A thin prism P1 with angle 4 degrees and made from glass of refractive index 1.54 is combined with a

Q: Why do we need two different types of glass?

If we used the same type of glass for both prisms, cancelling the deviation would also completely cancel the dispersion. By using glasses with different dispersive powers (like crown and flint), we can cancel the deviation for the mean color while still leaving a net spread of the other colors.

Q: What exactly does 'thin prism' mean in this context?

A thin prism is one with a very small refracting angle, typically less than $10^{\circ}$. For such prisms, the angle of deviation is practically independent of the angle of incidence and can be approximated by the simple formula $\delta = (\mu - 1)A$.

Q: Why are the prisms placed inverted relative to each other?

To achieve zero net deviation, the second prism must bend the light in the exact opposite direction to the first prism. Placing them base-to-apex ensures their deviations subtract from each other rather than add up.

Q: Does the emergent ray travel along the exact same line as the incident ray?

Not exactly. While the emergent ray is parallel to the incident ray (zero angular deviation), it does suffer a slight lateral displacement. However, for thin prisms, this lateral shift is usually negligibly small.

Q: If there is no deviation, how is there still dispersion?

Deviation refers to the bending of the *mean* ray (yellow). Dispersion refers to the *difference* in bending between the extreme rays (violet and red). Because the two glasses have different dispersive powers, they can be balanced to yield zero net deviation for yellow light, but they won't perfectly balance for violet and red, resulting in a net spectrum.

Question

A thin prism $P_1$ with angle $4^{\circ}$ and made from glass of refractive index $1.54$ is combined with another thin prism $P_2$ made from glass of refractive index $1.72$ to produce dispersion without deviation. The angle of the prism $P_2$ is

Options:

(A) $5.33^{\circ}$

(B) $4^{\circ}$

(C) $3^{\circ}$

(D) $2.6^{\circ}$

Sigma Percentile · Accepted Answer

The phenomenon of **dispersion without deviation** is a fascinating application of optics, primarily used in instruments like the direct vision spectroscope. Imagine you want to split white light into its constituent colors to study its spectrum, but you don't want the entire beam to bend away from your line of sight. How do you achieve this?

By combining two prisms made of different materials—typically crown glass and flint glass—we can perfectly balance their bending powers while exploiting their different dispersive powers.

### Analyzing the Setup

When a ray of light passes through a thin prism, it undergoes a deviation $\delta$. For a prism with a small refracting angle $A$ and refractive index $\mu$, this deviation is given by the simple relation:

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

If we place two such prisms base-to-apex (inverted relative to each other), their deviations will oppose each other. The net deviation $\delta_{	ext{net}}$ of the combination is the difference between the individual deviations:

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

### The Master Equation

The problem demands "dispersion without deviation", which means the net deviation for the mean ray (usually the yellow light) must be exactly zero. Setting $\delta_{	ext{net}} = 0$, we get:

$$
\delta_1 = \delta_2
$$

Substituting the formula for deviation for both prisms, we arrive at our master equation:

$$
(\mu_1 - 1)A_1 = (\mu_2 - 1)A_2
$$

This equation tells us that the bending power of the first prism must perfectly match the bending power of the second prism.

### Final Calculation

We are given the parameters for the first prism (crown glass): $\mu_1 = 1.54$ and $A_1 = 4^{\circ}$. For the second prism (flint glass), we know $\mu_2 = 1.72$ and we need to find its angle $A_2$.

Rearranging our master equation to solve for $A_2$:

$$
A_2 = \frac{(\mu_1 - 1)}{(\mu_2 - 1)} A_1
$$

Now, let's carefully substitute the given values into this expression:

$$
A_2 = \frac{(1.54 - 1)}{(1.72 - 1)} 	imes 4^{\circ}
$$

Subtracting the terms in the numerator and denominator:

$$
A_2 = \frac{0.54}{0.72} 	imes 4^{\circ}
$$

The fraction $\frac{0.54}{0.72}$ simplifies beautifully to $\frac{3}{4}$. Multiplying this by $4^{\circ}$ gives us our final result:

$$
A_2 = 3^{\circ}
$$

**Thus, the angle of the second prism must be $3^{\circ}$.** Even though the mean ray passes undeviated, the difference in the dispersive powers of the two glasses ensures that the white light is still spread out into a beautiful spectrum!

Animated Solution for Physics - Optics: A thin prism $P_{1}$ with angle $4^{\circ}$ and made from glass of refractive index $1.54$ is combined with another thin prism $P_{2}$ made from glass of refractive index $1.72$ to produce dispersion without deviation. The angle of the prism $P_{2}$ is

Select Answer:

Visualized Solution

$Visualizing the Setup$

$δ = (μ - 1) A$

$δ_{net} = 0$

(\mu_1 - 1)A_1 = (\mu_2 - 1)A_2

A_2 = \frac{(\mu_1 - 1)}{(\mu_2 - 1)}A_1

A_2 = \frac{(1.54 - 1)}{(1.72 - 1)}(4^{\circ})

A_2 = \frac{0.54}{0.72}(4^{\circ})

A_2 = 3^{\circ}

\text{Direct Vision Spectroscope}

The Sigma Insight: Refraction and Dispersion through Prism

Analyzing the Setup

The Master Equation

Final Calculation

Similar Questions

An isosceles prism of angle $12 0^{\circ}$ has a refractive index $1.44$ . Two parallel rays of monochromatic light enter the prism parallel to each other in air as shown. The rays emerging from the opposite face

The refractive index of glass is $1.520$ for red light and $1.525$ for blue light. Let $D_{1}$ and $D_{2}$ be angles of minimum deviations for red and blue light respectively in a prism of this glass. Then,

Monochromatic light is incident on a glass prism of angle $A$ . If the refractive index of the material of the prism is $μ$ , a ray incident at an angle $θ$ , on the face $A B$ would get transmitted through the face $A C$ of the prism provided