Contents
- 1 What are Characteristics of a Virtual Image? What is the Formula for Magnification of a Mirror?
- 1.1 Image Formation by Concave Mirror
- 1.2 Image Formation by Convex Mirror
- 1.3 Sign Convention for Spherical Mirror
- 1.4 Relation Among Object Distance, Image Distance And Focal Length
- 1.5 Conjugate Foci or Conjugate Points
- 1.6 Newton’s Equation
- 1.7 Magnification of an Image Formed by Spherical Mirrors
- 1.8 Magnification in terms of focal length, object distance and image distance:
- 1.9 Areal Magnificafion:
- 1.10 Longitudinal or axial magnification of the image of an object kept along the principal axis:
- 1.11 Formation of Image of a Virtual Object
The laws of Physics Topics are used to explain everything from the smallest subatomic particles to the largest galaxies.
What are Characteristics of a Virtual Image? What is the Formula for Magnification of a Mirror?
Ray tracing method: The position, nature and size of the image of an extended object, formed by a spherical mirror can be determined geometrically. Any extended object can be considered as the sum of point objects and the images of the point objects constitute as the image of the extended object. Any two of the following rays intersecting at a point will indicate the portion of the image.
- A ray parallel to the principal axis, after reflection passes through the focus, or appears to diverge from the focus.
- A ray passing through the focus, after reflection emerges parallel to the principal axis.
- A ray passing through the centre of curvature, after reflection retraces its path in the opposite direction.
Because the ray passing through the centre of curvature incidence on the mirror, is the normal incidence in this case.
Image Formation by Concave Mirror
1. Object at infinity: If an object is at infinity, the rays coming from it may be assumed to be parallel and hence after reflection will meet at the principal focus F. If the rays are oblique after reflection they will meet at a secondary focus F’ Fig.
Thus, the image of an object situated at infinity is formed on the focal plane. The image is real, inverted and very much dimin-ished in size.
2. Object placed between infinity and centre of curvature: An object PQ is placed beyond C on the principal axis of the concave mirror MOM’ [Fig.].
A ray PA starting from P and proceeding parallel to the principal axis, after reflection passes through F along AF.
Another ray PB passing through C is reflected back along BP. The two reflected rays AF and BP meet at p. So, the real image of P is formed at p. Normal pq is drawn on the principal axis, pq is the image of PQ. The image is situated between F and C. The image is real, inverted and diminished relative to the object.
3. Object at the centre of curvature: An object PQ stands at C [Fig.]. A ray PA parallel to the principal axis is reflected through F along AF. Another ray PB through F is reflected along BD parallel to the principal axis. The two reflected rays AF and BD meet at p. So, the real image of P is formed at p. pq is drawn normal on the principal axis, pq is the image of PQ.
The image is real, inverted and of the same size as the object and formed at the centre of curvature itself.
In case of a concave mirror if the object is placed at the centre of curvature the image is also formed at the centre of curvature. Hence only in this case, the intervening distance between object and image is minimum and is equal to zero.
4. Object placed between focus and centre of Curvature: An object PQ stands between F and C [Fig.]. A ray PA parallel to the principal axis is reflected through F along AF. Another ray PB incident normally at B goes back along BC. These two reflected rays meet at p. So, the image of P is formed at p. pq is drawn normal on the principal axis, pq is the image of PQ.
The image is real, inverted and magnified and situated between the centre of curvature and infinity.
5. Object at focus: An object PQ is placed at the focus F [Fig.]. A ray PA parallel to the principal axis is reflected through F along AF. Another ray PB incident normally at B goes back along BC. These two reflected rays being parallel to each other meet at infinity producing the image of P (not shown).
The image is real, inverted and magnified infinitely and situated at infinity.
6. Object placed between focus and pole: An object PQ is placed between the pole and focus [Fig.]. A ray PA parallel to the principal axis is reflected through F along AF. Another ray PB incident normally at B goes back along SC. These two/reflected rays which are divergent would not meet anywhere. But when they are produced backwards they meet at p. Thus they appear to diverge from p. pq is drawn normal on the principal axis. Thus pq is the image of PQ.
The image is virtual, erect, magnified and situated behind the mirror.
Image Formation by Convex Mirror
Consider an object PQ in front of a convex mirror MOM’ [Fig.]. A ray PA parallel to the principal axis goes back along AD. Ray AD appears to come from the focus. Another ray PB incident normally at B is reflected along BP. Reflected rays AD and BP, produced backwards appear to come from p. So, p is the virtual image of P. pq is drawn normal on the principal axis. Thus pq is the image of PQ.
