The relationship between the angle of minimum deviation (δm), the refractive index (n) of the prism, and the apex angle (A) of the prism is given by the formula:
sin((A + δm)/2) = n * sin(A/2)
Given that δm = A and n = 1.5, we can substitute these values into the formula:
sin((A + A)/2) = 1.5 * sin(A/2)
sin(3A/2) = 1.5 * sin(A/2)
Using the double angle identity for sine, sin(2θ) = 2sin(θ)cos(θ), we can rewrite the equation as:
2sin(A)cos(A) = 1.5 * 2sin(A/2)cos(A/2)
Simplifying further gives:
sin(2A) = 1.5 * sin(A)
Expanding sin(2A) using the double angle identity sin(2θ) = 2sin(θ)cos(θ):
2sin(A)cos(A) = 1.5 * sin(A)
Solving the equation gives:
2cos(A) = 1.5
cos(A) = 0.75
A = cos⁻¹(0.75)
A ≈ 41.41 degrees
Therefore, the apex angle of the prism is approximately 41.41 degrees.
