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Reflection — Plane & Spherical Mirrors and Mirror Formula

Master Reflection and Mirror Formula for JEE Main, JEE Advanced, NEET & Board Exams. Covers laws of reflection, plane mirror properties, concave and convex mirror terminology, sign convention, mirror formula derivation, magnification, all image formation cases and velocity of image with solved examples.

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Ray Optics (also called Geometrical Optics) treats light as rays that travel in straight lines and obey the laws of reflection and refraction. It provides the framework for understanding mirrors, lenses, prisms, and optical instruments. The study of reflection — particularly from curved (spherical) mirrors — is the starting point of this vast chapter. For JEE (Main & Advanced) and NEET, mirrors contribute direct numerical questions every year, testing the mirror formula, magnification, and image characteristics. For board exams, derivation of the mirror formula is a standard 3-mark question.

1. Basic Terminology of Ray Optics

Ray: The path along which light energy travels; represented as a straight line with an arrow.
Beam: A collection of rays. A parallel beam comes from a distant source; a convergent beam meets at a point; a divergent beam spreads out from a point.
Wavefront: The locus of all points in a wave that are in the same phase. Rays are always perpendicular to wavefronts.
Normal: A line perpendicular to the surface at the point of incidence.
Angle of Incidence ( $i$ ): Angle between the incident ray and the normal at the point of incidence.
Angle of Reflection ( $r$ ): Angle between the reflected ray and the normal at the point of incidence.

2. Laws of Reflection

The two laws of reflection hold for all types of reflecting surfaces — plane, concave, and convex:

First Law: The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
Second Law: The angle of incidence equals the angle of reflection: $i = r$ .

These laws hold for all wavelengths of light and are independent of the medium in which the light travels.

Reflection from a Plane Mirror — Key Results

Property	Detail
Image type	Virtual and erect
Image size	Same as object (magnification $m = + 1$ )
Image distance	Equal to object distance (image is as far behind mirror as object is in front)
Laterally inverted?	Yes — left and right are interchanged
Mirror rotated by $θ$	Reflected ray rotates by $2 θ$
Number of images (two mirrors at angle $θ$ )	$n = \frac{360 °}{θ} - 1$ (if $\frac{360 °}{θ}$ is even)
Minimum mirror length to see full image	Half the height of the person — independent of distance from mirror

3. Spherical Mirrors — Terminology

A spherical mirror is a mirror whose reflecting surface is a part of a hollow sphere. Two types:

Concave mirror: Reflecting surface is on the inner (cave) side of the sphere — converging mirror.
Convex mirror: Reflecting surface is on the outer side of the sphere — diverging mirror.

Important Terms

Term	Symbol	Definition
Centre of Curvature	C	Centre of the sphere of which the mirror is a part
Radius of Curvature	R	Radius of the sphere; distance from pole to C
Pole	P	Midpoint of the reflecting surface of the mirror
Principal Axis	—	Line joining pole P and centre of curvature C
Principal Focus	F	Point where parallel rays (close to principal axis) converge after reflection (concave) or appear to diverge from (convex)
Focal Length	f	Distance from pole to focus: $f = R / 2$
Aperture	—	Diameter of the circular boundary of the mirror

Key relation: $f = \frac{R}{2}$ — focal length is half the radius of curvature. This holds for both concave and convex mirrors.

4. Sign Convention

The New Cartesian Sign Convention is essential for applying the mirror formula correctly. All distances are measured from the pole P:

Distances measured in the direction of incident light (usually left to right) are positive.
Distances measured opposite to the direction of incident light are negative.
Heights measured above the principal axis are positive.
Heights measured below the principal axis are negative.

