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Spherical Mirrors

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Spherical Mirrors - Lesson Summary

  

Mirrors are the basic means of viewing our own beauty. Generally we can classify the mirrors into the following two types as

i.  Plane mirrors
ii. Curved mirrors.

Generally mirrors refer to plane mirrors. But if the surface of a mirror is curved it is said to be a curved mirror. If the curved mirror is a part of a huge sphere, then the mirror is a spherical mirror.



Terms Associated with Spherical Mirrors

  • Centre of curvature (C) is the centre of the sphere, of which the mirror is a part.
  • Radius of curvature (R) is the radius of the sphere, of which the mirror is a part.
  • Pole (P) is the geometric centre of the spherical mirror.
  • Principal axis is the line joining the pole and the centre of curvature.
  • Principal focus (F) is the point on the principal axis, where a parallel beam of light, parallel to the principal axis after reflection converges in the case of a concave mirror and appears to diverge from in the case of a convex mirror.
  • Focal length (f) is the distance of the principal focus from the pole of the mirror.                    

Spherical mirrors can be further classified into the following two types as

i.  Concave mirrors
ii . Convex mirrors.

The images formed by the mirrors are of two types they are

i.  Real images
ii. Virtual Images

Real images are those that can be caught on a screen while virtual images are those that cannot be caught on a screen.

Formation of Images by Spherical Mirrors
The image formed by a convex mirror is always erect, virtual, and diminished in size. The location of the object does not affect the characteristics of the image. Thus, as the object approaches the mirror, the image approaches the mirror too but not proportionately. This is why, the rear view mirrors of the cars and bikes are made of convex mirrors.

Hence, we have the caution “Objects seen in the mirror are closer than they appear” printed on the outside rear view mirrors of vehicles.
 
Unlike in a convex mirror, the nature and size of the image in a concave mirror depends on the distance of the object from the mirror.


Concave Mirror



If a part of a hollow glass sphere is cut and the cut part of the sphere is coated outside with silver or similar material, then its inner surface reflects the entire light incident on it, and thus, forms a mirror. Since the inner surface is a concave surface, the mirror so formed is called a concave mirror.

  • The geometric centre of a concave mirror is called its pole. 
  • The centre of the sphere from which concave mirror was cut is called the centre of curvature of the concave mirror. 
  • The distance from any point on the concave mirror to its centre of curvature is called the radius of curvature of the concave mirror. 
  • An imaginary line passing through the centre of curvature and the pole of the concave mirror is called principal axis of the concave mirror. 
  • The area of a concave mirror that is exposed to incident light is called the aperture of the concave mirror.
  • The length along the principal axis from the pole to the principal focus is called the focal length of the concave mirror.

If an object is placed close to a concave mirror such that the distance between the mirror and the object is less than its focal length, then a magnified and virtual image is formed. Due to this property, concave mirrors are used in many applications. A concave mirror can be used as a shaving mirror, and by dentists to view clearly the inner parts of the mouth.
Concave mirrors converge the light incident on them and hence are called converging mirrors. We can observe ourselves magnified when the mirror is placed close to our face. This is due the position of the object between the focus and the pole. As the object moves away from the mirror, the size of its image reduces along with its distance from the mirror. If an object is placed close to a concave mirror such that the distance between the mirror and the object is less than its focal length, then a magnified and virtual image is formed. Due to this property, concave mirrors are used as shaving mirrors, and by dentists to view clearly the inner parts of the mouth.

Reflection by Concave Mirrors                           Incident Ray                        Reflected Ray  Parallel to principal axis  Passes through focus  Passes through C  Retraces its path  Passes through focus  parallel to principal axis  Strikes the pole at an angle eith principal axis  Makes the same angle with principal axis

Image Formation by Concave Mirror

Depending on the position of the object in front of the concave mirror, the position, size and the nature of the image varies.


Object at infinity
A real, inverted, highly diminished image is formed at the focal point F, in front of the concave mirror.




Object beyond C
A real, inverted, diminished image is formed between C and F, in front of the concave mirror.
 



Object at C
A real, inverted, same sized image is formed at C, in front of the concave mirror.




Object between C and F
A real, inverted, enlarged image is formed beyond C, in front of the concave mirror.




Object at F
A real, inverted, highly enlarged image is formed at infinity, in front of the concave mirror.




Object between F and P
A virtual, erect and enlarged image is formed behind the concave mirror.


Image Formation by a Concave Mirror   Object Location   Image Location   Nature of Image   Infinity   At F • Real
• Inverted
• Highly Diminished   Beyond C   Beyond F and C • Real
• Inverted
• Diminished   At C   At C • Real
• Inverted
• Equal to size of object   Between C and F   Beyond C • Real
• Inverted
• Magnified   At F   Infinity • Real
• Inverted
• Highly Magnified   Between F and P   Behind the mirror • Virtual
• Erect
• Magnified



Uses of Concave Mirrors
  • Concave mirrors are used as shaving mirrors to see a larger image of the face.
  • Dentists use concave mirrors to view a magnified view of the interior parts of the mouth are concave.

  • ENT doctors use them for examining the internal parts of the ear, nose and throat.
  • They are used as reflectors in the headlights of vehicles, search lights and in torch lights to produce a strong parallel beam of light.
  • Huge concave mirrors are used to focus sunlight to produce heat in solar furnaces.

