Chapter 17.2, Problem 27SSC

To determine

To draw: A ray diagram that will show the positionand height of the image.

To verify: The answer using mirror and magnification equations.

Answer to Problem 27SSC

The ray diagram is shown in Figure 1.

The position of the image is $-6.46\text{ cm}$ .

The height of the image is $1.8\text{ cm}$ .

Explanation of Solution

Given:

The height of the object is $h_{\text{o}}=4.0\text{ cm}$ .

The position of the object is $d_{\text{o}}=14.0\text{ cm}$ .

The focal length is $f=-12.0\text{ cm}$

Formula used:

The expression for the mirror equation is,

  $\frac{1}{d_{\text{o}}}+\frac{1}{d_{\text{i}}}=\frac{1}{f}$

Here, $d_{\text{o}}$ is the position of the object, $d_{\text{i}}$ is the position of the image, and $f$ is the focal length.

The expression for the magnification equation is,

  $m=\frac{h_{\text{i}}}{h_{\text{o}}}=-\frac{d_{\text{i}}}{d_{\text{o}}}$

Here, $h_{\text{i}}$ is the height of the image and $h_{\text{o}}$ is the height of the object.

Calculation:

The procedure to draw the ray diagram

• Sketch the principal axis of the mirror as a horizontal line.
• Place the mirror at the center, place C at the left side of the mirror and place F at the other side.
• Sketch a vertical line at the mirror point to represent the mirror. This is the principal plane.
Sketch the object as an arrow and label its top $O_1$ .
• Draw ray 1, the parallel ray. It is parallel to the principal axis and reflects the principal plane and passes through F .
• Draw ray 2, the focus ray. It passes through F , reflects the principal plane, and is reflected parallel to the principal axis.
• The image is located where rays 1 and 2 cross after reflection. Label the point be $I_1$ .

Sketch the ray diagram as shown below.

  

Figure 1

The horizontal scale of 1 block is $1.2\text{ cm}$ .

The horizontal scale of 1 block is $0.38\text{ cm}$ .

The horizontal distance $\left(d_{\text{i}}\right)$ of the image from the mirror is,

  $\begin{array}{c}{d}_{\text{i}}=-\left(\begin{array}{l}\text{Number}\text{of}\text{blocks }\\ \text{from the mirror in }\\ \text{horizontal direction}\end{array}\right)\text{×}\left(\text{Horizontal}\text{Scale}\right)\\ =-5.38\text{blocks}\times 1.2\text{cm}\\ =-\text{6}\text{.46}\text{cm}\left(\begin{array}{l}\text{negative sign indicates}\\ \text{image is behind the mirror}\end{array}\right)\end{array}$

The vertical height $\left(h_{\text{i}}\right)$ of the image is

  $\begin{array}{c}{h}_{\text{i}}=\left(\begin{array}{l}\text{Number}\text{of}\text{blocks of image }\\ \text{in the vertical direction}\\ \text{from the axis}\end{array}\right)\text{×}\left(\text{Vertical}\text{Scale}\right)\\ =4.7\text{blocks}\times 0.38\text{cm}\\ =1.8\text{cm}\end{array}$

Using mirror and magnification equations.

The position of the image is,

  $\begin{array}{l}\frac{1}{d_{\text{o}}}+\frac{1}{d_{\text{i}}}=\frac{1}{f}\\ \frac{1}{d_{\text{i}}}=\frac{1}{f}-\frac{1}{d_{\text{o}}}\\ d_{\text{i}}=\frac{f{d}_{\text{o}}}{d_{\text{o}}-f}\\ d_{\text{i}}=\frac{\left(-12.0\text{ cm}\right)\left(14.0\text{ cm}\right)}{\left(14.0\text{ cm}\right)-\left(-12.0\text{ cm}\right)}\end{array}$

  $d_{\text{i}}=-6.46\text{ cm}$

The height of the image is,

  $\begin{array}{l}m=\frac{h_{\text{i}}}{h_{\text{o}}}=-\frac{d_{\text{i}}}{d_{\text{o}}}\\ h_{\text{i}}=-\frac{d_{\text{i}}{h}_{\text{o}}}{d_{\text{o}}}\\ h_{\text{i}}=-\frac{\left(-6.46\text{ cm}\right)\left(4.0\text{ cm}\right)}{\left(14.0\text{ cm}\right)}\\ h_{\text{i}}=1.8\text{ cm}\end{array}$

Conclusion:

Thus, ray diagram is shown in Figure 1. The position of the image is $-6.46\text{ cm}$ and the height of the image is $1.8\text{ cm}$ .