# Reflection of light

Difference between regular and diffuse reflection:

If a parallel beam of light is incident on a smooth surface, all reflected rays are parallel to each other after reflection such a reflection of light is called regular reflection. And if reflecting surface is not smooth but rough, the reflected rays may not be parallel to each other such a reflection is called irregular or diffuse reflection.

Difference between virtual images formed by plane, concave and convex mirrors;

The virtual image produced by a plane mirror is of same size as the object, where as virtual image produced by a concave mirror is magnified and that produced by convex mirror is diminished.

Focal length of plane mirror;

Focal length of plane mirrors is considered to be infinite, as the radius of curvature is infinite. A plane mirror can be considered to be the limit of either a concave or a convex spherical curved mirror as the radius, and therefore the focal length, becomes infinite.

Distinguish between real and virtual image;

A real image occurs where rays converge, whereas a virtual image occurs where rays only appear to converge. Real images can be produced by concave mirrors and converging lenses if and only if the object is placed further away from the mirror/lens than the focal point and this real image is inverted.

It possible to find out whether a mirror is plane, concave or convex from the nature of image;

It is possible to find whether a mirror is plane, concave or convex from the nature of image of an object. If the image is virtual and the size of the object is equal to the size of the image i.e. the magnification observed is always unity then the mirror is plane. If the size of the image is larger than the object when the mirror kept closer to a person is slightly displayed away, then the mirror is concave mirror. If the size of the image is smaller than the object when the mirror kept closer to a person is slightly displayed away, then the mirror is convex mirror.

The least distance between real object and real image in concave mirror:

The minimum distance between an object and its real image in the case of a concave mirror is 0.
This occurs when the object is at the 2F distance (the center of curvature); then the image is formed inverted at the same distance.

Focal length and radius of curvature of concave mirror:

Focal lengthReturn to “Telescopes from the Ground Up”Focal length. Focal length is the distance between the center of a convex lens or a concave mirror and the focal point of the lens or mirror — the point where parallel rays of light meet, or converge.

Radius of curvature of concave mirror is the radius of a circle which touches a curve at a given point and has the same tangent and curvature at that point.

The applications of concave mirror are;

1. Concave mirror is used for saving.

2. Concave mirror is used by doctors in ophthalmoscope.

3. Concave mirror is used in search light, in automobile headlights etc.

The applications of convex mirror are;

1. The convex mirror is used in vehicles as back view mirror.

2. The convex mirror is used in some lights used in laboratory.

3. The convex mirror is used in film shooting.

The sign conventions in curved mirror formula;

The sign conventions in curved mirror formula are described as follows.

 Types of mirror U real virtual f R Height of the object Height of the object real Height of the object virtual concave - - + - - + - + convex - no + + + + no +

The minimum size of a mirror to see full image of an object in front of it:

The minimum height of mirror required to be able to view a complete image of a person. Considered the following setup:

HF is the person. H denotes the head, F the feet, and E, the eyes. For the person to see his complete image, a ray each from H and F has to come and reflect into his eyes (E). Let HE=0.16m and HF=1.84m. KG is the minimum height if mirror required.

Now, since HI = IE = HE2 = 0.08m and FC = CE = EF2 = 0.92m, KG = 1m.

Thus minimum size of the mirror = $\frac{1}{2}{\rm{HF}}$

That is the required size of the mirror is the half the size of the man.

Angle of deviation:

The angle of deviation is the angle equal to the difference between the angleof incidence and the angle of refraction of a ray of light passing through the surface between one medium and another of different refractive index.

A ray of light that is incident on to the surface of a plane-mirror is reflected with the angle of incidence equal to the angle of reflection. Suppose that the ray had continued, through the mirror, in a straight line it would make an angle θ with the surface of the mirror. The total angle between the straight-line path and the reflected ray is twice the angle of incidence. This is called the deviation of the light and measures the angle at which the light has strayed from its initial straight-line path.

A ray of light that is incident on to the surface of a plane-mirror is reflected with the angle of incidence equal to the angle of reflection. Suppose that the ray had continued, through the mirror, in a straight line it would make an angle θ with the surface of the mirror. The total angle between the straight-line path and the reflected ray is twice the angle of incidence. This is called the deviation of the light and measures the angle at which the light has strayed from its initial straight-line path.

Law of rotation of light;

When the mirror rotates through an angle θ, then the reflected ray rotates through angle 2θ.

Let us consider a ray of light AO incident on a plane mirror XY at O. It is reflected along OB. Let α be the glancing angle with XY (as shown in figure). We know that the angle of deviation COB = 2α.

Suppose the mirror is rotated through an angle θ to a position X′Y′.

The same incident ray AO is now reflected along OP. Here the glancing angle with X′Y′ is (α + θ). Hence the new angle of deviation COP = 2 (α + θ). The reflected ray has thus been rotated through an angle BOP when the mirror is rotated through an angle θ.

