A very narrow beam of white light is incident at 56.10° onto the top surface of a rectangular block of flint glass 12.7 cm thick. The indices of refraction of the glass for red and violet light are 1.640 and 1.663, respectively. Calculate the dispersion angle (i.e., the difference between the directions of red and violet light within the glass block).
How wide is the beam when it reaches the bottom of the block, as measured along the bottom surface of the block?
To calculate the dispersion angle, we need to use Snell's law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the velocities of light in the two media.
First, let's find the angle of refraction for red light. We can use the following formula:
sin(angle of incidence) / sin(angle of refraction) = velocity of light in medium 1 / velocity of light in medium 2
For red light:
sin(56.10°) / sin(angle of refraction for red light) = velocity of light in air / velocity of light in glass
We can rearrange the formula to solve for the angle of refraction for red light:
sin(angle of refraction for red light) = sin(56.10°) * velocity of light in glass / velocity of light in air
Next, let's find the angle of refraction for violet light using the same formula:
sin(angle of refraction for violet light) = sin(56.10°) * velocity of light in glass / velocity of light in air
Now we have the angles of refraction for red and violet light within the glass block. The dispersion angle is the difference between these two angles:
dispersion angle = angle of refraction for violet light - angle of refraction for red light
To calculate the width of the beam when it reaches the bottom of the block along the bottom surface, we need to use the concept of refraction. The beam will bend as it enters and exits the glass block.
The formula to calculate the width of the beam is:
width of beam = width of incident beam * sin(dispersion angle)
Here, the width of the incident beam is the same as the width of the block, which is 12.7 cm.
Plug in the values for the dispersion angle and width of the incident beam to find the width of the beam when it reaches the bottom of the block along the bottom surface.
To calculate the dispersion angle, we can use Snell’s law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the indices of refraction of the two media:
n₁ * sin(θ₁) = n₂ * sin(θ₂),
Where:
n₁ = index of refraction of the incident medium (air/vacuum)
θ₁ = angle of incidence
n₂ = index of refraction of the flint glass
θ₂ = angle of refraction
First, let's calculate the angle of refraction for red light:
n₁ * sin(θ₁) = n₂ * sin(θ₂)
Using n₁ = 1 (since air/vacuum) and n₂ = 1.640 (index of refraction for red light):
1 * sin(56.10°) = 1.640 * sin(θ₂)
sin(θ₂) = (sin(56.10°)) / 1.640
θ₂ = sin^(-1)((sin(56.10°)) / 1.640)
θ₂ ≈ 38.88°
Similarly, let's calculate the angle of refraction for violet light:
n₁ * sin(θ₁) = n₂ * sin(θ₂)
Using n₁ = 1 (since air/vacuum) and n₂ = 1.663 (index of refraction for violet light):
1 * sin(56.10°) = 1.663 * sin(θ₂)
sin(θ₂) = (sin(56.10°)) / 1.663
θ₂ = sin^(-1)((sin(56.10°)) / 1.663)
θ₂ ≈ 38.17°
The dispersion angle (difference between the directions of red and violet light within the glass block) is the difference between θ₂ for red and violet light:
Dispersion angle = θ₂ (violet) - θ₂ (red)
Dispersion angle ≈ 38.17° - 38.88°
Dispersion angle ≈ -0.71° (rounded to two decimal places)
Now, let's calculate the width of the beam when it reaches the bottom of the block along the bottom surface. Given that the block of glass is 12.7 cm thick, we can use trigonometry and the dispersion angle calculated above.
Width of the beam at the bottom = thickness of the block * tan(dispersion angle)
Width of the beam at the bottom = 12.7 cm * tan(-0.71°)
Width of the beam at the bottom ≈ -0.156 cm or approximately -1.56 mm (rounded to two decimal places)
Note: The negative sign indicates that the beam becomes narrower towards the bottom of the block.