Reflection Of Light

**Reflection** is the phenomenon of light bouncing back into the same medium after striking a surface. Highly polished surfaces, like mirrors, are good reflectors of light. Reflection of light follows specific laws, known as the **laws of reflection**:

1. The angle of incidence is equal to the angle of reflection.
2. The incident ray, the normal to the reflecting surface at the point of incidence, and the reflected ray all lie in the same plane.

These laws apply to all types of reflecting surfaces, including plane mirrors and spherical mirrors.

You are already familiar with image formation by a **plane mirror**: the image is always virtual (cannot be formed on a screen), erect (upright), the same size as the object, located as far behind the mirror as the object is in front, and laterally inverted (left-right reversal).

Spherical Mirrors

**Spherical mirrors** are mirrors whose reflecting surface is part of a sphere. They can be curved inwards or outwards.

• A spherical mirror with its reflecting surface curved inwards (towards the center of the sphere) is called a **concave mirror**.
• A spherical mirror with its reflecting surface curved outwards is called a **convex mirror**.

Terms related to spherical mirrors:

• **Pole (P):** The center point of the reflecting surface of a spherical mirror.
• **Centre of Curvature (C):** The center of the sphere of which the mirror's reflecting surface is a part. For a concave mirror, C is in front; for a convex mirror, C is behind. C is not part of the mirror.
• **Radius of Curvature (R):** The radius of the sphere of which the mirror's reflecting surface is a part. The distance PC equals R.
• **Principal Axis:** An imaginary straight line passing through the pole and the center of curvature. It is normal to the mirror surface at the pole.
• **Principal Focus (F):** For a concave mirror, rays parallel to the principal axis after reflection converge at a point on the principal axis called the principal focus. For a convex mirror, rays parallel to the principal axis after reflection appear to diverge from a point on the principal axis behind the mirror, which is its principal focus.

• **Focal Length (f):** The distance between the pole (P) and the principal focus (F).
• **Aperture:** The diameter of the reflecting surface of a spherical mirror (e.g., distance MN in diagrams).

For spherical mirrors with small apertures, the radius of curvature R is twice the focal length f, i.e., **R = 2f**. This means the principal focus F is midway between the pole P and the center of curvature C.

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Image Formation By Spherical Mirrors

Spherical mirrors can form various types of images depending on the position of the object relative to the mirror. The image can be real or virtual, erect or inverted, and magnified, diminished, or the same size.

**Image Formation by Concave Mirror:**

The nature, position, and size of the image formed by a concave mirror vary significantly with the object's position. A summary is given in Table 9.1 (from text). Key points:

• When the object is placed far away (at infinity or beyond C), a real, inverted, and diminished image is formed.
• When the object is at C, a real, inverted, and same-sized image is formed at C.
• When the object is placed closer (between C and F, or at F), a real, inverted, and enlarged image is formed.
• When the object is placed very close (between P and F), a virtual, erect, and enlarged image is formed behind the mirror. This property is used in shaving mirrors and by dentists.

Summary of image formation by a concave mirror:

Position of the object	Position of the image	Size of the image	Nature of the image
At infinity	At the focus F	Highly diminished, point-sized	Real and inverted
Beyond C	Between F and C	Diminished	Real and inverted
At C	At C	Same size	Real and inverted
Between C and F	Beyond C	Enlarged	Real and inverted
At F	At infinity	Highly enlarged	Real and inverted
Between P and F	Behind the mirror	Enlarged	Virtual and erect

Uses of concave mirrors:

• Torches, searchlights, headlights (produce parallel beams of light when source is at F).
• Shaving mirrors, dentist mirrors (produce enlarged virtual image when object is between P and F).
• Solar furnaces (concentrate sunlight at F).

**Image Formation by Convex Mirror:**

Convex mirrors always form **virtual, erect, and diminished** images, regardless of the object's position. The image is always formed behind the mirror, between the pole (P) and the principal focus (F).