Quantity	Concave Mirror	Convex Mirror
Object distance $u$	Always $-$ ve (object in front)	Always $-$ ve (object in front)
Focal length $f$	$-$ ve (focus in front of mirror)	$+$ ve (focus behind mirror)
Radius of curvature $R$	$-$ ve	$+$ ve
Image distance $v$ (real image)	$-$ ve (in front of mirror)	Never forms real image
Image distance $v$ (virtual image)	$+$ ve (behind mirror)	$+$ ve (behind mirror)

5. Mirror Formula and Magnification

Mirror Formula

The relationship between object distance $u$ , image distance $v$ , and focal length $f$ :

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f} = \frac{2}{R}$

This formula is valid for all positions of the object and for both concave and convex mirrors, provided the sign convention is followed consistently.

Derivation of Mirror Formula (for Concave Mirror)

Consider a concave mirror with pole P, focus F, and centre C. For an object at point O beyond C, an image forms at I. Using similar triangles from the geometry of the ray diagram:

From similar triangles $△ A B F$ and $△ M P F$ :
$\frac{A B}{M P} = \frac{B F}{F P}$

Since $M P = A B^{'}$ (where $B^{'}$ is the image) and $F P = f$ , $B F = v - f$ (using magnitudes):
$\frac{A B}{A^{'} B^{'}} = \frac{B F}{F P} = \frac{v - f}{f}$ ... (1)

From similar triangles formed by object and image (using principal axis geometry):
$\frac{A B}{A^{'} B^{'}} = \frac{O B}{O B^{'}} = \frac{u}{v}$ ... (2)

From (1) and (2): $\frac{u}{v} = \frac{v - f}{f}$
$u f = v (v - f) = v^{2} - v f$
Dividing both sides by $u v f$ :
$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ ✓

Linear Magnification

The ratio of the height of the image ( $h^{'}$ ) to the height of the object ( $h$ ):

$m = \frac{h^{'}}{h} = - \frac{v}{u}$

Value of $m$	Image Characteristics
$m > 0$ (positive)	Virtual and erect image
$m < 0$ (negative)	Real and inverted image
$\| m \| > 1$	Image is enlarged (magnified)
$\| m \| < 1$	Image is diminished
$\| m \| = 1$	Image is same size as object

6. Ray Diagrams — Rules for Image Construction

Any two of the following standard rays are used to locate the image formed by a spherical mirror:

Ray 1 — Parallel to principal axis: After reflection, passes through F (concave) or appears to come from F (convex).
Ray 2 — Through/towards F: After reflection, emerges parallel to the principal axis.
Ray 3 — Through/towards C: Reflects back along the same path (normal incidence — angle of incidence = 0°).
Ray 4 — Incident at pole: Reflects making the same angle with the principal axis on the other side (like a plane mirror at that point).

7. Image Formation by Concave Mirror — All Cases

Object Position	Image Position	Nature	Size	Application
At infinity	At F	Real, inverted	Point-sized	Solar furnace, reflecting telescope
Beyond C (object $> 2 f$ )	Between F and C	Real, inverted	Diminished	Rear-view mirror (no, that's convex)
At C ( $u = 2 f$ )	At C	Real, inverted	Same size ( $m = - 1$ )	Used in optics experiments
Between F and C ( $f < u < 2 f$ )	Beyond C	Real, inverted	Enlarged	Projectors
At F ( $u = f$ )	At infinity	Real, inverted	Highly enlarged	Search lights, headlights, torches
Between P and F ( $u < f$ )	Behind mirror	Virtual, erect	Enlarged	Shaving/makeup mirrors, dentist mirrors

8. Image Formation by Convex Mirror

A convex mirror always forms a virtual, erect, and diminished image — regardless of object position. The image is always between P and F (behind the mirror).

Object Position	Image Position	Nature	Size
At infinity	At F (behind mirror)	Virtual, erect	Point-sized
Anywhere in front of mirror	Between P and F (behind mirror)	Virtual, erect	Diminished

Why Convex Mirrors are Used as Rear-View Mirrors

Always gives an erect image — easy to interpret while driving.
Diminished image gives a wider field of view — more area visible in a smaller mirror.
Image is always virtual — unambiguous and does not confuse the driver.

