Question

Obtain mirror formula for concave mirror.

Answer 1

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Obtain mirror formula for concave mirror.

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→ Check the attachment first provided with answer.

→ We need to Learn about Sign Conventions before deriving mirror formula as it involve "The sign conversations.

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$\Large\bold\star\underline{\underline\textbf{Mirror formula\:}}$

→ See the attachment .

→ Consider an object AB lying beyond the principal focus F of a concave mirror.

→ Let BK and BL be the two incident rays which meet at point B' after reflection from the mirror.

→ A'B' be the image of object AB.

→ Draw KN perpendicular to principal axis.

→ Now ∆ NKF and ∆ A'B'F are similar, therefore

» A'B' / KN = A' F /NF.....................(1)

→ When aperture of mirror is small then point N is very close to P. So, NF ≈ PF (approx) and KN ≈ AB

→ Equation (1) becomes,

A'B / AB = A'F / PF ..........................(2).

→ Also ∆ ABC and ∆ A'B'C are similar, therefore

A'B' / AB = A'C / AC ........................(3).

→ From equations (2) and (3), we have

A'F / PF = A'C / AC

(Or)

PA' - PF / PF = PC - PA' / PA -PC

..................(4)

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→ Using Sign Conventions , We have,

» PA' = - v

» PA' = - v » PF = - f

» PA' = - v » PF = - f» PC = R = - 2f

» PA' = - v » PF = - f» PC = R = - 2f» PA = - u

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→ So , Equation (4) becomes

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

» uv - 2vf - uf + 2f² = 2f² - vf

» uv - uf = vf.............(5)

Dividing equation (5) by uvf , we get

$\frac{1}{f} - \frac{1}{u} = \frac{1}{v}$

Or

$</p><p> \frac{1}{f } = \frac{1}{u} + \frac{1}{v}$

This equation is called mirror formula.

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Hope it helped u.

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