Reflection

aReflectionOptics : The branch of physics which deals with the phenomenon of light is called optics.Photometry: The branch of optics which deals with the measurement of light energy is called photometry.Concave and convex mirror

Concave Mirror: A spherical mirror in which reflection takes place in the inner side of the sphere of which the mirror is a part is called concave mirror.Convex Mirror: A spherical mirror in which reflection takes place in the outer side of the sphere of which the mirror is a part is called convex mirror.Real and Virtual ImageReal image : The image which is formed by actual meeting of light rays and can be formed on screen is called real image. It is always inverted.Virtual Image : The image which is formed without actual meeting of light rays and cannot be formed on screen is called virtual image. It is always erect.Terms in Spherical mirror• Pole : The center of the spherical reflecting surface of the mirror is called pole P.• Center of Curvature: The center of the sphere which is a part of the of the mirror is called centre of curvature C.• Radius of Curvature : The radius of the sphere which is a part the mirror is called radius of curvature R.• Prinpipal Axis: The line joining the pole and the center of curvature of the mirror is principal axis.• Principal focus : A point, where all the parallel beam of light after reflection, converse of appears to diverse is called the principal focus F. • Focal Length: The distance between the pole and principal focus of a mirror is called the focal length , f. • Aperture : the diameter of the boundary of the mirror is called aperture.Relation Between R and f

Let us consider a concave mirror of small aperture as shown in figure. When a ray of light OA is incident on the mirror it gets reflected as AB passing through focus F. From the law of reflection,∠ OAC = ∠ BAC & ∠ OAC = ∠ ACF (being alternate angles)So, ∠ BAC = ∠ ACFHence ∆ ACF is an isosceles triangleAnd AF = FCIf the aperture of the mirror is small , then AF≃ PFPF = FC = PC – PFPF = PC-PF2PF = PC2f = Rf=R

2${\displaystyle f=\frac{R}{2}}$Mirror Formula An expression showing the relation of focal length with object distance and image distance is known as mirror formula.1

f = 1

u +1

v${\displaystyle \frac{1}{f}=\frac{1}{u}+\frac{1}{v}}$Consideration:• The aperture is small• Object is at principal axis• Distance is measured from pole of mirror• Real distance is positive and virtual distance is negative• Focal length for concave mirror is positive and for convex lens is negative 1. Concave mirror forming real image Let us take a concave mirror of aperture mirror of aperture XY where a light ray AC is travelling parallel to principle axis from object AB to mirror at C and reflect through focus F and pass through A'. Let another light ray pass directly through focus from A to D at mirror and reflect through A'. A real and inverted image is formed when an object is placed on the principal axis beyond the principal focus F$\mathrm{F}$ of a concave mirror.

Let AB$\mathrm{A}\mathrm{B}$ be an object lying beyond the focus of a concave mirror as in figure. A ray of light BL after reflection from the mirror passes through the principal axis at F$\mathrm{F}$ along LB'. Another ray BC$\mathrm{B}\mathrm{C}$ from B passes through the centre of curvature C$C$ and incident normally on the mirror at point M. After reflection, this ray retraces its path and meets LB′$\mathrm{L}{\mathrm{B}}^{\prime }$ at B′${\mathrm{B}}^{\prime }$. So A′B′${\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }$ is the real image of the object AB$\mathrm{A}\mathrm{B}$. Draw LN perpendicular on the principal axis. Now 𝛥$𝛥$ 's NLF and A′B′F${\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }\mathrm{F}$ are similar, and thereforeSince aperture of the concave mirror is small, so the point N$\mathrm{N}$ lies very close to P$\mathrm{P}$.NF=PF$\mathrm{N}\mathrm{F}=\mathrm{P}\mathrm{F}$Also, NL=AB$\mathrm{N}\mathrm{L}=\mathrm{A}\mathrm{B}$Then, A′B′

AB=AF

PF$\frac{{\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }}{\mathrm{A}\mathrm{B}}=\frac{\mathrm{A}\mathrm{F}}{\mathrm{P}\mathrm{F}}$ .....................(i)Also △sABC$△s\mathrm{A}\mathrm{B}\mathrm{C}$ and A′B′C${\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }\mathrm{C}$ are similar, therefore, A′B′