9. Power of a Mirror

The power of a mirror is defined as the reciprocal of its focal length (in metres):

$P = \frac{1}{f}$ (unit: Dioptre, D)

Concave mirror: $f$ is negative → $P$ is negative.
Convex mirror: $f$ is positive → $P$ is positive.
Note: The sign convention for mirror power is opposite to that of lenses. For mirrors, the power is $- 1 / f$ in some conventions — always clarify with the sign convention used.

10. Important Derived Results

Object at C of Concave Mirror

When $u = - 2 f$ (object at centre of curvature):

$\frac{1}{v} + \frac{1}{- 2 f} = \frac{1}{- f}$ $\Rightarrow$ $v = - 2 f$

Image forms at C itself — same size, real, inverted. $m = - \frac{v}{u} = - \frac{- 2 f}{- 2 f} = - 1$ . ✓

Mirror Moved Towards/Away from Object

Differentiating the mirror formula with respect to time (for a moving mirror or moving object):

$- \frac{1}{v^{2}} \frac{d v}{d t} - \frac{1}{u^{2}} \frac{d u}{d t} = 0$ $\Rightarrow$ $v_{i m a g e} = - \frac{v^{2}}{u^{2}} \cdot v_{o b j e c t} = - m^{2} \cdot v_{o b j e c t}$

The velocity of the image = $- m^{2}$ times the velocity of the object. This result is used in JEE problems on moving objects/mirrors.

Number of Images in Two Inclined Mirrors

Angle $θ$ between mirrors	Number of images
$60 °$	$5$
$72 °$	$4$
$90 °$	$3$
$120 °$	$2$
$180 °$ (parallel mirrors)	$\infty$
$0 °$ (facing each other)	$\infty$

Mirror Problem — Foolproof Step-by-Step Method

Step 1 — Identify the mirror type. Concave ( $f$ negative) or convex ( $f$ positive)?
Step 2 — Assign signs to given quantities. Object is always in front → $u$ is always negative. Focal length: concave $= - | f |$ , convex $= + | f |$ .
Step 3 — Apply mirror formula. $\frac{1}{v} = \frac{1}{f} - \frac{1}{u}$ . Substitute with signs.
Step 4 — Interpret the sign of $v$ . $v$ negative → real image (in front of mirror). $v$ positive → virtual image (behind mirror).
Step 5 — Calculate magnification. $m = - v / u$ . Positive $m$ → virtual erect. Negative $m$ → real inverted. $| m | > 1$ → enlarged.
Step 6 — Cross-check with the image table. Verify your answer matches the expected image nature for that object position.

Quick Memory Aid — Concave Mirror Images

"Beyond C → diminished | At C → same size | Between F and C → enlarged | At F → infinity | Inside F → virtual erect enlarged"

JEE/NEET Power Tips — Avoid These Common Mistakes

$u$ is always negative for real objects. The object is always placed in front of the mirror (in the direction from which light comes). Under the New Cartesian Convention, this always gives $u$ a negative value.
$f = R / 2$ , NOT $f = 2 R$ . Focal length is half the radius of curvature. A very common reversal error.
For a concave mirror, $f$ is negative. Many students mistakenly use $f = + 15$ cm for a concave mirror of focal length 15 cm — this gives completely wrong answers.
Magnification formula is $m = - v / u$ , not $+ v / u$ . The negative sign is important — it correctly gives a negative $m$ for real (inverted) images and positive $m$ for virtual (erect) images.
A concave mirror acts like a convex mirror when the object is placed between P and F — it gives a virtual, erect, enlarged image. Do not assume concave always gives real images.
Convex mirror never gives a real image for any real object position. The image is always virtual, erect, and diminished.
When the mirror is rotated by $θ$ , the reflected ray rotates by $2 θ$ — not $θ$ . This is because the normal also rotates by $θ$ , and the angle of reflection changes by $θ$ , doubling the total rotation of the reflected ray.
Minimum mirror size to see full image = half the height of the person — this is independent of how far the person stands from the mirror. A very popular NEET conceptual question.

