Given that the angle of minimum deviation (δm) is equal to the apex angle (A) of the prism, we have δm = A.
The relation between the angle of minimum deviation, the refractive index (n), and the apex angle of the prism is given by the formula:
sin((A + δm)/2) = n * sin(A/2)
Substitute δm = A into the formula:
sin((A + A)/2) = 1.5 * sin(A/2)
sin(3A/2) = 1.5 * sin(A/2)
Since sin(3A/2) = sin(A + A/2) = sin(A)cos(A/2) + cos(A)sin(A/2), and sin(2A) = 2sin(A)cos(A),
We have: 2sin(A)cos(A/2) + cos(A)sin(A/2) = 1.5 * 2sin(A)cos(A)
2sin(A)cos(A/2) + 1.5sin(A)cos(A/2) = 3sin(A)cos(A)
3.5sin(A)cos(A/2) = 3sin(A)cos(A)
Dividing by sin(A)cos(A) gives:
3.5/sin(A) = 3/cos(A)
3.5/tan(A) = 3
tan(A) = 3.5/3
tan(A) = 1.1667
A = tan⁻¹(1.1667)
A ≈ 49.25 degrees
Therefore, the apex angle of the prism is approximately 49.25 degrees.
