Light – Reflection and Refraction · Lesson 2 of 15
Spherical Mirrors
“Mirrors get curvy, and suddenly your face has opinions.”
• Distinguish concave and convex spherical mirrors. • Identify the pole, centre of curvature, radius, principal axis and aperture. • Explain the principal focus and focal length of each mirror. • Relate radius of curvature to focal length for a small-aperture mirror. • Estimate the focal length of a concave mirror safely.
A spoon changes the appearance of your face because its reflecting surfaces are curved. The most common curved mirrors are made from a small part of a spherical surface. Their geometry gives us fixed reference points that let us predict where reflected rays will travel.
A mirror whose reflecting surface forms part of the surface of a sphere.
Concave and Convex Mirrors
A concave mirror has its reflecting surface curved inward, facing the centre of the sphere. A convex mirror has its reflecting surface bulging outward. In drawings, the non-reflecting back is usually shaded. The inner surface of a spoon approximates a concave mirror, while its outer surface approximates a convex mirror.
Geometrical Terms
The centre of the reflecting surface of a spherical mirror, represented by P.
The centre of the sphere of which the reflecting surface is a part, represented by C.
The radius of that sphere, represented by R; therefore PC = R.
The straight line passing through P and C. It is normal to the mirror at the pole.
The effective diameter of the circular reflecting surface of the mirror.
For a concave mirror, C lies in front of the reflecting surface. For a convex mirror, C lies behind it. These positions are not arbitrary; they follow from the location of the centre of the parent sphere.
Principal Focus and Focal Length
Rays arriving parallel to the principal axis meet at a point after reflection from a concave mirror. That real meeting point is its principal focus F. A convex mirror spreads parallel rays, but their backward extensions appear to meet at F behind the mirror. The distance PF is the focal length f.
A smaller focal length means stronger convergence or divergence because the reflected rays change direction more sharply. The relation R = 2f is used for mirrors whose aperture is small compared with the radius of curvature.
Estimating Focal Length
Never look directly at the Sun or at its reflection in a mirror. Concentrated sunlight can permanently damage the eyes. Perform the activity only under responsible supervision.
Direct a concave mirror toward the Sun and hold a sheet of paper in front of it. Move the paper until the smallest, sharpest bright spot is obtained. This spot is a tiny real image of the distant Sun. The distance from P to the paper is an approximate focal length. Do not hold the spot on the paper for long, because concentrated sunlight can heat and burn it.
Problem
A spherical mirror has a radius of curvature of 36 cm. Find its focal-length magnitude.
- 1.Use the small-aperture relation R = 2f.
- 2.Rearrange: f = R/2.
- 3.Substitute R = 36 cm: f = 36/2 = 18 cm.
- 4.The magnitude is 18 cm. Its sign would depend on whether the mirror is concave or convex under the sign convention.
Quiz
Where does the centre of curvature of a concave mirror lie?
What is the distance PC called?
Parallel rays reflected by a convex mirror appear to come from which point?
A mirror has R = 50 cm. What is the magnitude of f?
Why can paper burn near the focus of a concave mirror facing the Sun?
Practice Problems
- Classify a mirror whose reflecting surface bulges outward. Solution: It is a convex mirror.
- A concave mirror has f = 12 cm. Find R. Solution: R = 2f = 24 cm.
- A spherical mirror has R = 80 cm. Locate F relative to P by magnitude. Solution: f = R/2 = 40 cm, so F is 40 cm from P on the principal axis.
- Explain why the principal axis is normal to the mirror at P. Solution: The line through the sphere's centre and the surface point P is a radius, and a radius is perpendicular to the tangent surface at that point.
- Describe a safe method for estimating the focal length of a concave mirror. Solution: Face the mirror toward the Sun without looking at it, move a paper screen to obtain the smallest sharp spot, measure the P-to-screen distance briefly, and remove the paper before it overheats.
Key Takeaways
• A spherical mirror is part of a spherical reflecting surface. • Concave mirrors curve inward; convex mirrors bulge outward. • P, F and C lie on the principal axis. • The focal length is the distance PF and the radius of curvature is PC. • For a small-aperture spherical mirror, R = 2f. • A concave mirror converges parallel rays, while a convex mirror makes them diverge.