Light - Reflection and Refraction

For a concave mirror, the principal focus is the point on the principal axis where rays parallel to the principal axis actually meet after reflection.

For a spherical mirror, $R = 2f$.

Given $R = 20\text{ cm}$,

$$f = \frac{R}{2} = \frac{20}{2} = 10\text{ cm}$$

A concave mirror can form a virtual, erect and enlarged image when the object is placed between the pole and the principal focus.

We prefer a convex mirror as a rear-view mirror because it:

- always forms a virtual, erect and diminished image,
- gives a wider field of view,
- lets the driver see more area behind the vehicle, which helps in safe driving.

For a spherical mirror, $R = 2f$.

Given a convex mirror, the focal length is positive.

$$f = \frac{R}{2} = \frac{32}{2} = 16\text{ cm}$$

A real enlarged image by a concave mirror is possible when the object is placed between the pole and the focus. A three times magnified real image means magnification

$$m = -3$$

For mirrors,

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

Given $u = -10\text{ cm}$,

$$-3 = -\frac{v}{-10}$$

$$-3 = \frac{v}{10}$$

$$v = -30\text{ cm}$$

So the image is formed 30 cm in front of the mirror.

When light travels from air to water, it goes from a rarer medium to a denser medium. Therefore, it bends towards the normal because its speed decreases in water.

Refractive index,

$$n = \frac{c}{v}$$

Given $n = 1.50$ and $c = 3\times10^8\text{ m s}^{-1}$,

$$v = \frac{c}{n} = \frac{3\times10^8}{1.50} = 2\times10^8\text{ m s}^{-1}$$

From Table 9.3, the highest refractive index is for diamond ($2.42$), so it has the highest optical density.

The lowest refractive index is for air ($1.0003$), so it has the lowest optical density.

Light travels fastest in the medium with the lowest refractive index.

From Table 9.3:
- water: $1.33$
- kerosene: $1.44$
- turpentine oil: $1.47$

So light travels fastest in water.

The statement means

$$n = \frac{c}{v} = 2.42$$

So the speed of light in diamond is

$$v = \frac{c}{2.42}$$

That is, light travels 2.42 times slower in diamond than in vacuum.

1 dioptre is defined as the power of a lens whose focal length is 1 metre.

Since

$$P = \frac{1}{f}$$

if $f = 1\text{ m}$, then $P = 1\text{ D}$.

A real and inverted image equal in size to the object is formed by a convex lens when the object is placed at 2F.

Given image distance = 50 cm, so object is also at 50 cm on the other side of 2F? For a convex lens, when image is same size as object:

$$u = -2f, \quad v = +2f$$

Given $v = 50\text{ cm}$, so

$$2f = 50$$

$$f = 25\text{ cm} = 0.25\text{ m}$$

Thus object distance

$$u = -2f = -50\text{ cm}$$

So the needle is placed 50 cm in front of the lens.

Power:

$$P = \frac{1}{f} = \frac{1}{0.25} = +4.0\text{ D}$$

But the given Chapter example states same-size image at 2F, so if the image is at 50 cm, the object is also at 50 cm and the focal length is 25 cm. Hence the power is +4.0 D.

For a lens,

$$P = \frac{1}{f}$$

For a concave lens, focal length is negative.

Given $f = 2\text{ m}$,

$$f = -2\text{ m}$$

So,

$$P = \frac{1}{-2} = -0.5\text{ D}$$

A lens must be made of a transparent material. Clay is opaque, so it cannot be used to make a lens.

A concave mirror forms a virtual, erect and enlarged image only when the object is placed between the pole P and the principal focus F.