The Cartesian sign convensions are as follows.

(i) The object is always placed to the left of the mirror, i.e. the light from the object falls on the mirror from the left-hand side.

(ii) All distances parallel to the principal axis are measured from the pole of the mirror. (The principal axis is taken along the x-axis with pole as the origin.)

(iii) All the distances measured to the right of the pole are taken as positive while those measured to the left of the pole are taken as negative.

(iv) Distances measured perpendicular to and above the principal axis are taken as positive.

(v) Distances measured perpendicular to and below the principal axis are taken as negative.

Object distance *u* = −18 cm

Magnification =$\frac{1}{3}$ = $\frac{-v}{u}$

⇒ *v* = 6 cm (image distance)

v is positive and hence the image is formed on the other side of the mirror.

So we have a diminished erect virual image at the back of the mirror. So the mirror is a convex mirror.

Now let us use the mirror formula to calculate the focal length.

Mirror formula:

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

$\frac{1}{6}$ + $\frac{1}{-18}$ = $\frac{1}{f}$

⇒ *f* = 9 cm.

(The positive focal length confirms that the mirror is convex.)

OR

The bending of a light ray when it travels from one transparent medium to another is called refraction. Refraction is due to change in the speed of light that occurs when it enters from one transparent medium to another. The speed of light in a medium is dependent on the refractive index of the medium. The speed of light is higher in a rarer medium (medium with lower refractive index) than in a denser medium (medium with greater refractive index). Thus, a ray of light travelling from a rarer medium to a denser medium slows down and bends towards the normal. When it travels from a denser medium to a rarer medium, it speeds up and bends away from the normal.

Snell’s law states that the ratio of the sine of the angle of incidence to the sine of angle of refraction is a constant.

Mathematically it can be expressed as follows:

$\frac{\mathrm{sin}i}{\mathrm{sin}r}=\mathrm{cons}\mathrm{tan}t=$^{nab}

* *where ^{nab is the refractive index of medium B with respect to medium A. }

The refractive index of air with respect to glass is 2/3 and the refractive index of water with respect to air is 4/3. If the speed of light in glass is 2 × 10^{8} m/s, find the speed of light in (a) air, (b) water.

(i) Given: Refractive index of air with respect to glass *n*_{ag}= 2/3 = $\frac{{v}_{g}}{{v}_{a}}$

Speed of light in glass *v*_{g} = 2 × 10^{8} m/s

Therefore ${v}_{a}=\frac{{v}_{g}}{{n}_{\mathrm{ag}}}$ = 2×10^{8} m/s/(2/3)

= 3×10^{8} m/s

Speed of light in air = 3×10^{8} m/s

(ii) Refractive index of water with respect to air *n*_{wa} = 4/3 = $\frac{{v}_{a}}{{v}_{w}}$

Therefore ${v}_{w}=$3×10^{8} m/s /(4/3)

= 2.25×10^{8} m/s.

Speed of light in water = 2.25 ×10^{8} m/s