A layer of oil (n = 1.45) floats on an unknown liquid. A ray of light originates in the oil and passes into the unknown liquid. The angle of incidence is 64.8 degrees, and the angle of refraction is 68.0 degrees. What is the index of refraction of the unknown liquid?

To find the index of refraction of the unknown liquid, we can 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 indices of refraction of the two media involved.

Mathematically, Snell's Law is expressed as:

n1 * sin(theta1) = n2 * sin(theta2)

Where:
n1 = index of refraction of the first medium (oil)
n2 = index of refraction of the second medium (unknown liquid)
theta1 = angle of incidence
theta2 = angle of refraction

Given:

n1 = 1.45 (index of refraction of oil)
theta1 = 64.8 degrees (angle of incidence)
theta2 = 68.0 degrees (angle of refraction)

Using Snell's Law, we can rearrange the equation to solve for n2:

n2 = (n1 * sin(theta1)) / sin(theta2)

Substituting the given values:

n2 = (1.45 * sin(64.8°)) / sin(68.0°)

Calculating this expression:

n2 ≈ 1.45 * 0.9001 / 0.9271

n2 ≈ 1.4252

Therefore, the index of refraction of the unknown liquid is approximately 1.4252.

To determine the index of refraction of the unknown liquid, we can apply Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the two media involved.

Snell's law states that:

n1 * sin(theta1) = n2 * sin(theta2)

Where:
n1 = index of refraction of the first medium (oil in this case)
n2 = index of refraction of the second medium (unknown liquid in this case)
theta1 = angle of incidence
theta2 = angle of refraction

In this scenario, we know:
n1 = 1.45 (index of refraction of the oil)
theta1 = 64.8 degrees (angle of incidence)
theta2 = 68.0 degrees (angle of refraction)

Let's plug in these values into Snell's law and solve for n2 (index of refraction of the unknown liquid):

1.45 * sin(64.8 degrees) = n2 * sin(68.0 degrees)

To solve for n2, first calculate the sin of both angles in the equation:

0.907 * 1.45 = n2 * 0.927

Rearrange the equation:

0.907 * 1.45 / 0.927 = n2

Calculating the right side of the equation:

1.314015 / 0.927 ≈ 1.418

Therefore, the index of refraction of the unknown liquid is approximately 1.418.

sini/sinr=n2/n1

n2=n1(sini/sinr)=1.45(sin64.8/sin68.0)=...