A tank is filled with water to a height of 12.5 cm. The apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be 9.4 cm. What is the refractive index..?

A tank is filled with water to a height of 12.5 cm. The apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be 9.4 cm. What is the refractive index of water? If water is replaced by a liquid of refractive index 1.63 upto the same height, by what distance would be microscope have to be moved to focus on the needle again?

The microscope needs to be moved by approximately 1.73 cm to focus on the needle again.

To find the refractive index of water, we can use the apparent depth formula:

n_1/n_2 = d_real/d_apparent

Where:

• n_1 = Refractive index of the medium above the water (usually air, so n_1 = 1)
• n_2 = Refractive index of water
• d_real = Real depth of the needle (needle's actual depth in water)
• d_apparent = Apparent depth of the needle (as observed through the microscope)

Given:

• d_real = 12.5 cm
• d_apparent = 9.4 cm
• n_1 = 1

We can rearrange the formula to solve for n_2:

n_2 = (d_real/d_apparent) * n_1

n_2 = (12.5/9.4) * 1

n_2 ≈ 1.33

So, the refractive index of water is approximately 1.33.

Now, when water is replaced by a liquid of refractive index 1.63, the apparent depth of the needle will change. We can use a similar formula to find the new apparent depth, and then calculate the distance by which the microscope must be moved.

Let's denote the new apparent depth as d_apparent_new. We'll also denote the distance by which the microscope has to be moved as x.

d_apparent_new = (n_1/n_new) * d_real

Given:

• n_1 = 1
• n_new = 1.63
• d_real = 12.5 cm

d_apparent_new = (1/1.63) * 12.5

d_apparent_new ≈ 7.67 cm

Now, to find the distance x by which the microscope needs to be moved, we'll take the difference between the original apparent depth and the new apparent depth:

x = d_apparent - d_apparent_new

x = 9.4 - 7.67

x ≈ 1.73 cm

So, the microscope needs to be moved by approximately 1.73 cm to focus on the needle again.

Refraction of Light

Refraction of light is a phenomenon that occurs when light waves pass from one medium into another with a different optical density. Optical density is a measure of how much the material slows down the speed of light compared to its speed in a vacuum. When light passes through a medium such as air, water, glass, or any other transparent substance, it may change direction, or bend, due to the change in speed caused by the change in medium.

The bending of light during refraction can be understood through Snell's Law, which states:

n1 * sin(theta1) = n2 * sin(theta2)

Where:

• n1 and n2 are the refractive indices of the two media.
• theta1 and theta2 are the angles that the incident ray and the refracted ray make with the normal (an imaginary line perpendicular to the surface at the point of incidence).

Key points about refraction:

1. Change in speed: Light waves travel at different speeds in different materials due to differences in optical density. When light enters a denser medium, it slows down, causing it to bend towards the normal. When it exits to a less dense medium, it speeds up and bends away from the normal.

2. Change in wavelength: Refraction doesn't change the frequency of light but can change its wavelength. As light enters a denser medium, its wavelength decreases, while in a less dense medium, its wavelength increases.

3. Critical angle: If the angle of incidence is increased beyond a certain critical angle, the refracted ray will be bent so much that it travels along the boundary between the two media. This phenomenon is called total internal reflection and is the basis for optical fibers and mirages.

4. Dispersion: Different colors of light bend by different amounts when they pass through a prism due to differences in their wavelengths. This separation of colors is called dispersion and is responsible for the formation of rainbows and the color spectrum seen in a prism.

Refraction plays a crucial role in various optical devices and natural phenomena, from lenses and prisms to the behavior of light in the atmosphere, oceans, and other transparent media.