List the new Cartesian sign convention for reflection of light by spherical mirrors. Draw a diagram and apply these conventions for calculating the focal length and nature of a spherical mirror which forms a 1/3 times magnified virtual image of an object placed 18 cm in front of it.

OR

With the help of a ray diagram, state what is meant by refraction of light. State Snell's law for refraction of light and also express it mathematically.

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 $2 \times 10^8$ m/s, find the speed of light in (a) air, (b) water.

Part 1: New Cartesian Sign Convention for Spherical Mirrors

According to the New Cartesian Sign Convention:

1. The object is always placed to the left of the mirror. This implies that the light from the object falls on the mirror from the left-hand side.
2. All distances parallel to the principal axis are measured from the pole ($P$) of the mirror.
3. All distances measured in the direction of incident light (to the right of the origin, along the $+x$-axis) are taken as positive, while those measured opposite to the direction of incident light (to the left of the origin, along the $-x$-axis) are taken as negative.
4. Distances measured perpendicular to and above the principal axis (along the $+y$-axis) are taken as positive.
5. Distances measured perpendicular to and below the principal axis (along the $-y$-axis) are taken as negative.

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Part 2: Numerical Calculation (Mirror)

Given:

• Object distance, $u = -18 \text{ cm}$ (by sign convention)
• Magnification, $m = +\frac{1}{3}$ (positive because the image is virtual and erect)

We know that magnification for a spherical mirror is:

$$m = -\frac{v}{u}$$

Substituting the values:

$$\frac{1}{3} = -\frac{v}{-18}$$

$$\frac{1}{3} = \frac{v}{18} \implies v = +6 \text{ cm}$$

Using the mirror formula:

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

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

$$\frac{1}{f} = \frac{3 - 1}{18} = \frac{2}{18} = \frac{1}{9}$$

$$f = +9 \text{ cm}$$

Since the focal length $f$ is positive, the mirror is a convex mirror.

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OR Option

#### Refraction of Light

When a ray of light traveling obliquely in one transparent medium enters another transparent medium, its direction of propagation in the second medium changes. This phenomenon is called the refraction of light.

#### Snell's Law of Refraction

It states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, for the light of a given colour and for the given pair of media.

Mathematically:

$$\frac{\sin i}{\sin r} = \text{constant} = n_{21}$$

where $n_{21}$ is the refractive index of medium 2 with respect to medium 1.

#### Numerical Calculation (Refraction)

Given:

• Refractive index of air with respect to glass, $^{g}n_{a} = \frac{2}{3}$
• Refractive index of water with respect to air, $^{a}n_{w} = \frac{4}{3}$
• Speed of light in glass, $v_g = 2 \times 10^8 \text{ m/s}$

(a) Speed of light in air ($v_a$):

$$^{g}n_{a} = \frac{v_g}{v_a}$$

$$\frac{2}{3} = \frac{2 \times 10^8}{v_a}$$

$$v_a = 3 \times 10^8 \text{ m/s}$$

(b) Speed of light in water ($v_w$):

$$^{a}n_{w} = \frac{v_a}{v_w}$$

$$\frac{4}{3} = \frac{3 \times 10^8}{v_w}$$

$$v_w = \frac{3 \times 3 \times 10^8}{4} = 2.25 \times 10^8 \text{ m/s}$$