AB=A′C

AC$\frac{{\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }}{\mathrm{A}\mathrm{B}}=\frac{{\mathrm{A}}^{\prime }\mathrm{C}}{\mathrm{A}\mathrm{C}}$ ..........................(ii)Then, from these two equations, A F

PF=A′C

AC $\frac{{\mathrm{A}}^{}\mathrm{F}}{\mathrm{P}\mathrm{F}}=\frac{{\mathrm{A}}^{\prime }\mathrm{C}}{\mathrm{A}\mathrm{C}}$.....................(iii)Since all the distances are measured from the pole of the mirror (sign convention), so A′F=PA′-PF,A′C=PC-PA′${\mathrm{A}}^{\prime }\mathrm{F}=\mathrm{P}{\mathrm{A}}^{\prime }-\mathrm{P}\mathrm{F},{\mathrm{A}}^{\prime }\mathrm{C}=\mathrm{P}\mathrm{C}-\mathrm{P}{\mathrm{A}}^{\prime }$ and AC=PA-PC$\mathrm{A}\mathrm{C}=\mathrm{P}\mathrm{A}-\mathrm{P}\mathrm{C}$. Thus above equation becomes PA′-PF

PF=PC-PA′

PA-PC ................(iv)$$\frac{\mathrm{P}{\mathrm{A}}^{\prime }-\mathrm{P}\mathrm{F}}{\mathrm{P}\mathrm{F}}=\frac{\mathrm{P}\mathrm{C}-\mathrm{P}{\mathrm{A}}^{\prime }}{\mathrm{P}\mathrm{A}-\mathrm{P}\mathrm{C}}................(iv)$$Applying sign convention, PA=u$PA=u$ (object distance), PA′=v$P{A}^{\prime }=v$ (image distance), aPF=f,PC=R=2f$\begin{array}{l}PF=f,\\ PC=R=2f\end{array}$ Hence Eq. (iv) becomes a

v-f f=2f-v u-2f

uv-2vf-uf+2f2=2f2-vf

$$\begin{array}{l}\frac{\mathrm{v}-\mathrm{f}}{\mathrm{f}}=\frac{2\mathrm{f}-\mathrm{v}}{\mathrm{u}-2\mathrm{f}}\\ \mathrm{u}\mathrm{v}-2\mathrm{v}\mathrm{f}-\mathrm{u}\mathrm{f}+2{\mathrm{f}}^2=2{\mathrm{f}}^2-\mathrm{v}\mathrm{f}\end{array}$$ or uv=uf+vf$uv=uf+vf$Dividing by u v f$f$, we geta

uv uvf= uf uvf+vf uvf

1 f=1 u+1 v

$$\begin{array}{l}\frac{\mathrm{u}\mathrm{v}}{\mathrm{u}\mathrm{v}\mathrm{f}}=\frac{\mathrm{u}\mathrm{f}}{\mathrm{u}\mathrm{v}\mathrm{f}}+\frac{\mathrm{v}\mathrm{f}}{\mathrm{u}\mathrm{v}\mathrm{f}}\\ \frac{1}{\mathrm{f}}=\frac{1}{\mathrm{u}}+\frac{1}{\mathrm{v}}\end{array}$$ This equation is known as mirror formula. 2. Mirror Formula for Concave Mirror when Virtual Image is Formed When an object is placed between the pole and the focus of a concave mirror, virtual, erect and magnified image is formed behind the mirror.

The above figure shows the virtual image AB′$\mathrm{A}{\mathrm{B}}^{\prime }$ of a real object formed by a convex lens of focal length f$\mathrm{f}$ LN perpendicular on the principal axis is drawn. Now Δ${\displaystyle \Delta }$ NLF and Δ${\displaystyle \Delta }$ABF are similar, so A′B′

LN=A′F

NF$\frac{\mathrm{A}\mathrm{\prime }{\mathrm{B}}^{\prime }}{LN}=\frac{\mathrm{A}\mathrm{\prime }\mathrm{F}}{NF}$Since aperture of the mirror is small, point N$\mathrm{N}$ lies close to P $\mathrm{P}$. Therefore, NF=PF$\mathrm{N}\mathrm{F}=\mathrm{P}\mathrm{F}$ and NL=AB$\mathrm{N}\mathrm{L}=\mathrm{A}\mathrm{B}$and A′B′

