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Two rays are incident on a spherical mirror of radius R = 5 cm parallel to its optical axis at distances h1 = 0.5 cm and h2 = 3 cm.

Determine the distance Δx between the points at which these rays intersect the optical axis after being reflected at the mirror.

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Let 0 be the centre of the spherical surface of the mirror, ABC the ray incident at a distance BE from the mirror axis, and OB = R (Fig. 244).

From the right triangle OBE, we find that sin α = h/R. The triangle OBC is isosceles since ABO = OBC according to the law of reflection, and BOC = ABO as alternate-interior angles. Hence OD = DB = R/2. From the triangle ODC, we obtain

(C is the point of intersection of the ray reflected by the mirror and the optical axis).

For a ray propagating at a distance h1, the distance x1 ≈ R/2, with an error of about 0.5% since h12 << R2. For a ray propagating at a distance h2, the distance x2 = 3.125 cm. Finally, we obtain

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