∠BOP = ∠COP – ∠COB

∠BOP = 2 (α + θ) – 2α = 2θ

For the same incident ray, when the mirror is rotated through an angle, the reflected ray is rotated through twice the angle.

Principle focus and centre of curvature of a spherical mirror;

The principal focus of a spherical concave mirror is defined as a point on its principal axis where a beam of light parallel to the principal axis converges after being reflected by the mirror.

The concave mirror is a spherical mirror whose reflection surface is a section of an inner surface of a hollow sphere. The Centre of curvature of the concave mirror is the centre of the sphere of which the concave mirror is a part.

Figure;

Consider a ray of light AB, parallel to the principal axis, incident on a spherical mirror at point B. The normal to the surface at point B is CB and CP = CB = R, is the radius of curvature. The ray AB, after reflection from mirror will pass through F (concave mirror) or will appear to diverge from F (convex mirror) and obeys law of reflection, i.e., i = r.

From the geometry of the figure,

∠ BCP = θ = i

In ΔCBF, as θ = r

Bf = FC ( i = r)

If the aperture of the mirror is small, B lies close to P, \ BF = PF

or FC = FP = PF

or PC = PF + FC = PF + PF

or R = 2 PF = 2f

${\rm{f}} = \frac{{\rm{R}}}{2}$

Similar relation holds for convex mirror also

The mirror formula for a concave mirror:

Draw LN perpendicular on the principle axis.

Now Δ’s NLF and A’B’F’ are similar

$\frac{{{\rm{A'B'}}}}{{{\rm{LN}}}} = \frac{{{\rm{A'F}}}}{{{\rm{NF}}}} \ldots \ldots \ldots \ldots \ldots ..1$

Since aperture of the mirror is small, so point N lies close to P.

NF = PF and NL = AB

$\frac{{{\rm{A'B'}}}}{{{\rm{AB}}}} = \frac{{{\rm{A'F}}}}{{{\rm{PF}}}} = \frac{{{\rm{PF'}} - {\rm{PA}}}}{{{\rm{PF}}}} \ldots \ldots \ldots \ldots 2$

${\rm{\: \: }}$Also Δ’S ABC and A’B’C’ are similar, therefore

$\frac{{{\rm{A'B'}}}}{{{\rm{AB}}}} = \frac{{{\rm{A'C}}}}{{{\rm{AC}}}} = \frac{{{\rm{PC'}} + {\rm{PA'}}}}{{{\rm{PA}} - {\rm{PC}}}} \ldots \ldots \ldots \ldots 3$

From equation 1 and 3, we get

$\frac{{{\rm{PF}} - {\rm{PA'}}}}{{{\rm{PF}}}} = \frac{{{\rm{PC}} - {\rm{PA'}}}}{{{\rm{PA}} + {\rm{PC}}}} \ldots \ldots \ldots \ldots \ldots \ldots .4$

Applying sign conventions,

PF = f, PC = R = -2f

PA’ = -v, Pa = u

∴ Equation 4 becomes

$\frac{{ - {\rm{f}} + {\rm{v}}}}{{ - {\rm{f}}}} = \frac{{ - 2{\rm{f}} + {\rm{v}}}}{{{\rm{u}} - 2{\rm{f}}}}$

Or, $- {\rm{uf}} + 2{{\rm{f}}^2} + {\rm{\:uv}} - 2{\rm{vf}} = 2{{\rm{f}}^2} - {\rm{vf}}$

Or, u v = u f +v f

Diving by u v f, we get

$\frac{{{\rm{uv}}}}{{{\rm{uvf}}}} = \frac{{{\rm{uf}}}}{{{\rm{uvf}}}} + \frac{{{\rm{vf}}}}{{{\rm{uvf}}}}$