AB=A′F

PF$\frac{\mathrm{A}\mathrm{\prime }{\mathrm{B}}^{\prime }}{\mathrm{A}\mathrm{B}}=\frac{\mathrm{A}\mathrm{\prime }\mathrm{F}}{\mathrm{P}\mathrm{F}}$ ...............(i)Also △ABC$△ABC$ and A'B$A\text{'}B$ ' C are similar, therefore A′B′

AB=A′C

AC$\frac{A^{\prime }{B}^{\prime }}{AB}=\frac{A^{\prime }C}{AC}$ ..................(ii)Combining these two ratios, we get A′F

PF=A′C

AC$\frac{\mathrm{A}\mathrm{\prime }\mathrm{F}}{\mathrm{P}\mathrm{F}}=\frac{A^{\prime }C}{AC}$ ..........................(iii)Here, from figure, A′F$\mathrm{A}\mathrm{\prime }\mathrm{F}$ = PA′+PF$\mathrm{P}{\mathrm{A}}^{\prime }+\mathrm{P}\mathrm{F}$ , A′C$A^{\prime }C$ = PA′+PC$\mathrm{P}{\mathrm{A}}^{\prime }+\mathrm{P}\mathrm{C}$and AC$AC$ = PC-PA$\mathrm{P}\mathrm{C}-\mathrm{P}\mathrm{A}$Putting these values in eqn (iii) PA′+PF

PF=PA′+PC

PC-PA$\frac{\mathrm{P}{\mathrm{A}}^{\prime }+\mathrm{P}\mathrm{F}}{\mathrm{P}\mathrm{F}}=\frac{\mathrm{P}{\mathrm{A}}^{\prime }+\mathrm{P}\mathrm{C}}{\mathrm{P}\mathrm{C}-\mathrm{P}\mathrm{A}}$Applying sign convention image distance (PA′)=-v $PA\prime )=-v$(image being virtual)object distance ( PA) = ufocal length( PF)=f $PF)=f$and PC=R=2f$PC=R=2f$we have $$ -v+f

f=-v+2f

2f-u$\frac{-v+f}{f}=\frac{-v+2f}{2f-u}$or -2vf+uv+2f2-uf=-vf+2f2$-2vf+uv+2f^2-uf=-vf+2f^2$or uv=uf+vf$uv=uf+vf$Dividing both sides by u vf$vf$, we get uv

u∨f=uf

uvf+vf

uvf$\frac{u\mathrm{v}}{u\vee f}=\frac{uf}{uvf}+\frac{vf}{uvf}$ or 1

f=1

u+1

v$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$Thich is mirror formula. 3. Mirror Formula for Convex Mirror Let AB$AB$ be an object lying on the principal axis of the convex mirror of small aperture as shown in figure.

AB′$\mathrm{A}{\mathrm{B}}^{\prime }$ ' is the virtual image of the object lying behind the Draw LN perpendicular on the principal axis. Now △$△$ NLF$\mathrm{N}\mathrm{L}\mathrm{F}$ and △$△$A′B′F${\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }\mathrm{F}$ are similar∴$\therefore$ A′B′

LN=A′F

NF$\frac{A^{\prime }{B}^{\prime }}{LN}=\frac{A^{\prime }F}{NF}$Since aperture of the mirror is small, so NF=PF$\mathrm{N}\mathrm{F}=\mathrm{P}\mathrm{F}$ and LN=AB$\mathrm{L}\mathrm{N}=\mathrm{A}\mathrm{B}$and then A′B′

AB=A′F

PF$\frac{\mathrm{A}\mathrm{\prime }{\mathrm{B}}^{\prime }}{\mathrm{A}\mathrm{B}}=\frac{\mathrm{A}\mathrm{\prime }\mathrm{F}}{\mathrm{P}\mathrm{F}}$ ...............(i) Also △$△$ ABC$ABC$ and △A′B′C$△A^{\prime }{B}^{\prime }C$ are similar, A′B′

AB=A′C

AC $\frac{{\mathrm{A}}^{\prime }{\mathrm{B}}^{\prime }}{\mathrm{A}\mathrm{B}}=\frac{{\mathrm{A}}^{\prime }\mathrm{C}}{\mathrm{A}\mathrm{C}}$ ...................(ii)then, we haveA′F