The image is virtual, erect, diminished in size and situated behind the mirror.
For any portion of the object in front of a convex mirror, the image will always be formed behind the mirror. This image is virtual, erect and diminished in size. If the object is brought from infinity close to the pole, the size of the image will gradually increase from a mere point to nearly equal to object size.
Comparative Study of Real and Virtual Images in Case of Spherical Mirror
Characteristics of real image:
- It is formed on the same side of the mirror as the object.
- It is always inverted.
- Real image is not formed in a convex mirror (Exception: vide Section 1.11.4)
Characteristics of virtual image:
- It is always formed on the opposite side of the mirror as the object.
- It is always erect.
- The size of the virtual image becomes larger than the object or equal to it in case of a concave mirror. Whereas in case of a convex mirror it is smaller than the object or equal to it.
Sign Convention for Spherical Mirror
Cartesian sign convention:
- All distances are to be measured from the pole of the spherical mirror.
- All distances measured in a direction opposite to that of the incident rays are to be taken as negative and all distances measured in the same direction as that of the incident rays are to be taken as positive.
- If the principal axis of the mirror is taken as x -axis, the upward distance along positive y-axis is taken as positive while the downward distance along negative y -axis is taken as negative.
See the following Fig.(a) and (b) to understand the above rules.
The nature of sign of object distance (u), image distance (v) focal length (f), radius of curvature (r) and height of the image in case of the image formation of a real object by a spherical mirror is given in the following table.
Relation Among Object Distance, Image Distance And Focal Length
1. In case of concave mirror: Let O, F, C and OQ be the pole, focus, centre of curvature and principal axis of a concave mirror MOM’ [Fig.] respectively. PQ is an object placed perpendicularly on the principal axis in front of the mirror. A ray PA, parallel to the principal axis, after reflection passes through F. Another ray PA’, passing through C, after reflection goes back following the same path. These two reflected rays cuts each other at p. Hence p is the real image of P. pq is the image of PQ. AB is drawn perpendicular on the principal axis.
Triangles PCQ and pCq are similar.
∴ \(\frac{P Q}{p q}\) = \(\frac{C Q}{C q}\) …. (1)
Again, ΔABF and ΔpqF are similar.
∴ \(\frac{A B}{p q}\) = \(\frac{B F}{F q}\)
or, \(\frac{P Q}{p q}\) = \(\frac{B F}{F q}\) [∵ AB = PQ] …. (2)
From (1) and (2) we get, \(\frac{C Q}{C q}\) = \(\frac{B F}{F q}\)
Since the mirror is of small aperture, it can be assumed, BF ≈ OF.
∴ \(\frac{C Q}{C q}\) = \(\frac{O F}{F q}\) ….. (3)
Object distance, OQ = -u; image distance, Oq = -u; focal length, OF = -f; radius of curvature, OC = -r = -2f [Fig].
∴ CQ = OQ – OC = -u + 2f
Cq = OC – Oq = -2f + v
Fq = Oq – OF = -v + f
From (3) we get,
\(\frac{-u+2 f}{-2 f+v}\) = \(\frac{-f}{-v+f}\)
or, uv – uf – 2fv + 2f2 = 2f2 – fv
or, uv = uf + vf
Dividing both sides by uvf we get,
\(\frac{1}{f}\) = \(\frac{1}{v}\) + \(\frac{1}{u}\)
or, \(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\) = \(\frac{2}{r}\) …. (4)
2. In case of convex mirror: Let O, F, C and CQ be pole, focus, centre of curvature and principal axis of a convex mirror MOM’ respectively [Fig.]. PQ is an object placed perpendicularly on the principal axis in front of the mirror. A ray PA parallel to the principal axis and another ray PD proceeding to the centre of curvature, form a virtual image p of P after reflection from the mirror. pq is the image of PQ. AB is drawn perpendicular on the principal axis.
Triangles PCQ and pCq are similar.
∴ \(\frac{P Q}{p q}\) = \(\frac{C Q}{C q}\) …. (5)
Again Δ ABF and Δ pqF are similar.
∴ \(\frac{A B}{p q}\) = \(\frac{B F}{F q}\)
or \(\frac{P Q}{p q}\) = \(\frac{B F}{F q}\) [∵ AB = PQ] …. (6)
From (5) and (6) we get,
\(\frac{C Q}{C q}\) = \(\frac{B F}{F q}\)
Since the mirror is of small aperture, it can be assumed, BF ≈ OF.
