Question

Prove f = r/2 for spherical mirrors.

Answer 1

Answer:

Explanation:

Consider the  spherical mirror with the rays striking it:

The ray will strike the mirror and then get reflected. The reflected ray will form angles at two points C and F of the mirror. A line perpendicular to CP is subtended at right angle to form triangle MPF and MPC with different angles.

Let ‘’f’’ be the focal length with radius of curvature “R”.

So from the figure

∠MCP=θ And ∠MFP=2θ

tanθ=$\frac{M P}{P C}$

tan2θ=$\frac{M P}{P F}$

As PF = f and PC = R

tanθ=$\frac{M P}{R}$= MP=tanθR

And tan2θ=$\frac{M P}{f}$=MP=tan2θf

Equating  

tanθR=tan2θf

f=$\frac{R}{2}$.

Answer 2

Answer:

Consider the spherical mirror with the rays striking it:

The ray will strike the mirror and then get reflected. The reflected ray will form angles at two points C and F of the mirror. A line perpendicular to CP is subtended at right angle to form triangle MPF and MPC with different angles.

Let ‘’f’’ be the focal length with radius of curvature “R”.

So from the figure

∠MCP=θ And ∠MFP=2θ

tanθ=\frac{M P}{P C}

PC

MP

tan2θ=\frac{M P}{P F}

PF

MP

As PF = f and PC = R

tanθ=\frac{M P}{R}

R

MP

= MP=tanθR

And tan2θ=\frac{M P}{f}

f

MP

=MP=tan2θf

Equating

tanθR=tan2θf

f=\frac{R}{2}

2

R

.

Explanation:

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