Summary of image formation by a convex mirror:

Position of the object	Position of the image	Size of the image	Nature of the image
At infinity	At the focus F, behind the mirror	Highly diminished, point-sized	Virtual and erect
Between infinity and the pole P of the mirror	Between P and F, behind the mirror	Diminished	Virtual and erect

Uses of convex mirrors:

• Rear-view (wing) mirrors in vehicles: They provide an erect and diminished image, and because they are curved outwards, they offer a wider field of view than plane mirrors, helping the driver see a larger area of traffic behind them for safety.

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Representation Of Images Formed By Spherical Mirrors Using Ray Diagrams

Ray diagrams are useful for visualizing how spherical mirrors form images and determining the image's nature, position, and size. To draw ray diagrams, we consider the path of a few specific rays originating from the object, whose behaviour after reflection is known. The intersection of at least two reflected rays locates the image.

Convenient rays to consider:

1. A ray parallel to the principal axis: After reflection, it passes through the principal focus (concave mirror) or appears to diverge from the principal focus (convex mirror).

2. A ray passing through or directed towards the principal focus: After reflection, it emerges parallel to the principal axis.

3. A ray passing through or directed towards the centre of curvature (C): After reflection, it is reflected back along the same path because it strikes the mirror normally (perpendicularly).

4. A ray incident obliquely at the pole (P): It is reflected obliquely, following the laws of reflection (angle of incidence = angle of reflection with respect to the principal axis as the normal at P).

By drawing these diagrams, we can visually confirm the results summarized in the tables for concave and convex mirrors.

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Sign Convention For Reflection By Spherical Mirrors

To describe image formation mathematically and solve problems, a consistent set of rules for assigning signs (positive or negative) to distances is needed. The **New Cartesian Sign Convention** is used for spherical mirrors:

1. The **pole (P)** of the mirror is taken as the **origin** (0,0) of the coordinate system.
2. The **principal axis** is taken as the **x-axis**.
3. The **object is always placed to the left** of the mirror. Light from the object falls on the mirror from the left.
4. All distances are measured **from the pole** (P).
5. Distances measured in the direction of incident light (to the **right** along the principal axis) are taken as **positive (+)**.
6. Distances measured in the direction opposite to the direction of incident light (to the **left** along the principal axis) are taken as **negative (-)**.
7. Distances measured perpendicular to and **above** the principal axis (height upwards) are taken as **positive (+)**.
8. Distances measured perpendicular to and **below** the principal axis (height downwards) are taken as **negative (-)**.

According to this convention:

• Object distance (u) is usually negative (object on the left).
• Image distance (v) is negative for real images (formed on the left, in front of the mirror) and positive for virtual images (formed on the right, behind the mirror).
• Focal length (f) is negative for a concave mirror (F is in front) and positive for a convex mirror (F is behind).
• Object height (h) is usually positive (object is upright).
• Image height (h') is positive for erect images and negative for inverted images.

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Mirror Formula And Magnification

The relationship between object distance (u), image distance (v), and focal length (f) for a spherical mirror is given by the **mirror formula**:

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

This formula is valid for all spherical mirrors and all object positions. Numerical values for u, v, f must be substituted with their proper signs according to the New Cartesian Sign Convention.

**Magnification (m)** produced by a spherical mirror indicates how much larger or smaller the image is relative to the object. It is defined as the ratio of the height of the image (h') to the height of the object (h).

$$ m = \frac{\text{Height of the image} (h')}{\text{Height of the object} (h)} = \frac{h'}{h} $$

Magnification is also related to the object and image distances:

$$ m = -\frac{v}{u} $$

Combining these, $m = \frac{h'}{h} = -\frac{v}{u}$.

Interpretation of magnification sign:

• A positive value of m indicates that the image is **erect and virtual**.
• A negative value of m indicates that the image is **inverted and real**.
• If $|m| > 1$, the image is magnified (larger than the object).
• If $|m| = 1$, the image is the same size as the object.
• If $|m| < 1$, the image is diminished (smaller than the object).

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Refraction Of Light

**Refraction** is the phenomenon of the change in the direction of light when it passes from one transparent medium to another. This bending of light occurs because the speed of light is different in different media.

Examples of refraction in daily life:

• The bottom of a swimming pool appearing shallower than it is.
• Letters appearing raised when viewed through a glass slab.
• A pencil appearing bent when partly immersed in water.
• A lemon in water appearing bigger when viewed from the sides.

