Recall. "The Laws of Reflection"

(i) $\theta_i=\theta_r$

Angle of incidence = Angle of Reflection

(ii) RP , OP and PS fix in the plane ABCD

Raversibility of ray paths

Circular section of a hollow glass sphere, with a reflective coating on one surface.

Cross sections in the $x-y$ plane

$\theta \rightarrow i \text { mall }$

$0-5^{\circ}$

$tan \theta \approx \theta$,

$sin \theta \approx \theta$

for snall $\theta$ small aperture approximation

$tan \theta_i = \frac {MD}{CD} , tan 2\theta_i = \frac {MD}{QD}$

For peraxial rays, point $M$ is close to $P$

$\therefore C D \approx C P=R$, the radius of curvature llly $Q . D \approx Q . P$.

Also, $\tan \theta_i \approx \theta_i, \tan 2 \theta_i$ $\approx 2 \theta_i$

Gives $\theta_i=\frac{M D}{R}, 2 \theta_2=\frac{M P}{Q P} \rightarrow Q P=\frac{R}{2}$

$Q P=\frac{R}{2}$, constant

$M$ - is an arbitrarypoint

$\Rightarrow$ Any parallel raywill pass through $Q$

The point ' $Q$ ' is called the "principal focus" and' designate as ' $F$ '

The distance FP is celled "focal length', ' $f$ '

Thus $Q . P=F P=f$

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

Parallel Rays Inclined to the Principal Axis

Parallel rays remain parallel

Image of a point object: Plane mirror

$I$ is the image point

$\triangle O A B$ and $\triangle I A B$ ara congruent. $\Rightarrow O B=I B$

A "virtual images is formal at I.

Point $A$ is artibrary $OB=IB$ in all cases.

I is a "virtual image"

Intersection of any two rays from the abject is sufficient to determine the image.

The point $Q$ is arbitrary ; the object distance and image distance will satisfy a relation, independent of Q.