Question

Mathematical expression of mirror formula

Answer 1

Answer:

Mirror formula is the relationship between object distance (u), image distance (v) and focal length. Mirror formula is the relationship between object distance (u), image distance (v) and focal length. I.

Explanation:

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Answer 2

Answer:

Smitaharsh .....

AB be an object placed on the principal axis of a convex mirror of focal length f. u is the distance between the object and the mirror and v is the distance between the image and the mirror.Convex mirror formula:In ABC and A1B1C<ABC = <A1B1C (right angles)<ACB = <A1CB1<CAB = <CA1B (common angle)ABC is similar to A1B1CAB/A1B1 = BC/B1C........(1)similarly DEF issimilar to A1B1FDE/A1B1 = EF/B1F....(2)But DE = AB and when the aperture is very small EF = PFEquation (2) becomesAB/A1B1 = PF/B1F....(3)Frm equations (1) and (3) getPF/B1F = BC/B1CPF/PF-PB1 = PB + PC/PC - PB1f/f - v = -u + 2f/2f - v[PF = f, PB1 = v, PB = u, PC = 2f]2(2f - v)= (f-v)(2f-u)-vf + uf + 2 fv -vu=0fv+uf-vu=0....(4)Dividing both sides of equation(4) by uvf we getfv/uvf + uf/uvf - uv/uvf=01/u +1/v - 1/f=01/u + 1/v = 1/f

Mirror formula is the relationship between object distance (u), image distance (v) and focal length.

1v+1u=1f

proof of mirror formula:

In ΔABC and ΔA′B′C′

ΔABC ∼ ΔA′B′C [AA similarity]

ABA′B′=ACA′C′ ⋯ (1)

Similarly, In ΔFPE and ΔA′B′F′

EPA′B′=PFA′F

ABA′B′=PFA′F [ AB=EP] ⋯ (II)

From (i) &(ii)

ACA′C=PFA′F

=>A′CAC=A′FPF

=>(CP−A′P)(AP−CP)=(A′P–PF)/PF

Now, PF=−f;CP=2PF=−2f ; AP=−u ; and A′P=−v

Put these value in above relation:

⟹[(−2f)–(−v)](−u)−(−2f)=[(−v)–(−f)](−f)

⟹uv=fv+uf

⟹1f=1u+1v

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