Reflection at Plane and Spherical Surfaces

SOME IMPORTANT DEFINITIONS USED IN OPTICS

We define a varied collection of useful terms. Some of these definitions will be revisited when the appropriate topics are taken up.

(i) Ray: The path of light, as determined within the approximations of geometric optics, is a ray. In a homogeneous medium, it is a straight line.

(ii) Beam: A collection of rays, usually referred to as a bundle of rays, forms a beam. One may have a convergent, divergent or, a parallel beam. A convergent beam may converge to a point or, a line, a divergent beam may diverge from a point or, a line. A parallel beam consists of parallel rays.

(iii) Collimation: A process whereby a divergent (or, convergent) beam is rendered parallel usually through the use of lenses and/or mirrors.

(iv) Object & Image: The term object is used to refer to any object (being photographed or observed) that is a source of light, and its likeness (usually two dimensional) formed or observed by an optical system is the image.

(v) Optical System: An optical system consists of elements like lenses, mirrors, prisms, etc.

(vi) Axis: The axis of an optical system is frequently an axis of symmetry such that a ray directed along the axis continues in the same direction or, returns backwards (if reflected within the system)

(vii) Centre of Curvature: Most lenses and curved mirrors being manufactured spherical (i.e., their surfaces are spherical), the centre of curvature of the curved mirror or, the curved surface of a lens is important and is frequently denoted by the letter C. The radius of curvature is also equally important in the analysis.

(viii) Pole: The pole of a spherical surface (refracting or, reflecting) is the central point of the surface involved in the formation of the image. It is denoted by O or, P. The axis, for a single spherical surface, is the join of the pole with the centre of curvature; it is known as the principal axis.

(ix) Optical Centre: The optical centre of a thin lens is a point on the axis of the lens such at a ray directed towards that point emerges parallel to itself after passing through the lens.

(x) Paraxial Rays: It is observed that the formation of clear images by spherical surfaces takes place only with rays which are close to the principal axis and make very small angles with it.

(xi) Wave speed: It is the distance travelled by the wave disturbance in a unit time. It is denoted by the letter $\nu$.

(xii) Frequency: It is the number of vibrations made by the medium particle in 1 s. In other words, it is number of waves passing through a point of the medium in 1 second. It is generally represented by the letter n or $\nu$. . Its unit is hertz and it is represented as Hz or s^–1.

(xiii) Wavelength: It is the distance travelled by the wave in one complete period of a medium particle. In other words, distance between two consecutive crest or trough.

Relationship between the wavelength, wave speed and frequency: If the wave speed is $\nu$. and period is T, then by definition

Wavelength $\lambda$ = Distance travelled in one period

= Distance travelled in T s = $\nu$ x T

Or $\nu T=\lambda$

But $T=1/\nu \therefore \nu x1/\nu =\lambda$

Or $\nu =\nu \lambda$ i.e., Wave speed = Frequency X Wavelength.

REFLECTION AT PLANE SURFACE

(i) Reflection of Light

When light rays strike the boundary of two media such as air and glass, a part of light is turned back into the same medium. This is called Reflection of Light. The wavelength and the velocity of the light wave remains the same.

Laws of reflection are obeyed at every reflecting surface, i.e.

(ii) Laws of Reflection

(a) The incident ray (AB), the reflected ray (BC) and normal (BN) to the surface (SS') of reflection at the point of incidence (B) lie in the same plane. This plane is called the plane of incidence (or plane of reflection).

(b) The angle of incidence (the angle between normal and between the reflected ray and the normal are equal, i.e. $\angle i=\angle r$

(iii) Types of reflection

(a) Regular Reflection (b) Diffused Reflection or Scattering or Diffusion

(a) Regular Reflection: When the reflection takes place from a perfect plane surface it is called Regular Reflection.

(b) Diffused Reflection or Scattering or Diffusion: When the surface is rough light is reflected from the surface from bits of its plane surfaces in irregular directions. This is called diffusion. This process enables us to see an object from any position.

Characteristics of Reflection by a Plane Mirror

(i) Distance of object from mirror = Distance of image from the mirror.

(ii) The line joining the object point with its image is normal to the reflecting surface.

(iii) The image is laterally inverted (left right inversion).

(iv) The size of the image is the same as that of the object.

(v) For a real object the image is virtual and for a virtual object the image is real.

(vi) The minimum size of a plane mirror, required to see the full self image, is half the size of that person.

