Question

17 through 29 22 23, 29 More mirrors. Object 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 $\mathbf{\text{f}}$, (c) the radius of curvature $\mathbf{\text{r}}$, (d) the object distance $\mathbf{\text{p}}$, (e) the image distance $\mathbf{\text{i}}$, and (f) the lateral magnification m. (All distances are in centimeters.) It also refers to whether (g) the image is real $\left(R\right)$ or virtual$\left(V\right)$, (h) inverted (I) or noninverted $\left(\mathrm{NI}\right)$from$\mathit{O}$, and (i) on the same side of the mirror as the object $\mathit{O}$or the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.

Short answer

1. The type of mirror is convex.
2. Focal length is $-20\mathrm{cm}$.
3. The radius of curvature is $-40\mathrm{cm}$.
4. The object distance is $+5\mathrm{cm}$.
5. The image distance is$-40\mathrm{cm}$.
6. The magnification ratio is $+080$.
7. The image is virtual.
8. Non-Inverted.
9. The position of the image is on the opposite side.

Step-by-step solution

Given

$r=40\mathrm{cm}$

$i=4\mathrm{cm}$

Determining the concept

Here, the radius of curvature and image distance is given in the problem. It is given that the mirror is convex. Using that the focal distance, object distance, and magnification can be found. Using all these values, it can be decided if the image is virtual or real and the position of the image.

The formula is as follows:

$\begin{array}{l}\mathbf{r}\mathbf{=}\mathbf{2}\mathbf{f}\\ \frac{\mathbf{1}}{\mathbf{1}}\mathbf{=}\frac{\mathbf{i}}{\mathbf{+}}\frac{\mathbf{1}}{\mathbf{p}}\\ \mathbf{m}\frac{\mathbf{i}}{\mathbf{p}}\end{array}$

Where,

$\mathit{r}$= radius of curvature,

$\mathit{f}$= focal length,

$\mathit{p}$=object distance from the mirror,

$\mathit{i}$= image distance.

Determining the type of mirror

1. Type of mirror

It is given in the table $34-4$, that the type of mirror is convex.

Determining the Focal length

b. Focal length

Focal length $f$is calculated as,

$r=2\times f$,

As the mirror is convex, the radius of curvature must be negative, so$r=-40\mathrm{cm}$,

$\begin{array}{l}40=2\times f\\ f=-0\mathrm{cm}\end{array}$

Determining the Radius of curvature

c. Radius of curvature

As the mirror is convex, the radius of curvature must be negative

$r=-40\mathrm{cm}$

Determining the Object distance.

d. Object distance

As the mirror is convex, the image distance must be negative.

So, $i=-40\mathrm{cm}$,

The object distance is calculated by,

$\frac{1}{f}=\frac{1}{i}+\frac{1}{p}$

$$\begin{gathered} \frac{\text{1}}{\text{- 20}}\text{=}\frac{\text{1}}{\text{- 4}}+\frac{1}{p} \\ \frac{1}{p}=\frac{\text{1}}{\text{- 20}}\text{+}\frac{\text{1}}{\text{4}}\text{=}\frac{\text{1}}{\text{5}} \\ p=\text{+5.0cm} \end{gathered}$$

Determining the Image distance i

e. Image distance $\mathit{i}$

As the mirror is convex, the image distance must be negative.

$i=-40\mathrm{cm}$

Determining the lateral magnification

f. Magnification ratio

The magnification ratio is given as,

$$\begin{gathered} M=\frac{-i}{p} \\ M=\frac{\text{4.0}}{\text{5.0}} \\ M=\text{+ 0.80} \end{gathered}$$

Determining whether the image is virtual or real

g.Determine whether the image is virtual or real

Since the image distance is negative, the image is virtual.

Determining whether the image is inverted or not inverted

h. Whether inverted or not inverted

As the magnification is positive, so the image is non-inverted.

Determining the position of the image

I. Position of image

For spherical mirrors, virtual images form on the opposite side of the object. Since the image is virtual here, so it is formed on the opposite side of the mirror as the object $O$.

Here the basic formulae can be used to find the radius of curvature, image distance, and magnification ratio. Using that it can be decided that the image is virtual or real and on the opposite side of the same side as the object.