$\frac{1}{{\rm{f}}}$ = $\frac{1}{{\rm{v}}} + $$\frac{1}{{\rm{u}}} \\frac{1}{{\rm{f}}} = \frac{1}{{\rm{u}}} + \frac{1}{{\rm{v}}} \frac{1}{{\bf{f}}} = \frac{1}{{\bf{u}}} + \frac{1}{{\bf{v}}} for a convex mirror: When an object is placed between the pole and the focus of a concave mirror, virtual, erect and enlarged image is formed behind the mirror as shown in the figure. Draw LN perpendicular on the principle axis. Now Δ’s NLF and A’B’F’ are similar \frac{{{\rm{A'B'}}}}{{{\rm{NL}}}} = \frac{{{\rm{A'F}}}}{{{\rm{NF}}}} \ldots \ldots \ldots \ldots \ldots ..1 Since aperture of the mirror is small, so point N lies close to P. NF = PF and NL = AB. \frac{{{\rm{A'B'}}}}{{{\rm{AB}}}} = \frac{{{\rm{A'F}}}}{{{\rm{PF}}}} = \frac{{{\rm{PA'}} + {\rm{PF}}}}{{{\rm{PF}}}} \ldots \ldots \ldots \ldots 2 {\rm{\: \: }}Also Δ’S ABC and A’B’C’ are similar, therefore \frac{{{\rm{A'B'}}}}{{{\rm{AB}}}} = \frac{{{\rm{A'C}}}}{{{\rm{PC}}}} = \frac{{{\rm{PA'}} + {\rm{PC}}}}{{{\rm{PC}} - {\rm{PA}}}} \ldots \ldots \ldots \ldots 3 From equation 1 and 3, we get \frac{{{\rm{PA'}} + {\rm{PF}}}}{{{\rm{PF}}}} = \frac{{{\rm{PA'}} + {\rm{PC}}}}{{{\rm{PC}} - {\rm{PA}}}} \ldots \ldots \ldots \ldots \ldots \ldots .4 Applying sign convention PA’= -v, PF = f, PC = R = 2f ∴ Equation 4 becomes \frac{{ - {\rm{v}} + {\rm{f}}}}{{\rm{f}}} = \frac{{ - {\rm{v}} + 2{\rm{f}}}}{{2{\rm{f}} - {\rm{u}}}} Or, u v = u f +v f Diving by uvf, we get \frac{{{\rm{uv}}}}{{{\rm{uvf}}}} = \frac{{{\rm{uf}}}}{{{\rm{uvf}}}} + \frac{{{\rm{vf}}}}{{{\rm{uvf}}}} ∴\frac{1}{{\rm{f}}} = \frac{1}{{\rm{u}}} + \frac{1}{{\rm{v}}} Prove the relation \frac{1}{{\bf{f}}} = \frac{1}{{\bf{u}}} + \frac{1}{{\bf{v}}} for concave mirror: Draw LN perpendicular on the principle axis. Now Δ’s NLF and A’B’F’ are similar \frac{{{\rm{A'B'}}}}{{{\rm{LN}}}} = \frac{{{\rm{A'F}}}}{{{\rm{NF}}}} \ldots \ldots \ldots \ldots \ldots ..1 Since aperture of the mirror is small, so point N lies close to P. NF = PF and NL = AB \frac{{{\rm{A'B'}}}}{{{\rm{AB}}}} = \frac{{{\rm{A'F}}}}{{{\rm{PF}}}} = \frac{{{\rm{PF'}} - {\rm{PA}}}}{{{\rm{PF}}}} \ldots \ldots \ldots \ldots 2 {\rm{\: \: }}Also Δ’S ABC and A’B’C’ are similar, therefore \frac{{{\rm{A'B'}}}}{{{\rm{AB}}}} = \frac{{{\rm{A'C}}}}{{{\rm{AC}}}} = \frac{{{\rm{PC'}} + {\rm{PA'}}}}{{{\rm{PA}} - {\rm{PC}}}} \ldots \ldots \ldots \ldots 3 From equation 1 and 3, we get \frac{{{\rm{PF}} - {\rm{PA'}}}}{{{\rm{PF}}}} = \frac{{{\rm{PC}} - {\rm{PA'}}}}{{{\rm{PA}} + {\rm{PC}}}} \ldots \ldots \ldots \ldots \ldots \ldots .4 Applying sign conventions, PF = f, PC = R = -2f PA’ = -v, Pa = u ∴ Equation 4 becomes \frac{{ - {\rm{f}} + {\rm{v}}}}{{ - {\rm{f}}}} = \frac{{ - 2{\rm{f}} + {\rm{v}}}}{{{\rm{u}} - 2{\rm{f}}}} Or, - {\rm{uf}} + 2{{\rm{f}}^2} + {\rm{\:uv}} - 2{\rm{vf}} = 2{{\rm{f}}^2} - {\rm{vf}} Or, u v = u f +v f Diving by uvf, we get \frac{{{\rm{uv}}}}{{{\rm{uvf}}}} = \frac{{{\rm{uf}}}}{{{\rm{uvf}}}} + \frac{{{\rm{vf}}}}{{{\rm{uvf}}}} \frac{1}{{\rm{f}}} = \frac{1}{{\rm{v}}} +$$\frac{1}{{\rm{u}}}$

\$\frac{1}{{\rm{f}}}$ = $\frac{1}{{\rm{u}}} + \frac{1}{{\rm{v}}}$

The nature, position and image formed by a concave mirror in different positions:

Nature, position and image formed by a concave mirror in different positions are as given below

 Position of object Position of image Character of image At ∞ At f Real, zero size Between  ∞ and c Between f and c Real, inverted, diminished At c At c Real, inverted, same size Between c and f Between c and  ∞ Real, inverted, magnified At f At  ∞ Between f and V From - ∞ to V Virtual, upright, magnified At V At V Virtual, upright, same size

The nature, position and image formed by a convex mirror:

Image formation by convex mirrors;

 Position of object Position of image Character of image At ∞ At F Virtual, zero size Between ∞ and V Between F and V Virtual, upright, diminished At V At V Virtual, upright, same size

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