Combine φ=4r−2i with Snell's Law to express the angle φ as a function of i and n only.
(a) Evaluate φ(i,n) for the following values (give your answers in degrees):
i=3∘, n=1.35
2*(theta1)-4*arcsin((sin(theta1))/n)
How to put values
To combine φ=4r−2i with Snell's Law, we need to first express the angle φ in terms of i and n only.
Snell's Law relates the angles of incidence (i) and refraction (r) to the refractive indices (n) of two media:
n1 sin(i) = n2 sin(r)
In this case, we want to express the angle φ (which is related to r) in terms of i and n.
Let's start by solving Snell's Law for r:
sin(r) = (n1 / n2) * sin(i)
Now, we need to find the relation between φ and r. From the given equation φ = 4r - 2i, we can rearrange it to solve for r:
r = (φ + 2i) / 4
Substituting this into the equation for sin(r), we have:
sin((φ + 2i) / 4) = (n1 / n2) * sin(i)
Now, let's solve for φ by isolating it on one side of the equation. To do this, we'll follow these steps:
1. Take the inverse sine (sin^(-1)) of both sides of the equation.
2. Multiply both sides by 4 to remove the division.
3. Subtract 2i from both sides of the equation.
The resulting equation is:
φ = 4 * sin^(-1)((n1 / n2) * sin(i)) - 2i
Now, to evaluate φ(i,n) for the given values i=3∘ and n=1.35, we can substitute these values into the equation:
φ = 4 * sin^(-1)((n1 / n2) * sin(i)) - 2i
= 4 * sin^(-1)((1 / 1.35) * sin(3∘)) - 2 * 3∘
Using a scientific calculator, we can evaluate this expression to find the value of φ.