26Reflection of Plane and Curved Mirrors

📚
REFLECTION OF PLANE AND CURVED MIRRORS
▢ Reflection of Light:
❖ Definition: Returning of light into same medium after incidence on polished surface

Table 1: Laws of Reflection

Law
Statement
1st law
\(i=r\)
2nd law
Incident ray, reflected ray and normal at point of incidence lie in same plane
❖ Glancing Angle: Angle made by incident/reflected ray with polished surface
❖ Angle of Deviation: \(\delta=2g=180^\circ-2i=180^\circ-2r\)
❖ On Reflection:

Table 1: Changes During Reflection

Quantity
Change
Speed
Unchanged
Frequency
Unchanged
Wavelength
Unchanged
Amplitude
Decreases
Intensity
Decreases
Reflection from denser / rigid surface
Phase change by \(\pi\)
Reflection from rarer / free boundary
Phase unchanged
❖ Types:

Table 1: Regular vs Diffused Reflection

Feature
Regular reflection
Diffused / irregular reflection
Surface
Smooth plane surface
Rough surface
Parallel incident rays after reflection
Remain parallel in same order
Get diffused
Laws of reflection
Obeyed
Obeyed
▢ Plane Mirror:

Table 1: Plane Mirror Image

Object
Image
Real object
Virtual, erect, laterally inverted
Virtual object
Real, inverted

Table 2: Real and Virtual Object/Image

Condition
Type
Light diverges from object
Real object
Light appears to converge towards object
Virtual object
Light converges towards image
Real image
Light diverges from image
Virtual image

Table 3: Properties of Plane Mirror

Property
Value / Point
Image position
Behind mirror
Image distance
Equal to object distance
Relation
\(v=u\)
Magnification
\(m=\frac{v}{u}=1\)
Radius of curvature
\(R=\infty\)
Focal length
\(f=\infty\)
Power
0 dioptre
❖ Minimum Size of Mirror:

Table 1: Minimum Mirror Size

Purpose
Minimum mirror size
To see full image of oneself
\(\frac{1}{2}\) height of observer
Mirror on wall; observer at middle of room sees full wall behind
\(\frac{1}{3}\) height of wall
To see one's full image across end of room
\(\frac{1}{2}\) height of observer
▢ Deviation by Plane Mirror:

Table 1: Thin Plane Mirror

Quantity
Formula / Point
Reflection
\(i=r\)
Deviation
\(\delta=180^\circ-2i\)
Maximum deviation
\(180^\circ\)
Condition for maximum deviation
Ray falls parallel to normal
Depends on thickness
No
Depends on refractive index
No

Table 2: Two Inclined Plane Mirrors

Quantity
Formula / Point
Deviation by 1st mirror
\(\delta_1=180^\circ-2\alpha\)
Deviation by 2nd mirror
\(\delta_2=180^\circ-2\beta\)
Net deviation
\(\delta=360^\circ-2\theta\)
Depends on angle of incidence
No
▢ Rotation of Plane Mirror:

Table 1: Rotation Rules

Condition
Result
Mirror rotated by \(\theta\)
Normal rotates by \(\theta\)
Incident ray fixed; mirror rotated by \(\theta\)
Reflected ray rotates by \(2\theta\)
Mirror angular velocity = \(\omega\)
Normal angular velocity = \(\omega\)
Mirror angular velocity = \(\omega\)
Reflected ray angular velocity = \(2\omega\)
▢ Number of Images by Two Inclined Mirrors:

Table 1: Formula for Number of Images

Condition
Number of images
\(\frac{360^\circ}{\theta}\) even integer
\(n=\frac{360^\circ}{\theta}-1\)
Even integer case
Same formula whether object lies on bisector or not
\(\frac{360^\circ}{\theta}\) odd integer; object on bisector
\(n=\frac{360^\circ}{\theta}-1\)
\(\frac{360^\circ}{\theta}\) odd integer; object not on bisector
\(n=\frac{360^\circ}{\theta}\)
\(\frac{360^\circ}{\theta}\) fraction
Take integral part
Example
\(3.100\Rightarrow n=3\)
Example
\(3.999\Rightarrow n=3\)
❖ Special Cases:
  • •
    Two mirrors at \(90^\circ\): \(n=\frac{360}{90}-1=3\)
  • •
    At right angle, 3 images form; 2 are laterally inverted
  • •
    Two adjacent walls + ceiling mirrored → 7 images
▢ Displacement and Velocity of Image in Plane Mirror:
❖ **table:
    ◈ caption: Image Velocity
    ◈ data:
      1. Condition
      2. Formula
      1. Object velocity \(v_o\), mirror velocity \(v_m\)
      2. \(v_i=v_o+2v_m\)
      1. Relative velocity of image w.r.t. mirror
      2. \(v*{IM}=v_o+v_m\)
      1. Relative velocity of image w.r.t. object
      2. \(v*{IO}=2(v_o+v_m)\)
      1. Person approaches fixed plane mirror with speed \(v\)
      2. Relative velocity = \(2v\)
      1. Object moves with speed \(v\) at angle \(\theta\) with normal
      2. Relative velocity of image w.r.t. object = \(2v\cos\theta\)
▢ Spherical Mirrors:

Table 1: Nature of Image of Real Object

Mirror
Nature of image
Convex mirror
Virtual, erect, diminished
Concave mirror
Real, inverted, magnified/diminished OR virtual, erect of image
Convex mirror
Virtual, erect, diminished, magnified
❖ Ray Rules:
  • •
    Ray parallel to principal axis → passes through focus after reflection
  • •
    Ray passing through focus → becomes parallel to principal axis after reflection
  • •
    Ray passing through centre of curvature → retraces same path after reflection
❖ Mirror Formula: \(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\)
❖ Radius-Focal Relation: \(R=2f\)
❖ Distance Measurement: All distances \(u,v,f,R\) are measured from pole of mirror
❖ Focal Length Sign:
  • •
    Concave mirror: \(f=+ve\)
  • •
    Convex mirror: \(f=-ve\)
▢ Sign Convention Used in Notes:

Table 1: Mirror Sign Convention

Mirror / Case
\(u\)
\(v\)
\(f\)
\(m\)
Concave mirror / convex lens: real image
+
+
+
+
Concave mirror / convex lens: virtual image
+
-
+
-
Convex mirror / concave lens: virtual image
+
-
-
-
▢ Magnification:

Table 1: Types of Magnification

Type
Condition
Formula
Transverse / lateral / linear magnification
1D object perpendicular to principal axis
\(m_T=\frac{I}{O}=\frac{v}{u}=\frac{f}{u-f}=\frac{v-f}{f}=\sqrt{\frac{y}{x}}\)
Longitudinal / axial magnification
1D object parallel to principal axis
\(m_L=\frac{v_A-v_B}{u_A-u_B}=\frac{dv}{du}=-\left(\frac{v}{u}\right)^2=-m_T^2=-\frac{y}{x}\)
Superficial magnification
2D object perpendicular to principal axis
\(m_s=\frac{A_I}{A_O}=m_a m_b\)
For square object
\(m_a=m_b=m\)
\(m_s=m^2\)
▢ Image Velocity in Spherical Mirror:
❖ **table:
    ❖ Negative Sign: If object moves towards mirror, image moves away from mirror and vice versa
    ❖ Note: Speeds are related with axial magnification
    ▢ Concave Mirror Image Formation:

    Table 1: Location, Size and Nature of Images by Concave Mirror

    Object location
    Image location
    Magnification
    Nature
    Beyond C; \(u>2f\)
    Between F and C; \(f
    \(m<1\)
    Real, inverted, diminished, in front of mirror
    At C; \(u=2f\)
    At C; \(v=2f\)
    \(m=1\)
    Real, inverted, same size, in front of mirror
    Between C and F; \(f
    Beyond C; \(v>2f\)
    \(1
    Real, inverted, magnified, in front of mirror
    At focus; \(u=f\)
    At infinity; \(v=\infty\)
    \(m=\infty\)
    Real, inverted, highly magnified
    Between pole and focus; \(u
    Behind mirror
    \(m>1\)
    Virtual, erect, magnified
    At infinity; \(u=\infty\)
    At focus; \(v=f\)
    \(m<1\approx0\)
    Real, inverted, diminished
    At pole; \(u=0\)
    At pole; \(v=0\)
    \(m=1\)
    Virtual, erect
    ❖ Uses:
    • •
      Shaving mirror
    • •
      Ophthalmoscope
    • •
      Cinema projector
    • •
      Used when object is between pole and focus to get magnified virtual image
    ▢ Convex Mirror Image Formation:

