Question

Derive of mirror formula and lenses formula.

Answer 1

Derivation or proof of mirror formula:
Mirror formula is the relationship between object distance(u), image distance (v) and focal length.
$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

$In\Delta ABCand{A}^{\prime }{B}^{\prime }C<A=<A^{\prime }={90}^{\circ }<C=<C(Vert.opp.<s]\Delta ABC\Delta A^{\prime }{B}^{\prime }C\left[AAsimilarity\right]\frac{AB}{A^{\prime }{B}^{\prime }}=\frac{AC}{A^{\prime }C}...\left(I\right)$
Similarly,
$In\Delta ABCand{A}^{\prime }{B}^{\prime }C<A=<A^{\prime }={90}^{\circ }<C=<C(vert.opp.<s]\Delta ABC\Delta A^{\prime }{B}^{\prime }C\left[AAsimilarity\right]\frac{AB}{A^{\prime }{B}^{\prime }}=\frac{AC}{A^{\prime }C}...\left(1\right)$
Similarly, In \Delta FPE~ A'B'F
$\frac{EP}{A^{\prime }{B}^{\prime }}=\frac{PF}{A^{\prime }F}\frac{AB}{A^{\prime }{B}^{\prime }}=\frac{PF}{A^{\prime }F}[AB=EP]...\left(II\right)$
From (I) \& (II)
$\frac{AC}{A^{\prime }C}=\frac{PF}{A^{\prime }F}\Rightarrow \frac{A^{\prime }C}{AC}=\frac{PF}{A^{\prime }F}\Rightarrow \left(\frac{CP-A^{\prime }P}{AP-CP}\right)=\left(\frac{A^{\prime }P-PF}{PF}\right)$
Now, PF = - f; CP = 2PF = - 2f
AP = - u; and A'P = - v
Put these value in above relation:
$[\frac{(-2f)-\left(v\right)}{u-2f}=\frac{-v-f}{-f}]\Rightarrow uv=fv+uf\Rightarrow \frac{1}{f}=\frac{1}{u}+\frac{1}{v}$
Derivation or Proof of Lens formula:

Let AB is an object placed between f1 and f2 of the convex lens. The image A1 B1 is formed beyond $2F_2$ and is real and inverted.
OA = Object distance = u ; OA1 = Image distance $=v;O{F}_2=$ Focal length = f
In $\Delta OAB$ and $\Delta O{A}_1{B}_1$ are similar
$<BAO=<B_1{A}_{1}O={90}^{\circ }$
$<AOB=<A_{1}O{B}_1$ [vertically opp. < s]
$\Delta OAB\Delta O{A}_1{B}_1$
$\frac{A_1{B}_1}{AB}=\frac{O{A}_1}{OA}....\left(i\right)$
Similarly, $\Delta OC{F}_2\Delta F_2{A}_1{B}_1$
$\frac{A_1{B}_1}{OC}=\frac{F_2{A}_1}{O{F}_2}...........\left(ii\right)$
From equation (i) and (ii), we get
$\frac{O{A}_1}{OA}=\frac{O{A}_1-O{F}_2}{O{F}_2}\frac{v}{-u}=\frac{(v-u)}{f}vf=-u(v-f)$
Dividing equation (3) throughout by uvf
$\frac{1}{v}-\frac{1}{u}-\frac{1}{f}$