would anyone help me in solving this problem :

The refractive indices of acrown glass for red and blue light are 1.51 and 1.49 and those of flint glass are 1.77 and1.73 respectively. An isosceles prism of angle 6 degree is made of crown glass . A beam ofwhite light is incident on the prism at an small angle on the prism .The other flint glass isosceles prism is combined with the ceown glass prism in such a way that there is no deviation of the incidet ray .Determine the angle of the flint glass prism .and the net dispertion of the combined system .

please provide me the answer with correct explanation.

To tackle this problem, we need to analyze the behavior of light as it passes through the two prisms made of crown glass and flint glass. The goal is to find the angle of the flint glass prism that will ensure no net deviation of the incident ray, as well as to calculate the net dispersion of the combined system.

Understanding the Basics of Refraction

Refraction occurs when light passes from one medium to another, changing speed and direction due to differences in refractive indices. The refractive index (n) is a dimensionless number that describes how fast light travels in a medium compared to a vacuum. For our problem, we have:

• Crown glass: n_red = 1.51, n_blue = 1.49
• Flint glass: n_red = 1.77, n_blue = 1.73

Analyzing the Crown Glass Prism

First, let's consider the crown glass prism. The angle of the prism is given as 6 degrees. When white light enters the prism, it will split into its constituent colors due to different refractive indices for red and blue light. The angle of deviation (D) for a prism can be calculated using the formula:

D = (n - 1) * A

Where:

• D = angle of deviation
• n = refractive index of the material for the specific color
• A = angle of the prism

Calculating Deviation for Crown Glass

For red light:

D_red = (1.51 - 1) * 6 = 0.51 * 6 = 3.06 degrees

For blue light:

D_blue = (1.49 - 1) * 6 = 0.49 * 6 = 2.94 degrees

Setting Up the Flint Glass Prism

Next, we need to ensure that the combined system of the crown glass prism and the flint glass prism results in no net deviation. This means that the deviation caused by the flint glass prism must counteract the deviation caused by the crown glass prism.

Deviation for Flint Glass

Let’s denote the angle of the flint glass prism as θ. The deviation for the flint glass prism can be expressed similarly:

D_flint = (n - 1) * θ

For red light:

D_{flint, red} = (1.77 - 1) * θ = 0.77 * θ

For blue light:

D_{flint, blue} = (1.73 - 1) * θ = 0.73 * θ

Setting Up the Equation for No Deviation

To achieve no net deviation, we set the total deviation for red and blue light to zero:

D_red + D_{flint, red} = 0

Substituting the values:

3.06 + 0.77θ = 0

Solving for θ gives:

θ = -3.06 / 0.77 ≈ -3.97 degrees

Since we are looking for the angle of the prism, we take the absolute value:

θ ≈ 3.97 degrees

Calculating Net Dispersion

Now, let’s find the net dispersion of the combined system. The dispersion is the difference in deviation between red and blue light:

Dispersion = D_red - D_blue

For the crown glass prism:

Dispersion_crown = 3.06 - 2.94 = 0.12 degrees

For the flint glass prism:

Dispersion_flint = (0.77 * θ) - (0.73 * θ) = (0.77 - 0.73) * θ = 0.04 * θ

Substituting θ = 3.97 degrees:

Dispersion_flint = 0.04 * 3.97 ≈ 0.1588 degrees

Final Calculation of Net Dispersion

Now, we can find the total dispersion of the combined system:

Net Dispersion = Dispersion_crown + Dispersion_flint

Net Dispersion = 0.12 + 0.1588 ≈ 0.2788 degrees

In summary, the angle of the flint glass prism is approximately 3.97 degrees, and the net dispersion of the combined system is about 0.2788 degrees. This analysis illustrates how the properties of different materials can be utilized to manipulate light effectively.