An expression showing the relation between object distance, image distance and focal length of a mirror is called the mirror formula. To derive the formula following assumptions and sign conventions are made.
Mirror Formula for Convex Mirror when Real Image is formed
Let AB be an object lying beyond the focus of a concave mirror. A ray of light BL after reflecting from the concave mirror passes through the principal axis at F and goes along LB’. Another ray from B passes through the centre of curvature © and incident normally on the mirror at point M. after reflection, this ray retraces its path and meets LB’ at B’. So A’B’ is the real image of the object AB. Draw LN perpendicular on the principle axis.
\begin{align*} \text {Now} \: \Delta ‘s \: \text {NLF and A’B’F are similar, therefore} \\ \frac {A’B’}{NL} &= \frac {A’F}{NF} \dots (i) \\ \text {Since aperture of the concave mirror is small, so point N lies very close to P.} \\ NF &= PF \\ \text {Also} \\ NL &= AB \\ \text {equation} \: (i) \: \text {becomes,} \\ \frac {A’B’}{AB} &= \frac {A’F}{PF} \dots (ii) \\ \text {Also} \: \Delta ‘s \text {ABC and A’B’C are similar, therefore,} \\ \frac {A’B’}{AB} &= \frac {A’C}{AC} \dots (iii) \\ \text {From equation} \: (ii) \text {and} \: (iii), \: \text {we get} \\ \\ \frac {A’F}{PF} &= \frac {A’C}{AC} \dots (iv) \\ \text {Since all the distances are measured from the pole of the mirror, so} \\ \end{align*}
$$\left.\begin{aligned} A'F = PA' - PF \\ A'C = PC - PA' \\ AC = PA - PC\end{aligned} \right \} \dots (i)$$
\begin{align*} \text {Substituting the values of equation}\: (v)\text {in equation,}\: (iv) \text {we get} \\ \frac {PA’ – PF}{PF} &= \frac {PC – PA’}{PA - PC} \dots (vi) \\ \text {Applying sign convention,} \\ PA’ = v, PF = f, PC = R = 2f \: (\therefore R = 2f ) \\ PA &= u \\ \text {Hence equation} \: (vi) \: \text {becomes} \\ \frac {v - f}{f} &= \frac {2f – v}{u – 2f} \\ uv – 2fv – uf + 2f^2 &= 2f^2 – vt \\ uv &= uf + vf \\ \text {Dividing by uvf, we get} \\ \frac {uv}{uvf} &= \frac {uf}{uvf} + \frac {vf}{uvf} \\ \frac 1f &= \frac 1u + \frac 1v \\ \end{align*}
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, erect and enlarges image formed behind the mirror as shown in figure. Draw LN perpendicular on the principle axis.
\begin{align*} \text {Now} \: \Delta ‘s \:\text {NLF and A’B’F are similar, therefore} \\ \frac {A’B’}{NL} &= \frac {A’F}{NF} \dots (i) \\ \text {Since aperture of the concave mirror is small, so point N lies very close to P.} \\ \therefore NF = PF \: \text {and} NL = AB \\ \text {Also} \: \Delta ‘s \text {ABC and A’B’C are similar, therefore,} \\ \frac {A’B’}{AB} &= \frac {A’C}{AC}= \frac {PA’ + PC} {PC –PA}\dots (iii) \\ \end{align*}
\begin{align*} \text {From equation} \: (ii) \text {and} \: (iii), \: \text {we get} \\ \frac {PA’ + PF}{PF} &= \frac {PA’ + PC}{PC - PA} \dots (iv) \\ \text {From equation} \: (ii) \: \text {and} \: (iii), \text {we get} \\ \text {Applying sign convention} \\ PA’ = - v, PF = f, PC = R = 2f \: (\because R = 2f, PA = u) \\ \therefore \: \text {Equation} \: (iv) \: \text {becomes} \\ \frac {-v + f}{f} &= \frac {– v - 2f}{ 2f - u} \\ \text {or,} \: -2vf +uv + 2f^2 – uf &= -vf + 2f^2 \\ \text {or,} \: uv &= uf + vf \\ \end{align*}
\begin{align*} \text {Dividing by uvf, we get} \\ \frac {uv}{uvf} &= \frac {uf}{uvf} + \frac {vf}{uvf} \\ \frac 1f &= \frac 1u + \frac 1v \\ \end{align*}
Mirror Formula for Convex Mirror
Let AB be an object lying on the principle axis of the convex mirror of small aperture. A’B’ is the virtual image of the object lying behind the convex mirror as shown in the figure.
Draw LN perpendicular on the principal axis.
\begin{align*} \text {Now} \: \Delta ‘s \:\text {NLF and A’B’F are similar, therefore} \\ \frac {A’B’}{NL} &= \frac {A’F}{NF} \dots (i) \\ \text {Since aperture of the concave mirror is small, so point N lies very close to P.} \\ \therefore NF = PF \: \text {and} LN = AB \\ \text {Also} \: \Delta ‘s \text {ABC and A’B’C are similar, therefore,} \\ \frac {A’B’}{AB} &= \frac {A’C}{AC} = \frac {PC - PA’} {PA + PC}\dots (iii) \\ \end{align*}
\begin{align*} \text {From equation} \: (ii) \text {and} \: (iii), \: \text {we get} \\ \frac {PF- PA’}{PF} &= \frac {PC - PA’}{PA + PC} \dots (iv) \\ \text {Applying sign convention} \\ PA’ = - v, PF = - f, PC =- R = - 2f, PA = u \: (\because R = 2f,) \\ \therefore \: \text {Equation} \: (iv) \: \text {becomes} \\ \frac {-f + v }{f} &= \frac {- 2f + v}{ u - 2f} \\ \text {or,} \: -uf + 2f^2 + uv – 2vf = 2f^2 – vf \\ \text {or,} \: uv &= uf + vf \\ \end{align*}
\begin{align*} \text {Dividing by uvf, we get} \\ \frac {uv}{uvf} &= \frac {uf}{uvf} + \frac {vf}{uvf} \\ \frac 1f &= \frac 1u + \frac 1v \\ \end{align*}
To derive mirror formula assumptions and sign conventions are made.
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ASK ANY QUESTION ON Mirror Formula for Concave and Convex Mirror
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