Derive the equation for lateral magnification in spherical mirrors.

Explanation

(i) Consider an extended object OO' is kept perpendicular to the principal axis to the left of the single spherical surface. (ii) The image formed on the other side of the surface is II'. Consider a ray from O' in the first medium towards C in the second medium. (iii) As this ray is incident normal to the spherical surface, it goes undeviated in the second medium. The position of image may be located using the Equation (2). (iv) The lateral or transverse magnification m is defined as the ratio of height of the image to the height of the object.
` m = (II')/(OO') `
(v) From the two similar triangles `Delta`COO' and `Delta`CII', we can write,
` (II')/(OO') = (CI)/(CO)`
From the geometry,
` (CI)/(CO) = (PI - PC)/(PC + PO)`
Hence
` m = (II')/(OO') = (PI - PC)/( PC + PO)` .... (1)
Applying sign conventions in the above equation (1),
`II' = h_2 , OO' - h_1 , PI = +v, `
PC = +R, PO = u
(vii) Where, h, is the height of the object and `h_2` is the height of the image.
` m = (-h_2)/(h_1) = (v - R)/(R + (-u)) , m = h_2/h_1 = - ((v-R)/(R-u))`
After rearranging,
` m = h_2/h_1 = - (R - v)/(R-u) " " ....(2)`
(viii) We can also arrive at an equation for lateral magnification involving the refractive indices of the two media. (ix) Let us consider the equation for single spherical surface as,
` (n_2)/(v) - (n_1)/(u) = ((n_2-n_1))/(R)`
Further simplifying,
` (n_2u - n_1 v)/( vu) = ((n_2 - n_1))/(R)`
Rewriting for R, ` R = ((n_2 - n_1)vu)/(n_2 u - n_1 v)` ...(3 )
`R - v = (n_1v (v-u) )/(n_2 u - n_1 v) " " ....(4)`
(x) Substituting equations (3) and (4) in equation (2) we get the equation for lateral magnification as,
` m = h_2/h_1 = (n_1 v)/(n_2 v)`