An isosceles prism of angle 120 degrees has a refractive index 1.44. Two parallel rays of monochroma

Q: Why do the rays pass undeviated through the first face?

The rays are horizontal and the first face is perfectly vertical. This means the rays strike the surface along the normal (angle of incidence is $0^\circ$). According to Snell's law, if $i = 0^\circ$, then $r = 0^\circ$, so the light passes straight through.

Q: How did we find the angle of the normal to be $30^\circ$?

The prism is isosceles with an apex angle of $120^\circ$. By symmetry, the line bisecting the apex is horizontal. The angle between the slanted face and this horizontal axis is $120^\circ / 2 = 60^\circ$. The normal is perpendicular to the face, so it makes an angle of $90^\circ - 60^\circ = 30^\circ$ with the horizontal.

Q: Why is the deviation $\delta = r - i$?

The incident ray is horizontal. The normal is at an angle $i$ to the horizontal. The refracted ray is at an angle $r$ to the normal. Since the ray bends away from the normal (going from glass to air), its angle with the horizontal becomes $r - i$. This difference is the deviation $\delta$.

Q: Why do the rays converge instead of diverge?

The upper slanted face points downwards, so its normal points upwards. The ray bends away from the normal, which means it bends downwards towards the central axis. The lower ray bends upwards. Since they bend towards each other, they are converging.

Q: What if the refractive index was smaller, say 1.2?

If $\mu = 1.2$, then $\sin r = 1.2 \sin 30^\circ = 0.6$. The angle $r$ would be $\sin^{-1}(0.6) \approx 36.8^\circ$. The deviation would be $6.8^\circ$, and the rays would still converge, just at a smaller angle.

Question

An isosceles prism of angle $120^\circ$ 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

Options:

(A) are parallel to each other

(B) are diverging

(C) make an angle $2[\sin^{-1}(0.72) - 30^\circ]$ with each other

(D) make an angle $2\sin^{-1}(0.72)$ with each other

Sigma Percentile · Accepted Answer

### Visualizing the Prism
Imagine a beautiful isosceles glass prism with a wide apex angle of $120^\circ$. Two parallel rays of monochromatic light are traveling horizontally through the air and strike the flat, vertical face of this prism.

Because these rays hit the vertical face perfectly head-on (perpendicularly), their angle of incidence is $0^\circ$. According to Snell's law, they will pass straight through this first boundary without any deviation. They continue their horizontal journey deep inside the glass until they encounter the slanted faces at the back of the prism.

### The Geometry of the Slanted Faces
To understand what happens next, we need to analyze the geometry of the prism. The apex angle is $120^\circ$. By symmetry, the horizontal axis bisects this angle, meaning the slanted faces make an angle of $60^\circ$ with the vertical.

If we draw a normal (a perpendicular line) to the slanted face, it will make an angle of $90^\circ - 60^\circ = 30^\circ$ with the horizontal. Since our light rays are traveling perfectly horizontally, the angle they make with this normal is exactly $30^\circ$. Thus, the angle of incidence at the glass-air interface is $i = 30^\circ$.

### Snell's Law at the Boundary
Now, the light is trying to escape from the denser glass ($\mu = 1.44$) into the rarer air ($\mu = 1$). We apply Snell's law to find out how much it bends:

$$ \mu \sin i = 1 \cdot \sin r $$

Substituting our known values:

$$ 1.44 \sin 30^\circ = \sin r $$

Since $\sin 30^\circ = 0.5$, we get:

$$ \sin r = 1.44 	imes 0.5 = 0.72 $$

This gives us the angle of refraction: $r = \sin^{-1}(0.72)$. Because the light is entering a rarer medium, it bends *away* from the normal.

### The Final Convergence
How much did the ray actually deviate from its original horizontal path? The deviation $\delta$ is simply the difference between the angle of refraction and the angle of incidence:

$$ \delta = r - i = \sin^{-1}(0.72) - 30^\circ $$

Here is the fascinating part: the upper ray hits a face that slants downwards, so it bends downwards by an angle $\delta$. The lower ray hits a face that slants upwards, so it bends upwards by an angle $\delta$.

Instead of diverging, these two rays are actually bending towards each other! They are converging. The total angle between these two emerging rays is simply the sum of their individual deviations:

$$ 	ext{Total Angle} = 2\delta = 2[\sin^{-1}(0.72) - 30^\circ] $$

This elegant geometric dance of light perfectly matches option (c).

Animated Solution for Physics - Optics: 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

Select Answer:

Visualized Solution

Analyzing the Setup

Rays Inside the Prism

Geometry of the Prism

Angle of Incidence

Applying Snell's Law

Calculating Angle of Refraction

Deviation of Each Ray

Total Angle Between Rays

Conclusion

The Sigma Insight: Refraction and Dispersion through Prism

Visualizing the Prism Imagine a beautiful isosceles glass prism with a wide apex angle of $12 0^{\circ}$

The Geometry of the Slanted Faces To understand what happens next, we need to analyze the geometry of the prism

Snell's Law at the Boundary Now, the light is trying to escape from the denser glass ( $μ = 1.44$ ) into the rarer air ( $μ = 1$ )

The Final Convergence

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