_{0} = h_{1} ✕ h_{2}$

Note:

(1) If D>4f Two positions of the lens could be found to form the real image of the source on the screen.

(2) If D=4f One position of the lens could be found to form the real image of the source on the screen.

(3) If D<4f No position of lens could be found to form the real image of the source on the screen.

(4) So, for image form on the screen, $D > 4 f \Rightarrow f \leq \frac{D}{4}$

10.0Optical Power

It refers to the ability of a lens, mirror, or other optical system to converge or diverge light.
Optical power is also referred to as dioptric power, refractive power, focusing power, or convergence power.
Units of optical power is diopter $(m^{- 1})$

The optical power of the 1st lens is greater than the optical power of the 2nd lens.
The power of any converging optical system is positive.

Example : Convex lens in air, concave mirror.

The power of any diverging optical system is negative.

Example : Concave lens in air, convex mirror.

11.0Optical Power of Different Optical System

(1) Spherical surface

$P = \frac{μ _{2} - μ _{1}}{R}$

(2) Thin lens

$P = (μ_{l} - μ_{S}) [\frac{1}{R _{1}} - \frac{1}{R _{2}}]$

(3) Spherical Mirror:

$P = - \frac{1}{f}$

12.0Combination of lenses

Two or more than two lenses are placed in contact with each other.

$P_{e q} = P_{1} + P_{2}$

$\frac{1}{f _{e q}} = \frac{1}{f _{1}} + \frac{1}{f _{2}}$

If $P_{e q}$ is +ve then converging lens is formed.
If $P_{e q}$ is –ve then a diverging lens is formed.
If $P_{e q}$ is zero then the plane surface is formed.

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