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 non-inverted (NI) from O, and (i) on the same side of the mirror as the 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. The type of mirror is convex.
2. The focal length is, $-20 \text{cm}$.
3. The radius of curvature is,$-40 \text{cm}$ .
4. The object distance is,$+20 \text{cm}$.
5. The image distance is,$-10 \text{cm}$.
6. The magnification ratio is,$0.50$.
7. The image is virtual.
8. Non--inverted.
9. The position of the image is the opposite side.

Step-by-step solution

Step 1: Identification of the given data

The radius of curvature is,$-40 \text{cm}$ .

The image distance is, $-10 \text{cm}$.

Determining the concept

The radius of curvature is given, and from that, find the focal length and decide whether the mirror is convex or concave. Then, using the basic formulas, find the required values.

Formulas are as follows:

$\mathit{r}\mathbf{=}\mathbf{2}\mathit{f}$

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

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

Where, $m$is the magnification,$p$is the pole,$f$is the focal length.

(a) Determining the type of mirror

Type of the mirror-

The mirror is a convex type because the radius of curvature is negative which means the focal length is negative.

Therefore, the type of mirror is convex.

(b) Determining the focal length

The focal length-

$r=2\times f$

Substitute the values for the above equation

$\begin{array}{rcl}-40 \text{cm}& =& 2\times f\\ f& =& -20 \text{cm}\end{array}$

Therefore, the focal length of the image has been obtained.

(c) Determining the radius of curvature

The radius of curvature-

The radius of curvature is $-40\text{cm}$as given in the problem.

Therefore, the radius of curvature is $-40\text{cm}$.

(d) Determining the object distance.

Formula to find object distance is,

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

Substitute all the value in the above equation.

$\begin{array}{rcl}\frac{1}{-20 \text{cm}}& =& \frac{1}{-10 \text{cm}}+\frac{1}{p}\\ p& =& 20 \text{cm}\end{array}$

Therefore, the object distance is,$+20\text{cm}$

 Step 7: (e) Determining the image distance 

The image distance-$i$

The image distance isrole="math" localid="1663077525559" $i=-10\text{cm}$ as given in the problem.

Therefore, the image distance $i$is,-10 cm.

(f) Determining themagnification ratio

The magnification ratio,

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

Substitute all the value in the above equation.

$\begin{array}{rcl}M& =& \frac{10 \text{cm}}{20 \text{cm}}\\ & =& 0.50\end{array}$

Therefore, the magnification ratio is, 0.50

(g) Determining whether the image is virtual or real

Determine whether the image is virtual or real-

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

Therefore, the image is virtual.

(h) Determining whether inverted or non-inverted

Whether inverted or non-inverted-

The image is non-inverted because the magnification of the mirror is positive.

Therefore, the image is non--inverted.

(i) Determining the position of the image

Position of the image-

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

Therefore, the position of the image is on the opposite side.

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