How To Find Radius Of Curvature Of A Mirror

Video How to find the radius of curvature of a mirror

Curved mirror

We can define two general types of spherical mirrors. If the reflecting surface is the outer surface of the sphere, the mirror is said to be . convex mirror. If the inner surface is a reflective surface, it is called concave mirrorSymmetry is one of the main highlights of many optical devices, including mirrors and lenses. The axis of symmetry of such optical elements is often called the principal or optical axis. For a spherical mirror, the optical axis passes through the center of curvature of the mirror and the top of the mirror, as shown in Figure (PageIndex {1}). Read: how to find the radius of curvature of a mirrorFigure (PageIndex {1}). Spherical mirrors are made by cutting out a piece of a sphere and silvering the inner or outer surface. A concave mirror has a silver coating on the inner surface (think “cave”) and a convex mirror has a silver coating on the outer surface. ). According to the law of reflection, these rays are reflected so that they converge at a point, called focus. Figure (PageIndex {2b}) shows a spherical mirror large relative to its radius of curvature. For this mirror, the reflected rays do not cross each other at the same point, so the mirror does not have a well-defined focus. This is called wrong request and results in a blurred image of an enlarged object. Figure (PageIndex {2c}) shows a spherical mirror that is small relative to its radius of curvature. This mirror is an approximation of a parabolic mirror, so incident rays parallel to the optical axis will be reflected to a well-defined focal point. The distance along the principal axis from the mirror to the focal point is called the focal length of the mirror.Figure (PageIndex {2}): (a) A parallel ray reflected from a parabolic mirror reaches a point called focal point F. (b) A parallel ray reflected from a large spherical mirror does not cross each other at a common point. (c) If a spherical mirror is small relative to its radius of curvature, it approximates the center of the parabolic mirror, so parallel rays essentially intersect at a common point. The distance along the optical axis from the mirror to the focal point is the focal length f of the mirror. A convex mirror also has a focal point, as shown in Figure (PageIndex {3}). The incident ray parallel to the principal axis is reflected from the mirror and appears to originate from a point (F) at focal length (f) behind the mirror. Thus, the focus is virtual because no real rays actually pass through it; they just seem to derive from it.Figure (PageIndex {3}): (a) Rays reflected by a convex mirror: Incident light rays parallel to the principal axis are reflected from a convex mirror and appear to originate from a well-defined focus at the focal point. distance f is on the opposite side of the mirror. The focus is virtual because no real rays pass through it. (b) Image of a virtual image formed by a convex mirror. (credit b: modified work by Jenny Downing) Further reading: how to do au when driving a motorcycle How is the focal length of a mirror related to the radius of curvature of the mirror? Figure (PageIndex {4}) shows a single ray reflected by a concave spherical mirror. The incident ray is parallel to the optical axis. The point where the reflected ray passes through the principal axis is the focal point. Note that all incident rays parallel to the optical axis are reflected through the focal point — we only show one ray for simplicity. We want to find out how the focal length (FP) (denoted (f)) is related to the radius of curvature of the mirror, (R), whose length is[R=CF+FP. label{eq31}]The law of reflection tells us that the angles (angle OXC) and (angle CXF) are the same, and since the incident ray is parallel to the optical axis, the angles (angle OXC) and (angle XCP) are also equal. Thus, the triangle (CXF) is an isosceles triangle with (CF = FX). If angle (θ) is small, then[sin θ≈ θ label{sma}]called “small angle approx“), Then (FX≈FP) or (CF≈FP). Inserting this value into the reference equation {eq31} for the radius (R), we get Read More: How to shorten the bicycle chain (Updated 2020) | Top Q&A[begin{align} R &=CF+FP nonumber [4pt] &= FP + FP not number [4pt] &= 2FPnonumber [4pt] &= 2f end {align}]In other words, in a small angle approximation, the focal length (f) of a concave mirror is half its radius of curvature, (R):[f=dfrac{R}{2}.]In this chapter, we assume that small angle approx (also called paraaxial approximation) is always valid. In this approximation, all rays are paraxial, which means they make a small angle to the optical axis and at a distance much smaller than the radius of curvature relative to the optical axis. In this case, their angle of reflection (θ) is small, so[sin θ≈ tan θ≈ θ. label{smallangle}]Read more: how to create a maze in minecraftFigure (PageIndex {4}): Reflection in a concave mirror. In the small-angle approximation, a ray parallel to the optical axis CP is reflected through the focal point F of the mirror.