∴ \(\frac{C Q}{C q}\) = \(\frac{O F}{F q}\) …. (7)
Object distance, OQ = -u; image distance, Oq = + u; focal length, OF = +f; radius of curvature, OC = +r = +2f (Fig.).
∴ CQ = OQ + OC = -u + 2f
Cq = OC – Oq = +2f – v
Fq = OF – Oq = +f – v
From (7) we get,
\(\frac{-u+2 f}{+2 f-v}\) = \(\frac{+f}{+f-v}\)
or, 2f2 – vf = -uf + 2f2 + uv – 2vf
or, uv = uf + vf
Dividing both sides by uvf we get,
\(\frac{1}{f}\) = \(\frac{1}{v}\) + \(\frac{1}{u}\)
or \(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\) = \(\frac{2}{r}\) …. (8)
It may be noticed that using the same sign convention, the relation between the various distances is the same for both concave mirror and convex mirror. This relation viz., \(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\) = \(\frac{2}{r}\) is called the mirror equation or spherical mirror equation.
In case of a concave mirror if u and u are the object distance and the image distance of a real object and its real image respectively, then the u – v graph is a rectangular hyperbola
and the graph of their reciprocals (\(\frac{1}{u}\) – \(\frac{1}{v}\) graph) is a straight line.
Equation of plane mirror from the equation of spherical mirror: The mirror equation is
\(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\) = \(\frac{2}{r}\)
For a plane mirror, r → ∞
∴ \(\frac{1}{v}\) + \(\frac{1}{u}\) = 0 or, v = -u
This proves that in case of a plane mirror, the image is as far behind the mirror as the object Is in front of it (∴ v negative).
Effect of medium on the focal length and image distance for spherical mirror: Focal length of a spherical mirror is fixed, i.e., independent of the surrounding media. This is because, the law of reflection is invariant even if the medium changes.
For spherical mirror, If object distance u, image distance u and focal length f, then the relation between them is \(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\). For a fixed object distance u, the image distance v is fixed, because
f is fixed and independent of medium. So, if the position of object and mirror are kept fixed and the surrounding medium is changed, no change in position of image occur.
Conjugate Foci or Conjugate Points
We have the mirror equation:
\(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\)
If u and u are interchanged, the equation remains the same. This implies that if the object is placed at the position of the image, the image will be formed at the position of the object. These two points are cafled conjugate foci and the above equation is alternatively called the conjugate foci relation.
In case of virtual image, conjugate foci are situated on two opposite sides of the mirror and in case of real image, conjugate foci are situated on the same side of the mirror.
Newton’s Equation
Relation among u, v, f with reference to the pole is
\(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\) or \(\frac{u+v}{u v}\) = \(\frac{1}{f}\)
or, uv – uf – vf = 0
or, uu – uf – vf + f2 = f2
or, u(v – f) – f(v – f) = f2
or, (u – f)(v – f) = f2 ……(1)
Now, if the object distance and the image distance are measured from the focus, and taken equal to x and y respectively, then we can write,
u – f = x and v – f = y
So, from equation (1) we get,
xy = f2 ….. (2)
This equation is known as Newton’s equation. Since, f is constant, the graph of x versus y will be a rectangular hyper bola [Fig.].
Since, f2 is a positive quantity, x and y must have the same sign, i.e., the object and the image must be on the same side of the focus.
Magnification of an Image Formed by Spherical Mirrors
Linear or lateral magnification:
Definition: The ratio of the height of the image to the height of the object measured in planes that are perpendicular to the principal axis, is called the linear or lateral magnification of the image.
Linear or lateral magnification is denoted by m.
∴ m \(=\frac{\text { height of the image }}{\text { height of the object }}\) = \(\frac{I}{O}\) …. (1)
i) In case of a real image formed by a concave mirror: In Fig. the ray diagram for the formation of a real image by a concave mirror is shown. Here object distance, RQ = -u; image distance, Rq = -v, height of the object, PQ = 0 and height of the image, pq = -I. From the Fig., ΔPQR and ΔpqR are similar.
ii) In case of a virtual image formed by a concave mirror: In the Fig. the ray diagram for the formation of a virtual image by a concave mirror is shown.
Here object distance, PQ = -u; image distance, Rq = v, height of the object, PQ = 0 and height of the image, pq = I. From the Fig. ΔPQR and ΔpqR are similar.
iii) In case of a virtual image formed by a convex mirror: In the Fig. the ray diagram for the formation of a virtual image by a convex mirror is shown. Here object distance, RQ = -u; image distance Rq = v; height of the object, PQ = O and height of the image pq = I.
According to the Fig., ΔPQR and ΔpqR are similar.
