Spherical Mirrors

What is a Spherical Mirror?

A spherical mirror is a mirror that has the shape of a piece carved out of a sphere.

Spherical mirror is categorised into two forms, namely: Concave and Convex.

[Image will be Uploaded Soon]

  On this page, we'll learn about the following:

  •  Type of spherical Mirrors

  •  Application: usage, Examples

  • Derivations for a mirror formula with a ray diagram

  • Mirror images with their attributes: R, C, f, P, and Principal axis

Types of Spherical Mirrors

[Image will be Uploaded Soon]

  • Concave mirror: Outer surface: silvery polished, inner surface: reflective

  • Convex mirror: Outer surface: Reflective, Inner surface: Polished 

  Light reflected by convex and concave mirrors

[Image will be Uploaded Soon]

Applications

  • Concave Mirror  

Reflectors, Converging of light, Solar cooker, motor vehicle headlight, shaving, and makeup mirror, microscope, Telescope, Satellite dishes, dentist's mirrors.

  • Convex Mirror

 It is used in the security monitors, Side-view mirrors in cars, Security mirrors in ATMs, in buildings such as hotels, hospitals, stores, schools, etc.

  Key points: Consider a ray image of a Concave Mirror

 [Image will be Uploaded Soon]

Center of Curvature (C) = It is the center of the sphere from which the mirror is carved out.

Radius of Curvature  (R): It is the radius of the sphere from which the mirror is carved out. Measure of Radius is from point C to point P and DC = R.

Pole: The center of the mirror is called the Pole.

Focus or focal point (F): Rays of light parallel to the principal axis, after reflecting meet at a common point on the principal axis, that common point is the focal point (focus).

Since the rays actually meet at point F, that’s why it’s also known as Real focal point.

Focal length (f) = The distance between Focus and Pole.

Principal axis: The line joining the center of Curvature and the pole.

For a convex mirror, the rest of the definition would remain the same except that of Focus or focal point (F).

 Let's have a look at the ray diagram of the convex mirror:

[Image will be Uploaded Soon]

In this ray diagram, a ray of light coming from infinity, hits the polished surface and traces its path backward (reflects back), we produce the ray backside and they appear to meet (not actually) at point ‘F.’ 

So, The focal point or focus (F) = The ray of light parallel to the principal axis,after reflecting diverges, the diverging ray when produced to meet at common point, that 

The common point is known as Focal point or focus (F).

 Here we are dealing with mirrors of small apertures.

The rays thus formed are very close to the principal axis and thus considered to be ‘Paraxial.’

Derivation of Mirror Formula

Two terms to be discussed for this derivation are: ‘u’ and ‘v.’

Here, 

 f = focal length

 a =  The distance of object from the pole of the mirror.

 b =  The distance of the image formed from the mirror.

 Few conventions that are important to understand before deriving the equation

  • All distances are measured from the Pole ‘P.’

  • Distances measured in the direction of incidence of light are considered as positive, and those taken from the opposite direction are considered as negative.

  • All heights above the Principal axis are positive and those below it are negative.

  • Ray diagram determines the positive and negative image distance.

 Let's have a look at the ray diagram of a Concave Mirror

[Image will be Uploaded Soon]        

In the fig. B :

G P = b (distance of image from P)

AP = a (distance of object from P)

The point at which these two rays meet is ‘F.’

BD is a tangent drawn to the mirror.

Now, let’s take Δ OAF and Δ DPF

[Image will be Uploaded Soon]

Here, m (∠  OAF) = m (∠  DPF)

Because  ∠ OAF = ∠ DPF = 90° 

m (∠  OFA) =  m (∠  DFP)  (Vertically opposite angles)

And, m (∠ AOF) = m (∠  PDF), how?

 If two sets of corresponding angles are equal, then the third set has to be equal.

then,   m (∠ AOF) = m (∠  PDF)

 Hence, we proved that: Δ OAF Δ DPF  by AAA similarity.

  OA/ DP = AF/ PF= OF/ DF

  =  OA/ DP = AF/ PF

Here,         PF = f, AP = a

                   OA/ DP = AF/ f

                   AP = AF + FP = AF + f

           a = AF+ f

          AF = a - f

OA is the height of the object = ho

and,  DP = height of the image = hi

Now,  we have

ho/ hi = a - f/ f…….(1)

Now, consider Δ IGF and Δ BPF:

[Image will be Uploaded Soon]

Δ IGF 〜  Δ BPF by AAA similarity. 

Here, m (∠  IGF) = m (∠ BPF) = 90° 

 and, m (∠ IFG) = m (∠  BFP) ( Vertically opposite angles),  also

 m ( ∠ FIG) = m (∠  FBP)

IG/ BP = GF/ PF = IF/ BF

IG = hi and BP = ho

=  hi/ ho =  GF/ f

=  GP = GF + FP 

 =   v =  GF + f

=  GF = f - b

hi / ho = b - f / f  or  ho/ hi = f / b - f….(2)

Now, 

 Eq ..(1)  = Eq..(2)

= ( a - f) /  f  =  f /  (b - f)  as  hi / ho is common in both the equations (1) and (2):

= ab - bf - af  + f ^2  = f ^2

= ab = f * (a + b)

=   1/ f = (a + b)/ ab 

= 1/ f =  1/a + 1/ b…..Mirror equation.

Summary:

We summarised that the equation  1/f = 1/ a + 1/ b is same for both concave 

and convex mirrors.

The tangent BD we drew to the mirror (in Fig B) in which points B and D are considered as a single point on the mirror is of a small aperture.

Magnification (m) = hi/ ho = a/ b

Conceptual  Questions:

Q1: An object is placed at a distance of 30m from a concave mirror. It forms an image

is 1/ 3 of the size of the object. Calculate the position of the image from the mirror and focal length of the mirror.

Ans:  Given , a = 30 cm

      hi  =  1/ 3  of ho

Since, m =  hi/ ho =  a / b = 1/ 3 of ho / ho =  b/ 30

We get, b =  1/ 3 * 30 = 10 cm

 Using mirror formula:

1/b + 1/a = 1/f

1/10 + 1/30 = 1/f  

1/f = 4/ 30

Since, f = 30/4 = 7.5 m 

Therefore, the focal length of the concave mirror is 7.5 m 


Q2: An object is placed 20cm from a concave mirror. The focal length is 4 cm.  Determine (a) The image distance (b)  magnification of image.

Ans:  

The focal length (f) = 4cm 

Object distance (a) = 20cm

[Image will be Uploaded Soon]

Fig.  Ray Diagram for this question

1/f = 1/b + 1/a

1/ 4 = 1/b + 1/20

= 1/b =  1/ 4 - 1 /20

= 1/b = 4 / 20  

=  b = 20 / 4 = 5 cm

Now, Magnification (m) :

   m = - b /a  = - 5/ 20 = - 0.25