A thin prism of angle $$6^\circ$$ and refractive index for yellow light $$(n_Y) = 1.5$$ is combined with another prism of angle $$5^\circ$$ and $$n_Y = 1.55$$. The combination produces no dispersion. The net average deviation $$\delta$$ produced by the combination is $$\left(\frac{1}{x}\right)^\circ$$. The value of $$x$$ is _____

We need to determine the value of $$x$$ given that the net average deviation produced by a combination of two thin prisms is equal to $$\left(\frac{1}{x}\right)^\circ$$.

1. Identify the Given Parameters

• Angle of the first prism ($$A_1$$) = $$6^\circ$$
• Refractive index of the first prism for yellow light ($$n_1$$) = $$1.5$$
• Angle of the second prism ($$A_2$$) = $$5^\circ$$
• Refractive index of the second prism for yellow light ($$n_2$$) = $$1.55$$

2. Understand Deviation Through Thin Prisms

The average (or net) deviation ($\delta$) produced by a single thin prism for mean (yellow) light is given by the formula:

$$\delta = (n - 1)A$$

In this non-dispersive setup, the two prisms are placed inverted relative to each other (as shown in the diagram), meaning their individual deviations oppose one another. The net average deviation ($$\delta_{\text{net}}$$) is the difference between their individual deviations:

$$\delta_{\text{net}} = \delta_1 - \delta_2$$

3. Calculate Individual Deviations

First, let's find the deviation produced by the first prism ($$\delta_1$$):

$$\delta_1 = (1.5 - 1) \times 6^\circ = 0.5 \times 6^\circ = 3^\circ$$

Next, let's find the deviation produced by the second prism ($$\delta_2$$):

$$\delta_2 = (1.55 - 1) \times 5^\circ = 0.55 \times 5^\circ = 2.75^\circ$$

4. Calculate Net Average Deviation and Solve for $$x$$

Subtracting the two values gives the total net deviation of the combination:

$$\delta_{\text{net}} = 3^\circ - 2.75^\circ = 0.25^\circ$$

We can convert this decimal value into a fractional form to match the format given in the problem statement:

$$\delta_{\text{net}} = \left(\frac{25}{100}\right)^\circ = \left(\frac{1}{4}\right)^\circ$$

Comparing this result with the expression $$\left(\frac{1}{x}\right)^\circ$$ 

$$\frac{1}{x} = \frac{1}{4} \implies x = 4$$

Conclusion

The value of $$x$$ is 4.