Reflection at Curved Mirror

Derivation of the mirror formula (real image, concave mirror, small aperture)

Let an object AB be placed on the principal axis at distance $u$ from the pole P, in front of a concave mirror of focal length $f$ and radius of curvature $R=2f$ with centre of curvature C. Its real, inverted image A$'$B$'$ forms at distance $v$.

A ray from B reaching the pole P is reflected so that the angle of incidence equals the angle of reflection. The two triangles ABP and A$'$B$'$P are similar:

$\frac{A'B'}{AB}=\frac{PA'}{PA}=\frac vu\quad (1)$

A ray from B through C returns along itself; the triangles ABC and A$'$B$'$C are similar:

$\frac{A'B'}{AB}=\frac{CA'}{CA}=\frac{v-R}{R-u}\quad (2)$

Equating (1) and (2) (distances measured as positive in front of the mirror):

$\frac vu=\frac{R-v}{u-R}\;\Rightarrow\; v(u-R)=u(R-v)\;\Rightarrow\; 2uv=R(u+v)$

Dividing by $uvR$ gives $\dfrac2R=\dfrac1u+\dfrac1v$, and with $R=2f$:

$\boxed{\frac1f=\frac1u+\frac1v}$

Magnification $m=-\dfrac vu$ (real-is-positive convention gives the minus sign for an inverted image).

Nature of image for different object positions (concave mirror):

(c) Concave mirror

When the face is held closer than the focal length, a concave mirror forms an erect, virtual and magnified image, which shows fine detail for shaving and make-up. A plane mirror does not magnify and a convex mirror diminishes the image.

A concave mirror gives an image equal in size to the object only when the object is at the centre of curvature (the real, inverted image forms at the same point, $v=u$).

Here $u=v=40$ cm, so with the mirror formula,

$\frac1f=\frac1u+\frac1v=\frac1{40}+\frac1{40}=\frac1{20}$ $f=20\ \text{cm}$

(The radius of curvature is $40$ cm.) The focal length of the mirror is $20\ \text{cm}$.

(a) concave mirror

The field of view is the region of space that can be seen in the mirror. A convex mirror gives the widest field of view because it forms a diminished image of a large region. A plane mirror gives an intermediate field. A concave mirror, which magnifies when the object is near it, shows the smallest region, so its field of view is minimum.

For a real object, no. Whatever its position, a convex mirror forms only a virtual, erect and diminished image behind the mirror. With the mirror formula $\frac1v+\frac1u=\frac1f$, where $f>0$ for a convex mirror and $u<0$ for a real object, $\frac1v=\frac1f+\frac1{|u|}>0$, so $v$ is always positive (behind the mirror).

A real image is possible only for a virtual object: if a converging beam is incident on the mirror and would meet at a point behind the mirror closer than the focus ($u>0$, $u<f$), then $\frac1v=\frac1f-\frac1u<0$, so the reflected rays meet in front of the mirror and the image is real.

(a) A concave mirror reflects a beam of rays parallel to its principal axis so that they converge to meet at the principal focus. Hence it is called a converging mirror.

(b) A convex mirror is used as the rear-view (side) mirror of vehicles: it always gives an erect, diminished image and a wide field of view.

(d) convex mirror

A convex mirror always forms an erect, virtual, diminished image of a real object, whatever its position. A convex lens and a concave mirror can form real images, and a glass plate forms a virtual image of the same size.

(c) 2.4 ms$^{-1}$

The image in a plane mirror approaches the mirror at the same speed as the boy approaches it. So the boy and his image close in on each other at $1.2 + 1.2 = 2.4$ ms$^{-1}$.

(a) A concave mirror is a spherical mirror whose reflecting surface is curved inwards (the reflecting surface faces the centre of the sphere). It converges parallel rays of light.

Daily application: a shaving or make-up mirror. With the face between the pole and the focus, it forms an erect, magnified, virtual image.

(b)(i) The object is placed between the focus $F$ and the centre of curvature $C$. The image is then real, inverted and formed beyond $C$, with image distance greater than object distance. Since magnification $m=\dfrac{v}{u}>1$, the image is larger than the object.

(b)(ii) Using real-is-positive distances: $v=5\ \text{m}$ and $m=\dfrac{v}{u}=5$, so

$u=\frac{v}{5}=1\ \text{m}$

$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}=\frac{1}{1}+\frac{1}{5}=\frac{6}{5}$

$f=\frac{5}{6}\approx0.83\ \text{m}$

The focal length of the mirror is about $0.83\ \text{m}$.

Concave mirror: A concave mirror is a spherical mirror whose reflecting surface is curved inwards (the reflecting surface faces the centre of the sphere of which the mirror is a part). Its reflecting surface is on the same side as the centre of curvature.

Daily application: It is used as a shaving mirror — when the face is placed within the focus, the concave mirror forms a magnified, erect and virtual image, making shaving easier.

Because the object is placed between the principal focus (F) and the center of curvature (C) of the concave mirror. In this position, the reflected rays converge to form a real, inverted and magnified image on the screen. Since the image distance is greater than the object distance, the magnification $m = -v/u$ has magnitude greater than 1, so the image is larger than the object.

The image is real and formed on the screen at $v=5\ \text{m}$, and it is $5$ times the size of the object.

$m=\frac{v}{u}=5\ \Rightarrow\ u=\frac{5}{5}=1\ \text{m}$

Mirror formula (real-is-positive distances):

$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}=\frac{1}{1}+\frac{1}{5}=\frac{6}{5}$

$f=\frac{5}{6}\approx0.83\ \text{m}$

The focal length of the concave mirror is about $0.83\ \text{m}$ (about $83\ \text{cm}$).