    Table 1: Location, Size and Nature of Images by Convex Mirror

    Object location
    Image location
    Magnification
    Nature
    At infinity
    At focus
    \(m<1\)
    Virtual, erect, diminished, behind mirror
    Anywhere between infinity and pole
    Between focus and pole
    \(m<1\)
    Virtual, erect, diminished, behind mirror
    ❖ Use: Rear-view mirror in vehicles due to maximum field of view
    ▢ Field of Vision:

    Table 1: Field of View

    Mirror
    Field of view
    Convex mirror
    Maximum
    Plane mirror
    More than concave; less than convex
    Concave mirror
    Least
    ▢ Special Formulae:

    Table 1: Important Formulae

    Condition
    Formula / Point
    Apparent thickness of thick plane mirror
    \(t'=\frac{t}{\mu}=\frac{v+t-u}{2}\)
    Refractive index
    \(\mu=\frac{t}{t'}\)
    Newton's formula for real image
    \(f^2=x_1x_2\)
    Real image \(n\) times by concave mirror / convex lens
    \(u=\frac{n+1}{n}f\)
    Virtual image \(n\) times by concave mirror / convex lens
    \(u=\frac{n-1}{n}f\)
    Convex mirror / concave lens image \(\frac{1}{n}\) times
    \(u=(n-1)f\)
    Minimum distance between object and real image by concave mirror
    0
    Minimum distance between object and real image by convex lens
    \(4f\)
    Largest image distance from convex mirror of focal length \(f\)
    \(f\)
    ▢ Read and Digest:

    Table 1: Important Points

    Fact
    Answer
    Thick mirror forms multiple images
    Second image is brightest
    Convex mirror
    Always forms virtual, erect, diminished image
    Parabolic mirror
    Used in torch and vehicle headlights
    Focal length of spherical mirrors
    Same for all colours
    Spherical mirror immersed in liquid
    Focal length unchanged
    Plane mirror focal length
    Infinity
    Plane mirror power
    Zero
    3 mutually perpendicular plane mirrors
    Incident and final reflected rays are opposite; angle = 180°
    Man between 2 right-angle mirrors combing with right hand
    Seen combing with right hand in only one image
    Object moving from infinity to focus of concave mirror
    Image velocity first decreases then increases
    Object moving from focus to infinity of concave mirror
    Image velocity first increases then decreases
    Bird flying towards large concave mirror along principal axis
    Magnification first increases from 0 to infinity, then decreases from infinity to 1
    Q1.
    Power of a plane mirror in dioptre is
    📅MOE 2011
    Q2.
    The rear-view mirror in a car is
    📅KU 2014
    Q3.
    When a plane mirror is rotated through an angle 30° keeping incident ray constant, reflected ray is rotated through an angle
    📅KU 2008
    Q4.
    If an object is placed symmetrically between two plane mirrors inclined at an angle of 72°, the number of images formed is
    📅IE 2004
    Q5.
    An object is placed at a distance twice the focal length of a concave mirror. The image formed is
    📅IBP 2010
    Q6.
    A concave mirror has radius of curvature 20 cm. An object is placed 10 cm from the pole of the mirror. The image will be
    📅BPKIHS 2001
    Q7.
    An object is placed at 20 cm from a convex mirror of focal length 10 cm. The image formed is
    📅KU 2011
    Q8.
    A person approaches a plane mirror with velocity v. The relative velocity of approach of person and his image is
    📅BPKIHS 1996
    Q9.
    An object moves towards a plane mirror with velocity v making angle θ with normal. The velocity of image with respect to object is
    Q10.
    An ant moves towards a plane mirror with speed 2 m/s and the mirror moves towards the ant with same speed. The relative velocity between ant and its image is
    📅IOM 1998•MOE 2064
    Q11.
    Concave mirror
    📅KU 2014
    Q12.
    A virtual image larger than the object is formed by
    ▢ High-Yield Recall:

