Question

prove that for any spherical mirror 1/f=1/v+1/u where the symbols have their usual meaning​

Answer 1

Answer:

Refer to the mirror formula, 1v + 1u = 1f , where the symbols have their usual meanings. Which of the following graphs correctly depicts this variation

Explanation:

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Answer 2

Answer:

We proved that $\frac{1}{f} = \frac{1}{v} +\frac{1}{u}$ for  any spherical mirror .

Explanation:

First we draw a diagram of spherical mirror and

Let an object AB is place in front of the mirror .

Let u be the distance between the object and the mirror ,

let v be the distance between the image and mirror and

let f be the focal length .

 Now from the diagram we have B'P = -v , PF = -f and BP = -u

Now ,ΔB'A'F and ΔFPM are similar

∠A'B'F = 90 =∠MPF = 90  ∠A'FB'= ∠PFM  so the other angles is also equal .

$\frac{B'A'}{PM} = \frac{B'F}{PF}$

⇒$\frac{B'A'}{BA} =\frac{B'F}{PF}$  ......(i)       [PM = BA ]

Similarly , ΔA'B'P and ΔABP are similar ,

$\frac{B'A'}{BA} = \frac{B'P}{BP}$ ..........(ii)

From (i) and (ii) we get ,

$\frac{B'F}{PF} = \frac{B'P}{BP}$

⇒$\frac{B'P - PF }{PF} = \frac{B'P}{BP}$   .........(iii)      [B'F = B'P - PF ]

Now put the value of B'P , PF and BP in (iii) we get

⇒$\frac{-v +f }{-f} = \frac{-v}{-u}$

⇒$\frac{v -f }{f} = \frac{v}{u}$  

⇒ uv - uf = vf       [By cross multiplication]

Now divide both the side by uvf   we get ,

⇒$\frac{uv}{uvf} -\frac{fu}{uvf} =\frac{vf}{uvf}$

⇒$\frac{1}{f} -\frac{1}{v} =\frac{1}{u}$ ⇒$\frac{1}{f} = \frac{1}{v} +\frac{1}{u}$  proved .

Hence , here we proved that $\frac{1}{f} = \frac{1}{v} +\frac{1}{u}$.

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