(vii) For a light ray incident at an angle 'i' after reflection angle of deviation $\delta =\pi -2i$

(viii) If i = 0 then r = 0, this implies that a ray of light incident normally on a mirror retraces its path.

(ix) The eye always observes an object in the direction in which the rays enter the eye

(x) The laws of reflection holds good for all kinds of reflection.

(xi) Image of an object is the point at which rays after reflection (or reflection) actually converge or appear to diverge from that point.

(xii) If the direction of the incident ray is kept constant and the mirror is rotated through an angle$\theta$ about an axis in the plane mirror then the reflected ray rotates through an angle 2$\theta$.

(xiii) If an object moves towards (or away from) a plane mirror at a speed v, the image will also approach (or recede) at the same speed.

Further the relative speed of image to the object will be v – (–v) = 2v.

(xiv) If two plane mirror are inclined to each other at 90°, the emergent ray is always antiparallel to incident ray, if reflected from each mirror, irrespective of angle of incident.

(xv) When two plane mirror, inclined to each other at an angle $\theta$, the number of images formed can be determined as follows:

(a) If $\left[\frac{360}{\theta }\right]$ is an even integer, say 'p', then number of image formed say q = p – 1, for all position.

(b) If $\left[\frac{360}{\theta }\right]$ is an odd integer say 'q' then number of image formed say p = q, if the object is not on the bisector of the angle between mirrors. Also, p = q – 1, if the object is on the bisector.

(c) If $\left[\frac{360}{\theta }\right]$ is a fraction, then the number of image formed will be equal to its integral part.

(xvi) The images are laterally inverted.

(xvii) The linear magnification is unity.

Illustration 1: A ray of light on a plane mirror along a vector $\stackrel{\wedge }{i}+\stackrel{\wedge }{j}-\stackrel{\wedge }{k}$. The normal on incident point is along $\stackrel{\wedge }{i}+\stackrel{\wedge }{j}$. Find a unit vector along the reflected ray.

Solution: Reflection of a ray of light is just like an elastic collision of a ball with a horizontal ground. Component of incident ray along the inside normal gets reversed while the component perpendicular to it remains unchanged. Thus the component of incident ray vector $\overrightarrow{A}=\stackrel{\wedge }{i}+\stackrel{\wedge }{j}-\stackrel{\wedge }{k}$ parallel to normal, i.e., $\stackrel{\wedge }{i}+\stackrel{\wedge }{j}$gets reversed while perpendicular to it, i.e., $-\stackrel{\wedge }{k}$remains unchanged. Thus, the reflected ray can be written as

$\overrightarrow{R}=-\stackrel{\wedge }{i}-\stackrel{\wedge }{j}-\stackrel{\wedge }{k}$

$\therefore$ A unit vector along the reflected ray will be

$\therefore$ $\stackrel{\wedge }{r}=\frac{\overrightarrow{R}}{\left|R\right|}$= $\frac{-\stackrel{\wedge }{i}-\stackrel{\wedge }{j}-\stackrel{\wedge }{k}}{\sqrt{3}}$

REFLECTION-AT SPHERICAL MIRRORS

A spherical mirror is a reflecting surface which forms a part of a sphere (as shown in following a and b diagram). When the reflection takes place from the inner surface and outer surface is polished or silvered the mirror is known as concave mirror. Vice- versa, it is convex.

CHARACTERISTICS OF REFLECTION BY A SPHERICAL REFLECTING SURFACES (FOR SMALL APERTURE)

(i) Pole (P) is generally taken as the mid point of reflecting surface.

(ii) Centre of curvature (C) is the centre of the sphere of which the mirror is a part.

(iii) Radius of curvature is the radius of the sphere of which the mirror is a part. Distance between P and C.

(iv) Principal Axis is the straight line connecting pole P and centre of curvature C.

(v) Principal focus (F) is the point of intersection of all the reflected rays which strike the mirror (with small aperture) parallel to the principal axis. In concave mirror it is real and in the convex mirror it is virtual.

(vi) Focal length (f) is the distance from pole to focus.

(vii) Aperture is the diameter of the mirror.

(viii) If the incident ray is parallel to the principal axis, the reflected ray passes through the focus. (Fig (a))

(ix) If the incident ray passes through the focus, then the reflected ray is parallel to the principal axis (Fig.(b))

(x) Incident ray passing through centre of curvature will be reflected back through the centre of curvature. (Fig.(c))

(xi) $f=\frac{R}{2}$ where f = focal length R = Radius of curvature.