PF=A′C

AC...................(iii)$\frac{\mathrm{A}\mathrm{\prime }\mathrm{F}}{\mathrm{P}\mathrm{F}}=\frac{{\mathrm{A}}^{\prime }\mathrm{C}}{\mathrm{A}\mathrm{C}}...................(iii)$From Figure, A′F$\mathrm{A}\mathrm{\prime }\mathrm{F}$ = PF-PA′$\mathrm{P}\mathrm{F}-\mathrm{P}{\mathrm{A}}^{\prime }$A′C${\mathrm{A}}^{\prime }\mathrm{C}$= PC-PA′$\mathrm{P}\mathrm{C}-\mathrm{P}{\mathrm{A}}^{\prime }$AC$\mathrm{A}\mathrm{C}$= PA+PC$\mathrm{P}\mathrm{A}+\mathrm{P}\mathrm{C}$So eqn (iii) becomes PF-PA′

PF=PC-PA′

PA+PC $\frac{\mathrm{P}\mathrm{F}-\mathrm{P}{\mathrm{A}}^{\prime }}{\mathrm{P}\mathrm{F}}=\frac{\mathrm{P}\mathrm{C}-\mathrm{P}{\mathrm{A}}^{\prime }}{\mathrm{P}\mathrm{A}+\mathrm{P}\mathrm{C}}$Applying sign convention image distance (PA′)=-v $PA\prime )=-v$(image being virtual)object distance ( PA) = ufocal length(PF)=-f $PF)=-f$ (-ve for convex mirror)and PC=R=-2f$PC=R=-2f$ From these two equations, we get $\text{ From these two equations, we get }$a

-f+v -f=-2f+v u-2f

$\begin{array}{ll}& \frac{-\mathrm{f}+\mathrm{v}}{-\mathrm{f}}=\frac{-2\mathrm{f}+\mathrm{v}}{\mathrm{u}-2\mathrm{f}}\end{array}$ $$ or -uf+2f2+uv-2vf=2f2-vf$-\mathrm{u}\mathrm{f}+2{\mathrm{f}}^2+\mathrm{u}\mathrm{v}-2\mathrm{v}\mathrm{f}=2{\mathrm{f}}^2-\mathrm{v}\mathrm{f}$ or uv=uf+vf$\mathrm{u}\mathrm{v}=\mathrm{u}\mathrm{f}+\mathrm{v}\mathrm{f}$ Dividing by uvf, we get, uv

uvf=uf

uvf+vf

uvf$\frac{\mathrm{u}\mathrm{v}}{\mathrm{u}\mathrm{v}\mathrm{f}}=\frac{\mathrm{u}\mathrm{f}}{\mathrm{u}\mathrm{v}\mathrm{f}}+\frac{\mathrm{v}\mathrm{f}}{\mathrm{u}\mathrm{v}\mathrm{f}}$ ∴1

f=1

v+1

u$\therefore \frac{1}{\mathrm{f}}=\frac{1}{\mathrm{v}}+\frac{1}{\mathrm{u}}$ MagnificationMagnification of a mirror is defined as the ratio of the size of image formed by a spherical mirror to the size of the object. It is denoted by m.magnification (m) = size of image (I)

Size of object (O)${\displaystyle magnification(m)=\frac{sizeofimage(I)}{Sizeofobject(O)}}$Magnification can also be defined as the ratio of image distance to the object distance of a mirror.magnification (m) = image distance (v)

object distance (u)${\displaystyle magnification(m)=\frac{imagedistance(v)}{objectdistance(u)}}$so, m=I

O = v

u${\displaystyle so,m=\frac{I}{O}=\frac{v}{u}}$Note1. For real image magnification is positive.2. For virtual image mangification is negative.3. If the image is enlarged, magnification > 1.4. If the image is diminished, magnification < 1. Application of Spherical Mirrors1. A convex mirror is used as a reflector in street lamp to scatter light in wide area.2. A convex mirror is used in back mirror in vehicles since it produces errect image with lage field of view.3. A concave mirror is used in tourch light, head light of vehicles, search light, to focus light at a point.4. A concave mirror is used as saving mirror / make up mirror as it can form erect and magnified image.