Final answer:
The equation for refraction at a concave spherical interface is derived using Snell's law, considering a negative radius of curvature for a concave surface. The final equation relates the object distance, image distance, and the radii of curvature with the refractive indices of the two media.
Step-by-step explanation:
Derivation of the Spherical Interface Equation for Refraction at a Concave Surface
To derive the equation for refraction at a concave spherical interface, we consider the same geometric setup as for a convex surface but with a concave surface. We let R be the radius of curvature which will be negative for a concave surface, n1 be the refractive index of the medium containing the object, and n2 be the refractive index of the medium forming the concave surface. The derivation follows Snell's law, which relates the angles of incidence and refraction to the refractive indices of the two media.
The equation we arrive at is:
n1/(object distance) + n2/(image distance) = (n2-n1)/(-R)
Where the object distance and image distance are measured from the vertex of the surface along the principal axis, and the negative sign in the radius of curvature indicates that the surface is concave. The magnification can then be calculated using the formula magnification = - (n2 * image distance) / (n1 * object distance).