Question

17 through 29 22 23, 29 More mirrors. Object $\mathit{O}$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 $\mathit{m}$. (All distances are in centimeters.) It also refers to whether (g) the image is real localid="1662999140986" $\left(R\right)$or virtual $\mathbf{\left(}\mathit{V}\mathbf{\right)}$, (h) inverted $\mathbf{\left(}\mathit{I}\mathbf{\right)}$or non-inverted from $\mathbf{\left(}\mathit{N}\mathit{I}\mathbf{\right)}$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 concave
2. Focal length is 20 cm
3. The radius of curvature is 40 cm
4. The object distance is +60 cm
5. The image distance is +30 cm
6. The magnification ratio is -0.50
7. The image is real.
8. Inverted.
9. The position of the image is on the same side.

Step-by-step solution

Step 1: Given

$f=+20cm$.

$p=+60cm$.

Determining the concept

Here, the focal distance and object distance is given in the problem. Using that the radius of curvature and image distance can be found. Then, by using image distance and object distance, the magnification ratio 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:

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

Determining the type of mirror

a. Type of mirror

It is given that the image is on the same side as the object. Thus, the image is real. This implies that the mirror is concave.

Determining the Focal length

b. Focal length

As the mirror is concave, from the table 34-4, the focal distance is,

$f=+20cm$.

Determining the Radius of curvature

c. Radius of curvature

Use the following formula to find the radius of curvature,

$$\begin{gathered} r=2\times f \\ r=2\times 20 \\ r=40cm \end{gathered}$$

Determining the Object distance.

d. Object distance

The object distance is $p=+60cm$, as given in the table.

 Determining the Image distance

e. Image distance

It is known,

$$\begin{gathered} \frac{1}{f}=\frac{1}{i}+\frac{1}{p} \\ \frac{1}{20}=\frac{1}{i}+\frac{1}{60} \\ \frac{1}{i}=\frac{1}{20}-\frac{1}{60}=\frac{1}{30} \\ i=30cm \end{gathered}$$

The image distance is, $i=+30cm$.

 Determining the lateral magnification

f. Magnification ratio

The magnification ratio is given as,

$$\begin{gathered} M=\frac{-i}{p} \\ M=\text{-}\frac{\text{30}}{\text{60}} \\ M=\text{- 0.50} \end{gathered}$$

 Determining whether the image is virtual or real

g. Determine whether the image is virtual or real

Since the image distance is positive, the image is real.

Determining whether the image is inverted or not inverted

h. Whether inverted or not inverted

As the magnification is negative, the image is inverted.

Determining the position of the image

i. Position of image

For spherical mirrors, the real image is formed on the same side as the object. Since the image is real here, so it is on the same side of the mirror as the object $O$.