The extent of bending of light depends on the pair of media involved. Light bends at the interface (boundary) separating the two transparent media.

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Refraction Through A Rectangular Glass Slab

When a ray of light passes through a rectangular glass slab, it undergoes refraction twice: first at the air-glass interface and second at the glass-air interface.

Consider a ray of light incident obliquely on a rectangular glass slab:

1. At the first surface (air to glass): Light travels from a rarer medium (air) to a denser medium (glass). The light ray bends **towards the normal**.
2. Within the glass slab: Light travels in a straight line.
3. At the second surface (glass to air): Light travels from a denser medium (glass) to a rarer medium (air). The light ray bends **away from the normal**.

The bending at the two parallel surfaces (air-glass and glass-air) is equal and opposite. As a result, the emergent ray (the ray coming out of the glass slab) is **parallel** to the direction of the incident ray, but it is laterally shifted.

If a light ray is incident normally (perpendicularly) on the interface between two media, it passes straight through without any bending.

Refraction of light obeys specific laws:

1. The incident ray, the refracted ray, and the normal to the interface of two transparent media at the point of incidence, all lie in the same plane.
2. For a given pair of media and for light of a specific colour, the ratio of the sine of the angle of incidence ($\angle i$) to the sine of the angle of refraction ($\angle r$) is constant. This is **Snell's Law**:

   $$ \frac{\sin i}{\sin r} = \text{constant} $$

   This constant value is called the **refractive index** of the second medium with respect to the first medium.

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The Refractive Index

The **refractive index (n)** quantifies the extent to which a medium can bend light. It is related to the speed of light in different media.

• The refractive index of medium 2 with respect to medium 1 (n$_{21}$) is the ratio of the speed of light in medium 1 ($v_1$) to the speed of light in medium 2 ($v_2$).

  $$ n_{21} = \frac{v_1}{v_2} $$

• The refractive index of medium 1 with respect to medium 2 (n$_{12}$) is the ratio of the speed of light in medium 2 ($v_2$) to the speed of light in medium 1 ($v_1$).

  $$ n_{12} = \frac{v_2}{v_1} $$

  Note that $n_{21} = 1/n_{12}$.
• The **absolute refractive index** of a medium is its refractive index with respect to vacuum (or air, which is very close to vacuum). If c is the speed of light in vacuum and v is the speed of light in the medium, the absolute refractive index of the medium (n$_{medium}$) is:

  $$ n_{medium} = \frac{\text{Speed of light in vacuum (c)}}{\text{Speed of light in the medium (v)}} = \frac{c}{v} $$

  This is simply called the refractive index of the medium.

Optical density is related to refractive index. A medium with a higher refractive index is **optically denser** than a medium with a lower refractive index, which is considered **optically rarer**. The speed of light is higher in an optically rarer medium and lower in an optically denser medium. Light bends towards the normal when going from rarer to denser, and away from the normal when going from denser to rarer.

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Refraction By Spherical Lenses

A **lens** is a transparent material bounded by two surfaces, at least one of which is spherical. Lenses bend light rays, causing them to converge or diverge.

• A lens bounded by two spherical surfaces bulging outwards is a **double convex lens**, commonly called a **convex lens**. It is thicker in the middle than at the edges. Convex lenses are **converging lenses** because they converge parallel light rays after refraction.
• A lens bounded by two spherical surfaces curved inwards is a **double concave lens**, commonly called a **concave lens**. It is thicker at the edges than in the middle. Concave lenses are **diverging lenses** because they cause parallel light rays to diverge after refraction.

Terms related to spherical lenses:

• **Centres of Curvature (C$_1$, C$_2$):** The centers of the spheres of which the lens surfaces are parts. A lens has two centers of curvature.
• **Principal Axis:** An imaginary straight line passing through the two centers of curvature.
• **Optical Centre (O):** The central point of a lens. A ray of light passing through the optical centre goes undeviated.
• **Aperture:** The effective diameter of the circular outline of the lens.
• **Principal Focus (F$_1$, F$_2$):** A lens has two principal foci, one on each side. For a convex lens, rays parallel to the principal axis converge at F$_2$ on the other side. For a concave lens, rays parallel to the principal axis appear to diverge from F$_1$ on the same side.
• **Focal Length (f):** The distance between the optical centre (O) and either principal focus (F$_1$ or F$_2$).

