Question 370: Ray Optics

Question

A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length $$f,$$ as shown in the figure. Which of the figures shown in the four options qualitatively represent(s)  the shape of the image of the bent wire? (These figures are not to scale.)

Accepted Answer

Consider the point F, the focus of the concave mirror. All rays emanating from F will be reflected back parallel to the principal axis PF. Thus, virtual image F' of the point F is formed at $$\infty$$.
Consider the point L at an object distance u = $$-$$ | f | / 2. For convenience, we use symbol $$\widehat f$$ for | f | i.e., $$\widehat f$$ = | f |. Use mirror formula, 1/v + 1/u = 1/f, with f = $$-$$ $$\widehat f$$ to get the image distance v = $$\widehat f$$. Thus, the virtual image L' of the point L is formed on the left side of the mirror at a distance $$\widehat f$$.
The image of a point on LF will lie on the line L'F'. Now, consider the point M. Its object distance is u = $$-$$ $$\widehat f$$/2. Thus, the virtual image M' of the point M is also formed at a distance $$\widehat f$$ from the mirror. The distance of M' above the principal axis is given by h
M'
= $$-$$ (v/u) (LM) = $$-$$ ($$-$$2) ($$\widehat f$$/2) = $$\widehat f$$ (magnification by a spherical mirror).
Now consider a point N on the line segment FM. Let N
1
be the projection of N on the principal axis PF such that FN
1
= x. The object distance is u = $$-$$ ($$\widehat f$$ $$-$$ x). The mirror formula gives the image distance v = $$\widehat f$$ ($$\widehat f$$ $$-$$ x) / x. The distance of the image N' above the principal axis is given by h
N'
= $$-$$ (v/u) (N
1
N) = ($$\widehat f$$/x) (x) = $$\widehat f$$ (it is independent of x). Since, N is an arbitrary point on FM, the image of all points on FM will be formed at a distance $$\widehat f$$ above the optic axis.

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Ray Optics

A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length $f,$ as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image of the bent wire? (These figures are not to scale.)

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