Convex Mirror



If the cut part of the glass sphere is coated from inside with silver or a similar material, then its outer surface reflects the entire light incident on it, and thus forms a mirror. Since the outer surface is a convex surface, the mirror so formed is called a convex mirror.

  • The geometric centre of a convex mirror is called its pole. 
  • The centre of the sphere from which the mirror was cut is called the centre of curvature of the mirror. 
  • The distance from any point on the convex mirror to its centre of curvature is called the radius of curvature. 
  • An imaginary line passing through the centre of curvature and the pole of the mirror is called its principal axis.
  • The reflected rays, when projected backwards, appear to meet at a point on the principal axis. This point is called the principal focus. The length along the principal axis from the pole to the principal focus is called the focal length. 
  • The area of a convex mirror that is exposed to incident light is called the aperture. If the aperture of a convex mirror is small, then its focal length is equal to half its radius of curvature.

Convex mirrors diverge the light incident on them and hence they are called the diverging mirrors. Due to this they always form diminished, virtual and erect images irrespective of the position of the object in front of them. Thus, the magnification produced by these mirrors is always less than one. The field of view for a convex mirror is greater than that for a plane mirror, the aperture being the same. Hence, convex mirrors are used as rear-view mirrors in vehicles. It is also installed behind automated teller machines as a security measure.

The field of view for a convex mirror is greater than that for a plane mirror, the aperture being the same. Hence, convex mirrors are used as rear-view mirrors in vehicles. It is also installed behind automated teller machines as a security measure. The images formed by convex mirrors are always diminished, virtual and erect, irrespective of the position of the object.



Reflection by Convex Mirror

                        Incident Ray                     Reflected Ray   Parallel to principal axis  Appears to pass through focus  Directed towards the focus  Appears to pass parallel to principal axis  Strikes the pole at an angle eith principal axis  Makes the same angle with principal axis

 
Image Formation by Convex Mirror

Irrespective of the position of the object, a virtual, erect and diminished image is formed between F and P, behind the convex mirror. 




Uses of Convex Mirrors

  • Used as rear view mirrors in automobiles as it covers wide area behind the driver.
  • Used as reflectors for street light bulbs as it diverges light rays over a wide area.
  • Used as Rear view mirrors of vehicles and the ones used in ATM centres.

Sign Convention for Spherical Mirrors
  • Object is always considered at the left of mirror

  • Distances measured along y-axis above the principal axis are taken as positive and that measured along y-axis below the principal axis are taken as negative.

  • Distances measured in the direction of the incident ray are taken as positive and the distances measured in the direction opposite to that of the incident rays are taken as negative.

  • All distances are measured from the pole of the mirror.


Table Showing the Sign Convention Types of Mirror  u                     v  f  R Height of the Object  Height of the Image  Real  Virtual      Real  Virtual  Concave mirror  –        –     +  –   –              +        –       +  Convex mirror  –  No real image     +  +  +             +  No real image       +


Rules for Construction of Ray Diagrams for Spherical Mirrors

Rule 1: A light ray incident parallel to the principal axis, after reflection, either actually passes through the principal focus or appears to pass through the principal focus.
Rule 2: A light ray which first passes through the principal focus or appears to pass through the principal focus, after reflection, will travell parallel to the principal axis.  
Rule 3: A light ray which first passes through the centre of curvature or appears to pass through the centre of curvature, after reflection, retraces its initial path. 


Mirror Formula
The relation between the focal length (f), object distance (u) and the image distance (v) is given by

1 f =  1 v +   1 u   


Magnification

The ratio of the height of the image in a spherical mirror, to the height of the object is called magnification (m)

Magnification (M) =  Heigth of the Image (Hi) Height of the Object (Ho) =  Image Distance (v) Object  distance(u)


The distance from the principal focus to the pole of the mirror is the focal length of the mirror and is equal to half the radius of curvature, which is the distance between the centre of curvature and the pole.

For a real image object distance (u) and the image distance (v) are negative and the magnification is negative. If the  magnification of an image is negative it does mean that the image is real and inverted.
On the other hand for a virtual image object distance (u) is negative and image distance (v) is positive and hence the magnification is positive, i.e., the image is erect. If the  magnification of an image is positive it does mean that the image is virtual and erected.

If the magnification is less than 1 The image formed is diminished in size.
If the magnification is more than 1 The image formed is magnified in size.
If the magnification is equal to 1 The image formed is  equal to the object in size.


Differences Between Convex Mirror and Concave Mirror

                Convex Mirror                  Concave Mirror 1. Convex mirror is curved outwards.

2. The focal point of convex mirror is behind the mirror.

3. In convex mirror the image is always virtual, upright and smaller than the object.




4. Convex mirrors are used in cars (as passenger-side mirror since they provide upright and wide view), they are also used in camera phones, for safety measures there are also used in roads and driveways.
Besides these convex mirrors are found in many hospitals, schools etc. as hallway safety mirror.

5. It has a virtual focus. 1. Concave mirror is curved inwards.

2. The focal point of concave mirror is in front of the mirror.

3. In case of concave mirror different types of images are formed on different location of the object. The image is upside down (inverted) and far away but if we bring the object close to the mirror then image will be larger and upright.

4. Concave mirrors are used in telescope. These are also used as make up and shaving mirrors since these provide larger images. 
Besides these concave mirrors are used by dentists and also used in headlights of cars, solar devices, satellite dishes etc.


5. It has a real focus.

  

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