Question

17 through 29 22 23, 29 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to (a) the type of mirror, (b) the focal distance $\mathit{f}$, (c) the radius of curvature $\mathit{r}$ , (d) the object distance $\mathit{p}$, (e) the image distance $\mathit{i}$, and (f) the lateral magnification m. (All distances are in centimeters.) It also refers to whether (g) the image is real (R) or virtual (V), (h) inverted (I) or noninverted (NI) from O, and (i) on the same side of the mirror as object O or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.

Short answer

1. Type of mirror is convex.
2. Focal length is $-30\text{cm}$.
3. Radius of curvature is role="math" localid="1663139922964" $-60\text{cm}$.
4. Object distance is $+120\text{cm}$.
5. Image distance is role="math" localid="1663140008408" $-24\text{cm}$
6. Magnification ratio is$+0.20$
7. Image is virtual.
8. Non Inverted.
9. Position of image is opposite side

Step-by-step solution

Step 1: Given

$M=+0.20\text{cm}$

$f=30\text{cm}$

Determining the concept

From mirror properties, the type of mirror can be decide. Then to find focal length, use magnification ratio and formula for focal length. Then, by using focal length the object distance and image distance can be found. Then, from image distance it can be identify which type of image it is.

The formula is as follows:

$\begin{array}{rcl}\mathit{r}& \mathbf{=}& \mathbf{2}\mathit{f}\\ \frac{\mathbf{1}}{\mathbf{f}}& \mathbf{=}& \frac{\mathbf{1}}{\mathbf{i}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{p}}\\ \mathit{m}& \mathbf{=}& \frac{\mathbf{-}\mathbf{i}}{\mathbf{p}}\end{array}$

where,

r= radius of curvature,

f= focal length,

p= object distance from mirror,

i= image distance.

(a) Determining the type of mirror

Type of mirror

Magnification ratio is as follows,

$m=\frac{-i}{p}$

The positive value of $m$is given, which means image distance must be negative because object distance is always positive.

As image distance is negative, it means the image is virtual and a virtual image is possible only for convex mirrors.

(b) Determining the Focal length

Focal length

The focal length is given as$f=30\text{cm}$ .

As the mirror is convex, the focal length should be negative.

So,$f=-30\text{cm}$.

(c) Determining the Radius of curvature

Radius of curvature

Use the following formula to find the radius of curvature,

$\begin{array}{rcl}r& =& 2f\\ r& =& 2\left(30\right)\\ r& =& 60\text{cm}\end{array}$

Step 6:(d) Determining the Object distance.

Object distance

It is known,

$\begin{array}{rcl}m& =& \frac{-i}{p}\\ i& =& -mp\\ \frac{\text{1}}{f}& =& \frac{\text{1}}{i}+\frac{\text{1}}{p}\\ \frac{\text{1}}{f}& =& \frac{\text{1}}{-mp}+\frac{\text{1}}{p}\end{array}$

role="math" localid="1663141272017" $\begin{array}{rcl}\frac{1}{f}& =& \frac{1}{p}\left(1-\frac{1}{m}\right)\\ p& =& f\left(1-\frac{1}{m}\right)\\ p& =& -30\left(1-\frac{1}{0.2}\right)\\ p& =& 120\text{cm}\end{array}$

So object distance is 120 cm.

(e) Determining the Image distance

Image distance

Image distance is as follows

$\begin{array}{rcl}i& =& -mp\\ i& =& -0.2\times 120\\ i& =& -24\text{cm}\end{array}$

(f) Determining the Magnification ratio

Magnification ratio

Magnification ratio is$M$ .

Magnification ratio is +0.20 as given in the table.

(g) Determining whether the image is virtual or real

Determine whether image is virtual or real

Since image distance is negative, image is virtual.

(h) Determining whether the image is inverted or not inverted

Whether inverted or no inverted

As magnification is positive so image is non- inverted.

(i) Determining the position of the image

An image is formed on opposite side of mirror from the object.

The basic formulas to find radius of curvature, image distance and magnification ratio can be used; then from that, it is decided whether the image is virtual or real, on same side or opposite side.