Practice Questions (JEE / NEET / Board Level)

Q1: An object is placed 30 cm in front of a concave mirror of focal length 20 cm. The image distance and nature of the image are:

A) $v = - 60$ cm, real and inverted
B) $v = + 60$ cm, virtual and erect
C) $v = - 60$ cm, virtual and inverted
D) $v = + 30$ cm, real and erect

Answer: A) $v = - 60$ cm, real and inverted.

Explanation:

Using the standard sign convention: $u = - 30$ cm (object in front), $f = - 20$ cm (concave mirror).

Apply the mirror formula: $\frac{1}{v} = \frac{1}{f} - \frac{1}{u}$

$\frac{1}{v} = \frac{1}{- 20} - \frac{1}{- 30}$
$= - \frac{1}{20} + \frac{1}{30}$
$= \frac{- 3 + 2}{60} = \frac{- 1}{60}$

$v = - 60$ cm.

A negative $v$ indicates a real image formed in front of the mirror.
Magnification: $m = - \frac{v}{u} = - \frac{- 60}{- 30} = - 2$
Since $m$ is negative and $| m | > 1$ , the image is real, inverted, and enlarged (2×).

Q2: A convex mirror of focal length 15 cm forms an image at 5 cm behind the mirror. The object distance is:

A) 7.5 cm
B) 10 cm
C) 12 cm
D) 15 cm

Answer: A) 7.5 cm.

Explanation:

For a convex mirror, $f = + 15$ cm. A virtual image behind the mirror means $v = + 5$ cm.

Apply the mirror formula rearranged for $u$ :
$\frac{1}{u} = \frac{1}{f} - \frac{1}{v}$

$\frac{1}{u} = \frac{1}{15} - \frac{1}{5}$
$= \frac{1}{15} - \frac{3}{15} = \frac{- 2}{15}$

$u = - 7.5$ cm.

The object distance is 7.5 cm (the negative sign confirms it is placed in front of the mirror, as expected).

Q3: An object 2 cm tall is placed 15 cm from a concave mirror of radius of curvature 20 cm. The height of the image is:

A) +4 cm (erect)
B) -4 cm (inverted)
C) +2 cm (erect)
D) -2 cm (inverted)

Answer: B) -4 cm (inverted).

Explanation:

Radius of curvature $R = - 20$ cm, so focal length $f = \frac{R}{2} = - 10$ cm (concave mirror).
Given object distance $u = - 15$ cm and object height $h = + 2$ cm.

Find the image distance ( $v$ ):
$\frac{1}{v} = \frac{1}{f} - \frac{1}{u}$
$\frac{1}{v} = \frac{1}{- 10} - \frac{1}{- 15} = - \frac{1}{10} + \frac{1}{15}$
$\frac{1}{v} = \frac{- 3 + 2}{30} = \frac{- 1}{30}$
$v = - 30$ cm.

Calculate magnification ( $m$ ):
$m = - \frac{v}{u} = - \frac{- 30}{- 15} = - 2$

Calculate image height ( $h^{'}$ ):
$h^{'} = m \times h = - 2 \times 2 = - 4$ cm.

The image is 4 cm tall and inverted (a real image).

Q4: Which of the following mirrors is used by a dentist to examine teeth?

A) Plane mirror
B) Convex mirror
C) Concave mirror
D) Both plane and concave

Answer: C) Concave mirror.

Explanation:

A dentist holds the mirror close to the tooth (placing the object between the pole and the focal point). A concave mirror in this configuration produces a virtual, erect, and enlarged image, making the tooth appear larger and easier to examine.

A plane mirror would only give an image of the same size, and a convex mirror would produce a diminished (smaller) image — neither of which is useful for close dental examination.

Q5: A person stands 40 cm from a plane mirror. If the mirror is moved 10 cm towards the person, by how much does the image shift?

A) 10 cm towards person
B) 20 cm towards person
C) 10 cm away from person
D) 20 cm away from person

Answer: B) 20 cm towards person.

Explanation:

Initially: The person is 40 cm in front of the mirror, so the image is 40 cm behind the mirror. The absolute position of the image from the person is $40 + 40 = 80$ cm.

