The speed of light in a certain piece of clear plastic is 1.95 108 m/s. A ray of light strikes the plastic at an angle of 21°. At what angle is the ray refracted?
To determine the angle at which the ray of light is refracted, we can use Snell's Law, which relates the angles of incidence and refraction to the ratio of the speeds of light in two different media.
Snell's Law can be expressed as:
n₁ sin(θ₁) = n₂ sin(θ₂)
Where:
- n₁ is the refractive index of the medium from which the light is coming (in this case, air),
- n₂ is the refractive index of the medium into which the light is entering (in this case, the clear plastic),
- θ₁ is the angle of incidence, and
- θ₂ is the angle of refraction (the value we're trying to find).
Since we are given the speed of light in the plastic, we can calculate the refractive index using the formula:
n = c/v
Where:
- c is the speed of light in a vacuum (approximately 3.00 x 10^8 m/s), and
- v is the speed of light in the medium.
Let's calculate the refractive index of the clear plastic:
n₂ = c/v = (3.00 x 10^8 m/s)/(1.95 x 10^8 m/s) = 1.54
Now that we have the refractive indices, we can rearrange Snell's Law to solve for θ₂:
sin(θ₂) = (n₁/n₂) * sin(θ₁)
Substituting the given values:
sin(θ₂) = (1/1.54) * sin(21°)
Taking the inverse sine of both sides:
θ₂ = sin^(-1)((1/1.54) * sin(21°))
Using a calculator, we find:
θ₂ ≈ 11.8°
Therefore, the ray of light is refracted at an angle of approximately 11.8° when it enters the clear plastic.
To find the angle at which the ray is refracted, 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 speeds of light in the two mediums.
The formula for Snell's Law is:
n1 * sin(theta1) = n2 * sin(theta2)
Where:
- n1 is the refractive index of the first medium (air/vacuum in this case, which is approximately 1),
- theta1 is the angle of incidence,
- n2 is the refractive index of the second medium (plastic in this case),
- theta2 is the angle of refraction.
We are given:
- n1 = 1
- theta1 = 21°
- n2 (refractive index of plastic) is not provided.
Since the speed of light in the plastic is given, we can use it to find the refractive index using the formula:
speed of light in vacuum / speed of light in medium = n1 / n2
Let's calculate the refractive index of plastic first.
speed of light in vacuum = speed of light in vacuum = 3 * 10^8 m/s (approximately)
refractive index of plastic (n2) = speed of light in vacuum / speed of light in plastic
= 3 * 10^8 m/s / (1.95 * 10^8 m/s) (using the given speed of light in plastic)
= 3/1.95
Now, substitute the values into Snell's Law to find theta2:
n1 * sin(theta1) = n2 * sin(theta2)
1 * sin(21°) = (3/1.95) * sin(theta2)
sin(theta2) = (1 * sin(21°)) / (3/1.95)
sin(theta2) = sin(21°) / (3/1.95)
Using a calculator, find the sin(theta2) by dividing sin(21°) by (3/1.95).
[Calculate sin(21°), then divide it by (3/1.95)]
Once you have the value of sin(theta2), use the inverse sine function (sin^-1) to find the angle theta2.
theta2 = sin^-1[(1 * sin(21°)) / (3/1.95)]
Calculate theta2 using the value of sin(theta2) obtained from the previous step.
Finally, round the value of theta2 to the nearest degree to obtain the angle at which the ray is refracted.