∴ \(\frac{p q}{P Q}\) = \(\frac{q R}{Q R}\)
∴ \(\frac{I}{O}\) = \(\frac{v}{-u}\) = \(-\frac{v}{u}\) …. (4)
From equations (2), (3) and (4) it follows that the magnification produced by both kinds of mirror is given by
m = \(\frac{I}{O}\) = \(-\frac{v}{u}\) ….. (5)
Some useful hints:
1. The relation m = \(-\frac{v}{u}\) is applicable both for concave and convex mirrors.
2. In solving numerical problems, values of u, u and f should be put with appropriate signs in mirror equation. No sign for the unknown quantity should be used.
3. Using appropriate sign of u and u, if
- m becomes negative, the image will be inverted
- m becomes positive, the image will be erect
4. If |m| > 1; size of the image > size of the object
If |m| < 1; size of the image < size of the object
If |m| = 1; size of the image = size of the object
Magnification in terms of focal length, object distance and image distance:
From the mirror equation we get.
\(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\)
or \(\frac{u}{v}\) + 1 = \(\frac{u}{f}\) or, \(\frac{u}{v}\) = \(\frac{u-f}{f}\) or, \(\frac{v}{u}\) = \(\frac{f}{u-f}\)
Since, m = \(-\frac{v}{u}\)
∴ m = \(-\left(\frac{f}{u-f}\right)\)
or, m = \(\frac{f}{f-u}\) …. (6)
Again the equation can be written as,
1 + \(\frac{v}{u}\) = \(\frac{v}{f}\) or, \(\frac{v}{u}\) = \(\frac{v-f}{f}\)
since m = \(-\left(\frac{\nu-f}{f}\right)\) = \(\frac{f-v}{f}\)
Areal Magnificafion:
Definition: Areal magnification for spherical mirror is the ratio between the image of an area of a plane, and the area of that plane placed perpendicular to the principal axis of the mirror.
Let us consider, the length and breadth of a plane of a rectangular object are l and b respectively.
∴ Area of the plane, A = lb
If linear magnification of image of the object by spherical mirror be m, then
length of the image, l’ = m × l
and breadth, b’ = m × b
∴ Area of the image, A’ = l‘b’ = m2lb = m2A
Therefore, areal magnification,
m’ = \(\frac{A^{\prime}}{A}\) = m2
Longitudinal or axial magnification of the image of an object kept along the principal axis:
Definition: If any object is placed along the principal axis of a spherical mirror then the ratio of the image length and object length is the longitudinal or axial magnification of that
image.
Let an object ADEB is placed in front MM’ [Fig.].
From the figure, the distance of farther point A of an object, OA = u1 and that of the nearer point B of the object, OB = u2.
Now, the distance of farther point of the image, OB’ = v1 and that of the nearer point, OA’ = v2.
Longitudinal magnification, m” = \(\frac{v_1-v_2}{u_1-u_2}\) = \(\frac{\Delta v}{\Delta u}\)
Here, Δu and Δv are the lengths of the object and its image respectively, along the principal axis of the mirror. For very small magnitude of Δu and Δv, these can be considered as du and
dv respectively.
So, m” = \(\frac{d v}{d u}\)
Differentiating the equation \(\frac{1}{v}\) + \(\frac{1}{u}\) = \(\frac{1}{f}\) with respect to u , we get,
\(-\frac{1}{v^2} \frac{d v}{d u}-\frac{1}{u^2}\) = 0 [∵ f is constant]
or, \(\frac{d v}{d u}\) = \(-\frac{v^2}{u^2}\)
∴ m” = \(\frac{d v}{d u}\) = -m2
∴ longitudinal magnification = -(linear magnification)2
i) Note that, if object length and image length are very small then, m” = \(\frac{d v}{d u}\)
ii) m may be positive or negative but m” is always negative. This implies that irrespective of whether the object is virtual or real, the image is formed along the principal axis, with opposite alignment. This phenomenon is called axial inversion.
Formation of Image of a Virtual Object
In Fig.(a) and (b), a beam of converging rays is incident on the mirror. In the absence of the mirror in either case, the
converging beam of rays would meet at P behind the mirror. But the beam of rays meet at P’ after reflection. Here the point P is the virtual object and P’ is the real image of point P. Obviously, object distance OP is positive. Thus, a convex mirror can also form a real image, but only if the object is virtual.
Now consider the case, where the virtual object distance OP is greater than the focal length OF of a convex mirror. In this case, image P’ will be virtual [Fig.(c)].
In the case for concave mirror, real image always be formed for virtual object and this image is situated between pole and focus of the mirror.