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Understanding Physics Topics is essential for solving complex problems in many fields, including engineering and medicine.
What are Paraxial Rays? What do you Mean by Aperture?
Mirrors used in torches, headlights or in view finders of vehicles are curved mirrors. Curved mirrors may be spherical, cylindrical or parabolic. In this chapter however, we shall restrict our discussion only to spherical mirrors.
The laws of regular reflection are equally applicable to curved surfaces as well. But in this case the position of the image and its size differ widely, as we shall see later.
Spherical Mirror
A spherical mirror is a part of a hollow sphere or a spherical surface. There are two types of spherical mirrors—
- concave mirror and
- convex mirror.
When the inner surface of a spherical mirror acts as a reflector, it is a concave mirror [Fig. (a)]. When the outer surface of a spherical mirror acts as a reflector, it is a convex mirror [Fig.(b)].
Some Related Terms
Pole: The centre of the spherical reflecting surface is called the pole of the mirror. In Fig. (a) and (b) O is the pole.
Centre of curvature: The centre of the sphere of which the spherical mirror is a part is called the centre of curvature of the mirror. In Fig. (a) and (b), C is the centre of curvature of the mirror MOM’. Obviously, the centre of curvature of the concave mirror is in front of the reflecting surface while in case of the convex mirror it is behind the reflecting surface.
Radius of curvature: It is the radius of that sphere of which the mirror is a part. In Fig(a) and (b), OC is the radius of curvature.
Principal axis: The line passing through the centre of curvature and the pole of the mirror is called the principal axis of the mirror. In Fig. (a) and (b), XX’ is the principal axis.
Aperture: The line joining the two extreme points on the periphery of a spherical, mirror is called the aperture of the mirror. The angle subtended at the centre of curvature by the line is called the angular aperture of the mirror. In Fig. (a) and (b), the line MM’ is the aperture of the spherical mirror and ∠MCM’ is its angular aperture. The discussion in this chapter will be confined to spherical mirrors of small apertures not exceeding 10°, although for the sake of clarity, illustrative diagrams will indicate larger apertures.
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Paraxial and non-paraxial or marginal rays:
Paraxial rays: Rays which are incident very close to the pole and form a very small angle with the principal axis of a spherical mirror are called paraxial rays [Fig.].
Non-paraxial or marginal rays: Rays which are incident very far away from the pole or near the margin of a spherical mirror and form a comparatively large angle with the principal axis are called non-paraxial or marginal rays [Fig.].
All rays incident on a spherical mirror whose aperture is negligibly small compared to its radius of curvature, are considered to be paraxial rays. For further discussions we will assume all spherical mirrors to be of a small aperture and a comparatively large radius of curvature.
Principal focus: If rays of light parallel to the principal axis are incident on a spherical mirror, the rays after reflection from the mirror converge to a point on the principal axis in case of a concave mirror and appear to diverge from a point on the principal axis behind the mirror in the case of a convex mirror, this point is called the principal focus or simply focus of the mirror. In Fig.(a) and Fig.(b) F is the principal focus of
the concave mirror and convex mirror respectively. So, focus of concave mirror is real and that for the convex mirror is virtual.
It can alternatively be defined as the point on the principal axis of a spherical mirror at which the image of an object placed at infinity is formed.
According to the principle of reversibility of light rays it can be said—
i) The rays diverging from the principal focus of a concave mirror proceed parallel to the principal axis after reflection from the mirror [Fig.(a)].
ii) The rays appearing to converge to the principal focus of a convex mirror proceed parallel to the principal axis after reflection from the mirror [Fig.(b)].
Focal length: The distance between the principal focus and the pole of the mirror is called the focal length of the mirror. In Fig.(a) and (b) OF is the focal length.
Focal length of a spherical mirror does not depend on the colour of incident light.
Focal plane and secondary focus: Focal plane of a spherical mirror is the imaginary plane passing through the principal focus at right angles to the principal axis of the mirror [Fig.(a), (b)]
If a parallel beam of rays is incident on a spherical mirror such that it is inclined to the principal axis then, after reflection the reflected rays converge to the point F1 [Fig.(a)] on the focal plane in case of a concave mirror and appear to diverge from the point F1 [Fig.(b)] on the focal plane in case of a convex mirror. The point F1 is called the secondary focus of the spherical mirror. The principal focus is a fixed point on the principal axis. But the secondary focus is not a fixed point. If the angle of inclination of the parallel rays with the principal axis is changed, the position of the secondary focus also changes. But a secondary focus always lies on the focal plane.
Relation between Focal Length and Radius of Curvature
1. In case of concave mirror: Let MOM’ be a concave mirror of small aperture [Fig.]. C, F, O are the centre of curvature, focus, and pole of the M mirror respectively. Ray PQ is parallel to the principal axis, hence passes through F after reflection, CQ being the radius of curvature is perpendicular to the mirror at Q.
∴ ∠PQC = ∠FQC [angle incidence = angle of reflection]
Since, PQ and CO are parallel,
∴ ∠PQC = ∠QCF [alternate angles]
∴ ∠FQC = ∠QCF,i.e., ΔQCF isanisoscelesthangle.
Hence, FQ = FC
Since, the aperture of the mirror is very small, Q and O are very close to each other. So, FQ = FO.
∴ FO = FC or, FO = \(\frac{1}{2} c\)OC
i.e., f = \(\frac{r}{2}\), where f is the focal length and r the radius of curvature.
2. In case of convex mirror: Let MOM’1 be a convex mirror of small aperture [Fig.]. C, F and O are the centre of curvature, focus and pole of the mirror respectively. OC is the radius of curvature. A ray PQ parallel to the principal axis is incident at Q of the mirror. After reflection, the ray QR appears to come from F. The points C and Q are joined and is extended to N. Since CQ = CO = radius of curvature of the mirror, QN is normal at incidence point Q on the mirror.
∴ ∠PQN = ∠RQN
[angle of incidence = angle of reflection]
But ∠RQN = ∠CQF [vertically opposite]
∴ ∠PQN = ∠CQF
Since PQ and OC are parallel.
∴ ∠PQN = ∠FCQ [corresponding angles]
∴ ∠FCQ = ∠CQF
Hence, FQ = FC
Since, the aperture of the mirror is small, Q and O are very close to each other. So, FQ = FO.
∴ FO = FC or FO = \(\frac{1}{2}\)OC
i.e., f = \(\frac{r}{2}\)
Hence the focal length of a spherical mirror having small aperture is equal to half of its radius of curvature.
Reflecting power: Reflecting power of a spherical mirror, D = \(\frac{1}{f}\) = \(\frac{2}{r}\). As both focal length and radius of curvature of a plane mirror are infinite, so power of a plane mirror is zero.