Question

Derive the formula for spherical mirror 1/f=1/v+1/u,Answer the given question in detail.

Answer 1

To Prove ⇒ $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$

In this, f = Focal length of the lens.

Image Distance = v

Object Distance = u

Mirror Formula can be derived from any of the mirror, either from the concave mirror or convex mirror. Result in both the cases will be the same.

Now, Let us derive that by using the Concave Mirror.

Proof ⇒

Refers to the Ray Diagram in the attachment.

u, v, f, R , all are negative, since all lies in the negative side of the Cartesian plane.

From the attachment,

In ΔABC and ΔCXY

∠ ABC = ∠ CXY [Alternate angles]

∠ BAC = ∠ CYX [Alternate angles]

∠ ACB = ∠ YCX [Vertically Opposite angles]

By A.A. Postulate, Both the Δ's are similar.

∴ ΔABC ≈ ΔYXC

∴ $\frac{AB}{YX} = \frac{BC}{XC}$ -----eq(i)

Now, In ΔABP and ΔXYP,

∠ ABP = ∠ PXY = 90°

∠ APB = ∠ XPY

By A.A. Postulate,

Δ APB ≈ ΔYPX

∴ $\frac{AB}{XY} = \frac{PB}{PX}$ -----eq(ii)

Now, From eq(i) and eq(ii),

$\frac{BC}{XC} = \frac{PB}{PX}$

From the attachment,

BC = PB - PC

XC = PC - PX

Putting this in the above equation,

$\frac{PB - PC}{ PC - PX} = \frac{PB}{PX}$

∴ $\frac{-u - (-R)}{-R - (-v)} = \frac{-u}{-v}$

⇒ $\frac{R - u}{v - R} = \frac{u}{v}$

⇒ v(R - u) = u(v - R)

⇒ Rv - uv = uv - Ru

⇒ Rv + Ru = uv + uv

⇒ R(v + u) = 2vu

⇒ R/2 = vu/(v + u)

⇒ R/2 = 1/u + 1/v

We know, R/2 = f (focal length)

∴ 1/f = 1/u + 1/v

∴ $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$

Hence, Proved.

Hope it helps.