Question

34.13 Dental Mirror. A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she wants an erect image with a magnification of 2.00 when the mirror is 1.25cm from a tooth. (Treat this problem as though the object and image lie along a straight line.) (a) What kind of mirror (concave or convex) is needed? Use a ray diagram to decide, without performing any calculations. (b) What must be the focal length and radius of curvature of this mirror? (c) Draw a principal-ray diagram to check your answer in part (b).

Short answer

a) The dentist must use concave mirror.

b) The focal length is 2.5cm and radius of curvature is 5cm.

c) The principal ray diagram is shown below

Step-by-step solution

Lateral magnification and object image relation for spherical mirror

Lateral magnification for spherical message is given by:

$\mathbf{m}\mathbf{=}\frac{\mathbf{-}\mathbf{v}}{\mathbf{u}}\mathbf{=}\frac{{\mathbf{h}}_{\mathbf{i}}}{{\mathbf{h}}_{\mathbf{o}}}$

where, m is the magnification,

u is object distance from the vertex of the mirror,

u is (+) when the object is in front of the reflecting surface of mirror,

v is image distance from the vertex of the mirror,

v is (+) when the image is in front of the reflecting surface of mirror,

is the height of the image, is the height of the object,

m is (+) when the image is erect and (-) when the image is inverted

Object image relation for spherical mirror is given by:

$\frac{\mathbf{1}}{\mathbf{u}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{v}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{R}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{f}}$

Where, u= object distance from the vertex of the mirror,

u is (+) when the object is in front of the reflecting surface of mirror,

v is image distance from the vertex of the mirror,

v is (+) when the image is in front of the reflecting surface of mirror,

R is the radius of curvature of the mirror, f= focal length,

R is (+) in concave mirror and (-) in convex mirror,

u and v are negative if the case is opposite to what is mentioned above

Remember that the focal length of a mirror depends only on the curvature of the mirror and not on the material from which the mirror is made because the formation of the image results from rays reflected from the surface of the material.

Apply: The above rule can be applied on either concave and convex mirrors or even plane mirror when we consider the radius to be infinity. The equation gives one the exact location of an image by knowing the radius or the focal length of a mirror and the location of the object. In addition, the sign for the distance of the image can be negative if the image in the opposite direction of the outgoing rays which means the image is not real image but is virtual one and is located inside the mirror.

Determine the kind of mirror needed

a) Only a concave mirror can produce an upright and magnified image. Convex mirror can produce upright image, but it is diminished.

Hence the dentist must use concave mirror to get magnification 2.

Determine the focal length and radius of curvature of mirror

b) By lateral magnification,

$$\begin{gathered} m=\frac{-v}{u} \\ v=-mu \\ v=-2.\left(1.25\right)=-2.5cm \end{gathered}$$

Now, using object image relation for spherical mirror,

Now,$\frac{1}{\mathrm{u}}+\frac{1}{\mathrm{v}}=\frac{1}{\mathrm{f}}$

Solve for f to get

$f=\frac{uv}{u+v}$

Substitute$v=-2.5cm$ and$u=1.25cm$

$f=\frac{1.25.\left(-2.5\right)}{1.25-2.5}=2.5cm$

Positive sign confirms the mirror is concave.

$R=2f=2.2.5=5cm$

Hence, the focal length is 2.5cm and radius of curvature is 5cm.

Draw a principal-ray diagram

The figure below shows the principal ray diagram where image is virtual and magnified by 2.