Derivation of Formula for Curved Mirrors:
Observe the below figure. The ray from the top of the object AB is emanating and passing parallel to the principal axis and striking the mirror at X. After reflection it passes through Focus and another ray starting from B passes through the Center of Curvature (C) and strikes the mirror at Y. After reflection it returns back in the same direction.
The two rays XB' and YB' are meeting at B'. So B' is the image of B. Hence the image of AB is A'B'.
From the given side figure, ∆ABC and ∆A'B'C' are similar triangles,
AB / A'B' = AC / A'C ..................................... ①
Draw a line XP' perpendicular to the principal axis
Similarly, ∆P'XF and A'B'F are also similar triangles
P'X / A'B' = P'F / A'F...................................... ②
From the figure, we can say that P'X = AB
Hence equation ② becomes
AB / A'B' = P'F / A'F ......................................③
From the equations ① and ③, we can write
AC / A'C = P'F / A'F ...................................... ④
If the paraxial rays (rays that are travelling very near to the principal axis) are considered, we can say that P' coincides with P
Then P'F = PF
AC/ A'C = PF / A'F ........................................ ⑤
we can observe from the figure, that
AC = PA - PC
A'C = PC - PA'
A'F = PA' - PF
by substituting these in equation ⑤
PA - PC / PC - PA' = PF / PA' - PF ............... ⑥
we know that PA = u,
PC = R = 2f,
PA' = v
PF = f
u - 2f / 2f - v = f / v - f
(u - 2f)(v- f) = f(2f - v)
uv - uf - 2vf + 2f2 = 2f2 - vf
uv = 2f2 -vf + uf + 2vf - 2f2
uv = uf + vf ...................................................... ⑦
Divide equation ⑦ with uvf
uv / uvf = uf / uvf + vf / uvf
1/f = 1/v + 1/u
This is known as Mirror Formula
Note: While using Mirror Formula, we have to apply Sign Convention in every situation
Sign Convention for the parameters related to the mirror formula:
- All distances should be measured from the pole.
- The distances measured in the direction of incident light to be taken positive and those measured in the direction opposite to the incident light to be taken negatively.
- Height of object (ho) and height of the image (hi) are positive if measured upwards from the axis and negative if measured downwards.
Magnification: Relation between the size of the object and the size of the image
Magnification is the increase in image size produced by the spherical mirrors with respect to the object size.
It is defined as the ratio of the height of the image (hi) to the height of the object (ho) and is denoted by m
magnification, m = height of the image (hi) / height of the object (ho) = -v / u (on applying sign convention)
Magnification is also equal to the ratio of image distance to the object distance (-v / u).
Points to Remember in magnification formula:
Magnification (m) is negative (-ve) for Real and Inverted images whereas it is positive (+ve) for Virtual images. So magnification is always -ve for the convex mirror, while it depends on the position of the object in a concave mirror.
If, (i) m<1 , image is diminished
(ii) m>1, image is enlarged
(iii) m=1, image is of the same size as of that of an object.
Making of Solar Cooker:
- Make a wooden/iron frame in the shape of a TV dish. Cut acrylic mirror sheets into 8 or 12 pieces in the shape of isosceles triangles with a height equal to the radius of your dish antenna.
- The bases of 8 or 12 triangles together make the circumference of the dish.
- Stick the triangle mirror to the dish as shown in the above figure.
- Your Solar Cooker/heater is ready.
- Arrange it so that the concave part faces the sun. Find its focal point/Focus (F) and place a vessel at that point. The vessel gets heated enough to cook rice.
Uses/Application of Spherical Mirrors:
Uses of Concave Mirror:
- It is used as a reflector to concentrate light in the solar cooker.
- It is used as a shaving mirror.
- It is used by dentists to observe cavities in the teeth.
- It is used in a telescope.
- It is used in an ophthalmoscope to examine the eye, nose, throat and ear of a person.
- It is used in searchlight, headlight in automobiles etc.
Uses of Convex Mirror:
- Used as rearview mirror.
- Used as a device to check theft in shops.
- Used to bring the view of corners which are not directly accessible.
- Used to light a large area.
Important Points:
- Real images are those where the light actually converges.
- Virtual images are locations from where light appears to have converged.
- Real images are formed when the objects are placed outside the focal length of a converging mirror.
- Virtual images are formed by placing an object inside the focal length of a converging lens.
- The image distance (i) is positive for real images and negative for virtual images.
- The height of the image and object are positive if measured upwards from the principal axis and negative if measured downwards from the principal axis.
- When light rays shine on a smooth metal surface they are reflected in a way that produces a clear image. This is called Specular Reflection.
- You can see your own reflection in a mirror. The image that you see is very clear because the light rays bounce off the smooth mirror in a precise way.
- When light shines on a rough surface the rays are reflected at many different angles and the image becomes distorted. This is called Diffuse Reflection.
What we have learnt:
- Normal to the curved surface at a point on it is the line joining the centre of curvature and that point.
- Mirror Formula: 1/f = 1/v +1/u
- Magnification: m = size of the image / size of the object = (hi) / (ho)
= image distance / object distance = -v/u
- The image formed by the actual intersection of reflection rays is called a Real Image. This can be caught on screen.
- The image formed by controlled rays of reflection is called a Virtual Image. It cannot be caught on screen.