After moving: The mirror moves 10 cm towards the person. The new object distance is 30 cm. Therefore, the new image is 30 cm behind the new mirror position. The absolute position of the new image from the person is $30 + 30 = 60$ cm.

The total shift of the image is $80 - 60 = 20$ cm towards the person.

Shortcut rule: When a plane mirror moves a distance $d$ towards a stationary object, the image shifts by a distance of $2 d$ towards the object ( $2 \times 10 = 20$ cm).

Q6: For a concave mirror, the object is placed such that a virtual, erect image twice the size of the object is formed. If the focal length is 20 cm, the object distance is:

A) 10 cm
B) 20 cm
C) 30 cm
D) 40 cm

Answer: A) 10 cm.

Explanation:

A virtual, erect, and enlarged image means magnification $m = + 2$ .
Since $m = - \frac{v}{u} = 2$ , we get $v = - 2 u$ .

Using the standard mirror formula:
$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Substitute $v = - 2 u$ and $f = - 20$ cm (focal length of a concave mirror is negative):
$\frac{1}{- 2 u} + \frac{1}{u} = \frac{1}{- 20}$

Find a common denominator to add the terms on the left:
$\frac{- 1 + 2}{2 u} = \frac{1}{- 20}$
$\frac{1}{2 u} = \frac{1}{- 20}$

Solving for $u$ gives $2 u = - 20 ⟹ u = - 10$ cm.

The object distance is 10 cm (placed between the pole and focus, which aligns with the condition for a virtual image).

Q7 (JEE Advanced type): An object moves with speed $v_{0}$ towards a concave mirror of focal length $f$ along the principal axis. When the object is at distance $u$ from the mirror, the speed of its image is:

A) $\frac{f^{2}}{(u - f)^{2}} v_{0}$
B) $\frac{f^{2}}{u^{2}} v_{0}$
C) $\frac{u^{2}}{f^{2}} v_{0}$
D) $\frac{(u - f)^{2}}{f^{2}} v_{0}$

Answer: A) $\frac{f^{2}}{(u - f)^{2}} v_{0}$ .

Explanation:

Starting from the mirror formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ .

Differentiate both sides with respect to time ( $t$ ):
$- \frac{1}{v^{2}} \cdot \frac{d v}{d t} - \frac{1}{u^{2}} \cdot \frac{d u}{d t} = 0$
$\frac{d v}{d t} = - (\frac{v^{2}}{u^{2}}) \frac{d u}{d t}$

Here, $\frac{d v}{d t}$ is the image velocity and $\frac{d u}{d t}$ is the object velocity ( $v_{0}$ ).

Rearrange the mirror formula to find the ratio $\frac{v}{u}$ :
$\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{u - f}{f u} ⟹ v = \frac{f u}{u - f}$
Therefore, $\frac{v}{u} = \frac{f}{u - f}$

Substitute this back into the velocity equation:
$Velocity of image = - {(\frac{f}{u - f})}^{2} v_{0}$

Taking the magnitude (speed), we get $\frac{f^{2}}{(u - f)^{2}} v_{0}$ .

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Reflection — Plane & Spherical Mirrors and Mirror Formula

1. Basic Terminology of Ray Optics

2. Laws of Reflection

Reflection from a Plane Mirror — Key Results

3. Spherical Mirrors — Terminology

Important Terms

4. Sign Convention

5. Mirror Formula and Magnification

Mirror Formula

Derivation of Mirror Formula (for Concave Mirror)

Linear Magnification

6. Ray Diagrams — Rules for Image Construction

7. Image Formation by Concave Mirror — All Cases

8. Image Formation by Convex Mirror

Why Convex Mirrors are Used as Rear-View Mirrors

9. Power of a Mirror

10. Important Derived Results

Object at C of Concave Mirror

Mirror Moved Towards/Away from Object

Number of Images in Two Inclined Mirrors

Mirror Problem — Foolproof Step-by-Step Method

Quick Memory Aid — Concave Mirror Images

Practice Questions (JEE / NEET / Board Level)

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Practice Questions

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