    Table 1: Reflection One-Liners

    Fact
    Answer
    Reflection
    Returning back of light in same medium
    Law of reflection
    \(i=r\)
    Plane mirror deviation
    \(\delta=180^\circ-2i=2g\)
    On reflection
    Speed, frequency, wavelength unchanged
    Reflection from denser medium
    Phase change \(\pi\)
    Regular reflection
    Smooth surface
    Diffused reflection
    Rough surface
    Plane mirror image
    Virtual, erect, laterally inverted
    Plane mirror magnification
    1
    Plane mirror focal length
    \(\infty\)
    Plane mirror power
    0
    Minimum mirror for full image
    \(\frac{1}{2}\) height
    Mirror rotated by \(\theta\)
    Reflected ray rotates by \(2\theta\)
    Two inclined mirror deviation
    \(360^\circ-2\theta\)
    Object fixed, mirror moves by \(z\)
    Image moves by \(2z\)
    Mirror fixed, object moves by \(z\)
    Image moves by \(z\)
    Relative speed in fixed plane mirror
    \(2v\)
    Mirror formula
    \(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\)
    Radius-focus relation
    \(R=2f\)
    Concave mirror focal length
    Positive
    Convex mirror focal length
    Negative
    Convex mirror image
    Always virtual, erect, diminished
    Concave mirror object at focus
    Image at infinity
    Concave mirror object at C
    Image at C, same size
    Concave mirror object between P and F
    Virtual, erect, magnified
    Transverse magnification
    \(m_T=\frac{v}{u}=\frac{f}{u-f}\)
    Longitudinal magnification
    \(m_L=-m_T^2\)
    Superficial magnification
    \(m_s=m^2\)
    Newton's mirror formula
    \(f^2=x_1x_2\)
    Maximum field of view
    Convex mirror
    Parabolic mirror
    Torch and headlights
    Focal length of mirror in liquid
    Unchanged
    Q1.
    Two mirrors are at 60°, the number of image formed is
    📅BP 2012/2016
    Q2.
    An object is placed at a distance twice the focal length of concave mirror. Then image formed is:
    📅BP 2010
    Q3.
    A plane mirror is rotated by an angle θ. The change in deviation of a ray produced by the mirror is
    📅MOE 2012
    Q4.
    Power of a plane mirror in Dioptre is
    📅MOE 2011
    Q5.
    A ray of light falls on the surface of a spherical glass paper weight making an angle α with the normal and is refracted in the medium at an angle β. The angle of deviation of the emergent ray from the direction of the incident ray is
    📅IOM 2009
    Q6.
    Concave mirror
    📅KU 2014
    Q7.
    The rear view mirror in a car is
    📅KU 2014
    Q8.
    An object is placed at 20 cm from a convex mirror of focal length 10 cm. The image formed by the mirror is:
    📅KU 2011
    Q9.
    An ant moves towards the plane mirror with speed of 2 m/s & the mirror is moved towards the ant with the same speed. What is the relative velocity between the ant and its image?
    📅IOM 98/MOE 2064
    Q10.
    When a mirror is rotated through an angle 30° keeping incident ray constant then reflected ray is rotated through an angle
    📅KU 08
    Q11.
    Two mirrors A and B are inclined at angle θ. A ray of light incident in mirrors B is deviated to 62° and the angle of emergence is 20°, then find the angle of inclination.
    📅IOM 2066
    Q12.
    If an object is placed symmetrically between two plane mirrors inclined at an angle of 72°. The number of images will be
    📅IE-04
    Q13.
    A concave mirror has radius of curvature 20 cm. An object is placed 10 cm from the pole of the mirror. The image will be at
    📅BPKTHS 01
    Q14.
    A person approaches a plane mirror with velocity v then the relative velocity of approach of person and his image is
    📅BPKIHS-96
    Q15.
    An object is moving towards a plane mirror with a velocity v making a certain angle θ with normal of a plane mirror. The velocity of image w.r.t object is
    📅
    Q16.
    A cubical room is formed with 6 plane mirrors. An object started to move along the diagonal of floor. The velocity of image in two adjacent walls is 20√2 cm/s, then the velocity of image along the diagonal of the roof is:
    📅
    Q17.
    Two plane mirrors are inclined at a certain angle undergoes a deviation of 300°. The number of images observe is
    📅
    Q18.
    Two plane mirrors inclined at an angle θ form 9 images of an object placed symmetrically between them. Then the angle θ is:
    📅
    Q19.
    A ray of light is incident on a plane mirror at an angle of incidence 30°. The ray after reflection is deviated through
    📅
    Q20.