(xii) Sign convention and magnification. We follow Cartesian coordinate system convention according to which:

(a) The pole of the mirror is the origin

(b) The direction of the incident rays is considered as positive x-axis.

(c) Vertically up is positive y-axis

Note: Radius of Curvature and Focal Length of:

(a) Concave mirror is positive

(b) Convex mirror is positive

(xiii) (a) Linear Magnification or lateral magnification or transverse magnification $=M=\frac{h_2}{h_1}=-\frac{v}{u}$

$h_2=y$coordinate of image; $h_1=y$ coordinate of the object

(both perpendicular to the principle axis of mirror)

(b) Longitudinal Magnification $=\frac{v_2-v_1}{u_2-u_1}$(for any size of the object) $=-\frac{v^2}{u^2}$(for short object)

Tracing for spherical mirror

REAL AND VIRTUAL SPACES

A mirror, plane or spherical divides the space into two:

(i) A side, where the reflected rays exist. This is real space.

(ii) The other side where the reflected rays do not exist. This is virtual space

Object

Object is decided by incident rays only. The point object for mirror is that point.

(a) From which the rays actually diverge to be incident on the mirror (Real object).

Or

(b) Towards which the incident rays appear to converge (virtual object).

Image

Image is decided by reflected or refracted rays only. The point image for a mirror is that point.

(a) Towards which the rays reflected from the mirror, actually converge (real image).

Or

(b) From which the reflected rays appear to diverge (virtual image).

NEW CARTESIAN SIGN CONVENTION – FOR MIRRORS AND LENSES

There are several sign-conventions. Students should try to follow any one of these. In this package, the new Cartesian sign convention has been used wherever required.

(i) All distance are measured from the pole.

(ii) Distances in the direction of incident rays are taken as positive.

(iii) Distances in the direction of incident rays are taken as negative.

(iv) Distances above the principal axis are taken as positive.

(v) Distances below the principal axis are taken as negative.

(vi) Angles measured from the normal, in anti-clockwise direction are positive, while in clockwise direction are negative.

MIRROR FORMULA

Consider the shown figure where O is a point object and I is corresponding image. CB is normal to the mirror at B.

By laws of reflection, $\angle$OBC = $\angle$PBC = $\theta$

$\alpha$ + $\theta$ = $\beta$, $\beta$ + $\theta$ = $\gamma$ $\Rightarrow$ $\alpha$ + $\gamma$ = 2$\beta$

For small aperture of the mirror, $\alpha$, $\beta$, $\gamma$ $\to$ 0

$\Rightarrow$ $\alpha \approx \tan \alpha ,\beta \approx \tan \beta ,\gamma \approx \tan \gamma ,P^{\prime }\to P$

$\Rightarrow$ tan $\alpha$ + tan $\gamma$ = 2 tan $\beta$

$\Rightarrow \frac{B{P}^{\prime }}{O{P}^{\prime }}+\frac{B{P}^{\prime }}{I{P}^{\prime }}=2\frac{B{P}^{\prime }}{C{P}^{\prime }}$

Applying sign convention, u = -OP, v = -IP, R = -CP

$\Rightarrow -\frac{1}{u}+\left(-\frac{1}{v}\right)=-\frac{2}{R}$

If u = $\infty$, $\frac{1}{v}=\frac{2}{R}$, but by definition, if u = $\infty$, v = f.

Hence, $f=\frac{R}{2}and\frac{1}{u}+\frac{1}{\nu }+\frac{1}{f}$

For convex mirrors, an exactly similar formula emerges.

Illustration 2: An object is placed 21cm infront of a concave mirror of radius of curvature 10cm. A glass slab of thickness 3cm and refractive index 1.5 is then placed closed to the mirror in the space between the object and mirror. Find the position of the final image formed. (You may take the distance of the near surface of the slab from the mirror to be 1cm)

Solution: Given that radius of curvature = 10 cm

$\therefore$ focal length f = 5 cm

Here $u=x-t\left(1-\frac{1}{\mu }\right)=21-3\left(\frac{1}{3}\right)=20cm.$

Where 1.5 = refractive index of slab

Using the formula for concave mirror

\begin{align}  \dfrac{1}{f}=\dfrac{1}{u}+\dfrac{1}{v} \\  \dfrac{1}{-5}=\dfrac{1}{-20}+\dfrac{1}{v} \\ \end{align}

Solving we get $v=-\frac{20}{3}cm.$

Applying the correction due to the slab, we have final image distance = |v| + 1

$=\frac{20}{3}+1=\frac{23}{3}cm.$