For thin lenses with small apertures and centers of curvature equidistant from the optical center (symmetrical lenses), there are two principal foci, F$_1$ and F$_2$, located at equal distances from the optical center on opposite sides.

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Image Formation By Lenses

Lenses form images by refracting light. The nature, position, and size of the image depend on the type of lens and the position of the object.

**Image Formation by Convex Lens:**

A convex lens can form both real and virtual images, and the image characteristics vary with object position. A summary is given in Table 9.4 (from text).

Summary of image formation by a convex lens:

Position of the object	Position of the image	Relative size of the image	Nature of the image
At infinity	At focus F2	Highly diminished, point-sized	Real and inverted
Beyond 2F1	Between F2 and 2F2	Diminished	Real and inverted
At 2F1	At 2F2	Same size	Real and inverted
Between F1 and 2F1	Beyond 2F2	Enlarged	Real and inverted
At focus F1	At infinity	Infinitely large or highly enlarged	Real and inverted
Between focus F1 and optical centre O	On the same side of the lens as the object	Enlarged	Virtual and erect

**Image Formation by Concave Lens:**

A concave lens always forms a **virtual, erect, and diminished** image, regardless of the object's position. The image is always formed on the same side of the lens as the object, between the optical centre (O) and the principal focus (F1).

Summary of image formation by a concave lens:

Position of the object	Position of the image	Relative size of the image	Nature of the image
At infinity	At focus F1	Highly diminished, point-sized	Virtual and erect
Between infinity and optical centre O of the lens	Between focus F1 and optical centre O	Diminished	Virtual and erect

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Image Formation In Lenses Using Ray Diagrams

Similar to mirrors, ray diagrams for lenses use specific rays whose paths after refraction are known. Any two of these rays can be used to locate the image.

Convenient rays to consider for lenses:

1. A ray of light from the object, parallel to the principal axis: After refraction, passes through the principal focus on the other side (convex lens) or appears to diverge from the principal focus on the same side (concave lens).

2. A ray of light passing through or directed towards a principal focus: After refraction, emerges parallel to the principal axis.

3. A ray of light passing through the optical centre (O): It emerges without any deviation.

Ray diagrams for convex lenses:

Ray diagrams for concave lenses:

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Sign Convention For Spherical Lenses

The New Cartesian Sign Convention is used for spherical lenses, similar to mirrors, but with distances measured from the optical centre (O).

1. The **optical centre (O)** is taken as the **origin**.
2. The **principal axis** is the **x-axis**.
3. The **object is always placed to the left** of the lens.
4. All distances are measured **from the optical centre** (O).
5. Distances in the direction of incident light (to the **right** along the principal axis) are **positive (+)**.
6. Distances opposite to incident light (to the **left** along the principal axis) are **negative (-)**.
7. Heights above the principal axis (upwards) are **positive (+)**.
8. Heights below the principal axis (downwards) are **negative (-)**.

According to this convention:

• Object distance (u) is usually negative.
• Image distance (v) is positive for real images (formed on the right) and negative for virtual images (formed on the left).
• Focal length (f) is positive for a convex lens and negative for a concave lens.
• Object height (h) is usually positive.
• Image height (h') is positive for erect images and negative for inverted images.

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Lens Formula And Magnification

The relationship between object distance (u), image distance (v), and focal length (f) for a spherical lens is given by the **lens formula**:

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

This formula is valid for all spherical lenses and object positions, provided distances are used with proper signs.

**Magnification (m)** produced by a lens is the ratio of image height (h') to object height (h):

$$ m = \frac{h'}{h} $$

Magnification is also related to object and image distances:

$$ m = \frac{v}{u} $$

So, $m = \frac{h'}{h} = \frac{v}{u}$.

Interpretation of magnification sign and value is similar to mirrors: positive m for erect/virtual, negative m for inverted/real; $|m|>1$ for magnified, $|m|=1$ for same size, $|m|<1$ for diminished.

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