# Light through the corner of a glass block

• T7
In summary, the conversation discusses proving algebraically that light cannot pass across the corner of a right-angled block of glass with a refractive index of 1.5. Two different methods are presented and it is ultimately concluded that there is no valid solution that would allow the light to escape the glass.
T7
Hi,

I am trying to prove algebraically that light cannot pass across the corner of a right-angled block of glass of refractive index n=1.5. Would someone be willing to let me know whether the method I have used below is valid or not?

'i1' denotes the angle of incidence that the light ray makes with the corner of the glass on entry, 'r1' its angle of refraction. 'i2' denotes the angle of incidence the light within the glass makes against the side perpendicular to the side of its entry, and 'r2' denotes the angle of refraction.

If the light ray is to escape, r2 < 90.

Now n = (sin i1)/(sin r1) ... [1]

And (1/n) = (sin i2)/(sin r2) ... [2]

Since i2 is simply made by the refracted ray with the normal of the perpendicular side,

i2 = 90 - r1.

Therefore sin i2 = cos r1.

From [2], sin r1 = sin i1 / n
From [1], cos r1 = sin r2 / n

Given that sin^2 r1 + cos^2 r1 = 1

(sin r2)^2/n^2 + (sin i1)^2/n^2 = 1

Therefore

sin r2 = sqr(n^2 - (sin i1)^2)

But there are no values of i1 (the angle of incidence) such that 0 <= sin r2 < 1 therefore no value of i1 will produce an angle such that 0 <= r < 90.

Not an elegant proof, perhaps, but is it valid? Anyone care to suggest a better one (just using algebra)?

Cheers.

Hi,

I like your answer, but it's a bit hard to follow the layout. From what I can see, it sees mathematically valid, but I think the 'easier' answer is something like this: Critical angle for glass is:
$$C=\sin^{-1}(\frac{1}{1.5})$$
$$C=41.8^{\circ}$$
Therefore the refracted ray inside the block must meet the edge of the glass to air interface at less than or equal to 41.8 degrees. What is the angle of incidence required for this?
$$1 \sin \theta = 1.5 \sin 41.8$$
$$\theta = \sin^{-1}(1.5 \sin 41.8)$$

This is not possible, since Sine is always between 1 and 0. If we were to decrease the angle of incidence, we will increase the angle at which the light meets the glass to air boundary, so there is no solution that gives a possible answer - the ray must totally internally refract instead.

## 1. How does light travel through the corner of a glass block?

Light travels through the corner of a glass block by undergoing refraction. When light enters the glass block at an angle, it changes speed and bends towards the normal line, which is an imaginary line perpendicular to the surface of the glass block. This change in speed and direction is what allows light to travel through the corner of the glass block.

## 2. Why does light bend when it enters the glass block at an angle?

This bending of light, also known as refraction, occurs because light travels at different speeds in different mediums. In this case, light travels slower in the glass block than in air, causing it to change direction and bend towards the normal line.

## 3. Does the thickness of the glass block affect the path of light through the corner?

Yes, the thickness of the glass block does affect the path of light. The thicker the glass block, the more times the light will undergo refraction as it travels through the block, resulting in a more complex path.

## 4. How does the angle of incidence impact the path of light through the corner of a glass block?

The angle of incidence, or the angle at which light enters the glass block, determines the angle at which light will bend. The greater the angle of incidence, the greater the angle of refraction. This means that light entering the glass block at a steeper angle will bend more sharply.

## 5. What factors influence the speed at which light travels through the corner of a glass block?

The speed of light through the corner of a glass block is influenced by the refractive index of the glass, which is a measure of how much the speed of light is reduced when it passes through the material. It is also affected by the wavelength of light, with longer wavelengths traveling slower than shorter wavelengths.

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