    A person is approaching a plane mirror with speed 10 cm/s. If the initial distance between person and mirror is 2m, then the distance between person and his image after 2.5 sec will be
    📅
    Q21.
    It is desired to photograph the image of an object placed at a distance 3m from a plane mirror. The camera which is at a distance of 4.5m from the mirror should be focused at a distance of:
    📅
    Q22.
    A ray is reflected in turn by three plane mirrors mutually at right angles to each other. The angle between the incident and reflected rays is:
    📅
    Q23.
    Two inclined plane mirrors are inclined at an angle 60° with each other. A ray of light travelling horizontally is reflected first from one mirror and then from the other mirror. Then the resultant deviation is
    📅
    Q24.
    A ray of light incident to the first mirror and parallel to the second mirror is reflected from the second mirror parallel to the first mirror. The angle between two mirrors is
    📅
    Q25.
    A point object is placed on principal axis of a concave mirror of focal length 20cm at distance 30cm from pole. The image is formed at distance
    📅
    Q26.
    How far should an object be held from concave mirror of focal length 40 cm so as to obtain a real image twice the size of the object?
    📅
    Q27.
    A bright spot situated at 60cm in front of convex mirror forms a virtual image 20cm behind the mirror. The focal length of the mirror is:
    📅
    Q28.
    A shaving mirror of focal length f produces an image x times the size of the object. If the image is real, then the distance of the object from the mirror is:
    📅
    Q29.
    What is the magnification when the object is placed at 2f from the pole of a convex mirror?
    📅
    Q30.
    A spherical mirror produces an image of magnification 3. Then the distance of the object from the mirror may be, if the focal length of spherical mirror is 24cm
    📅
    Q31.
    A convex mirror of focal length 20cm produces an image 1/4 times the size of the object. Then the distance in between the object and its image is
    📅
    Q32.
    A spherical mirror produces an image 3 times of the size of object. If the image is erect and the distance between the object and its image is 100cm then the focal length of the spherical mirror is:
    📅
    Q33.
    A short linear object of length 'b' is placed along the axis of a concave mirror. The distance of object from the pole of a concave mirror is u. Then the size of the image is equal to:
    📅
    Q34.
    A luminous object is placed 50cm from surface of a convex mirror and a plane mirror is set so that virtual images in two mirrors coincide. If plane mirror is at a distance of 30cm from object, then focal length of convex mirror is:
    📅
    Q35.
    A rod of length 10cm is placed parallel to the principal axis of a concave mirror. The nearest point of a rod is at a distance of 50cm from the mirror. The focal length of the mirror is 30cm. Then the length of the image is
    📅
    Q36.
    A thin rod of length f/2 lies along the axis of concave mirror of focal length f such that its real elongated image just touches one end of the rod. The length of the magnified image is:
    📅
    Q37.
    A thin rod of length f/2 lies along the axis of concave mirror of focal length f such that its real diminished image just touches the end of the rod. The length of its image is:
    📅
    Q38.
    A square of side 3cm is placed at a distance of 25cm from a concave mirror of focal length 10cm. The centre of a square is passing through the principal focus and plane is normal to the principal axis. Then area of image is:
    📅
    Q39.
    The focal length of a concave mirror is f and the distance of the object to the principal focus is p. The ratio of the size of the image to the size of the object is:
    📅
    Q40.
    Two plane mirror parallel to each other are 10m apart. An object is placed at a distance of 4m from one of the mirrors. What is the distance between two second images formed by the two mirrors?
    📅
    Q41.
    A man is 180 cm tall and his eyes are 10 cm below the top of his head. In order to see his entire height right from the feet to the head he uses a plane mirror at a distance of 1m from him. The minimum height of the plane mirror required is
    📅KU 2015
    Q42.
    If the object is real, a convex mirror always forms
    📅IOM 2015
    Q43.
    Focal length of convex mirror is 30cm. If image is 1/5 times magnified. Then object distance will be?
    📅IOM 2015