Problem 14 - Physics

Question

At illustrates the concepts pertinent to this problem. A convex mirror has a focal length of \(-40.0 \mathrm{~cm}\). A \(12.0\) -cm-tall object is located \(40.0 \mathrm{~cm}\) in front of this mirror. Using a ray diagram drawn to scale, determine the (a) location and (b) size of the image. Note that the mirror must be drawn to scale.

Step-by-Step Solution

Answer

The image is located 20 cm behind the mirror and is 6 cm tall.

Step 1: Identify Known Variables

We are given that the object height is \( h_o = 12.0 \) cm, the object distance is \( d_o = 40.0 \) cm, and the focal length \( f = -40.0 \) cm of the convex mirror.

Step 2: Use the Mirror Equation

The mirror equation is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Substitute the known values: \( \frac{1}{-40} = \frac{1}{40} + \frac{1}{d_i} \).

Step 3: Solve for Image Distance

Rearrange the formula to find \( d_i \): \( \frac{1}{d_i} = \frac{1}{-40} - \frac{1}{40} \). Simplifying gives \( \frac{1}{d_i} = -\frac{2}{40} = -\frac{1}{20} \), so \( d_i = -20 \) cm.

Step 4: Use the Magnification Formula

The magnification \( m \) is given by \( m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \). Substitute the values: \( m = -\frac{-20}{40} = \frac{1}{2} \).

Step 5: Calculate the Image Height

Using \( m = \frac{h_i}{h_o} \), we get \( \frac{h_i}{12} = \frac{1}{2} \) which yields \( h_i = \frac{1}{2} \times 12 = 6 \) cm.

Step 6: Analyze the Results

The image is located 20 cm behind the mirror and is 6 cm tall. Being behind the mirror confirms it is virtual and being smaller than the object confirms it is reduced in size.

Key Concepts

Convex Mirror

Convex mirrors are a type of spherical mirror where the reflective surface bulges outward. These mirrors are commonly used in situations where a wider field of view is necessary, such as in side-view mirrors on vehicles or security mirrors in stores.
A key characteristic of convex mirrors is that they diverge light rays, never allowing them to converge. This results in images that are always virtual, erect, and reduced in size compared to the actual object. Because of these properties, convex mirrors are particularly useful in providing a wide-angle perspective.
One important thing to note is the focal length of a convex mirror. It is considered to be negative because the focal point is behind the mirror, further emphasizing the nature of the virtual images they produce.

• Reflective surface bulges outward
• Diverges light rays
• Produces virtual, smaller images
• Used for wide field of view

Ray Diagram

Ray diagrams are a visual tool used to understand how images are formed by mirrors and lenses. They help in tracing how a set of key light rays behave in relation to the mirror.
In the case of convex mirrors, a typical ray diagram might include the following rays:

• A ray parallel to the principal axis, which upon reflection, appears to diverge from the focal point.


• A ray directed toward the center of curvature, which reflects back on itself.


• A ray directed toward the focal point before reflection, which then travels parallel to the principal axis.

These diagrams show that the three reflected rays appear to diverge from a virtual image behind the mirror. This aids in the understanding of how the virtual image is formed and how its properties such as size and location are derived relative to the mirror's geometry.

Mirror Equation

The mirror equation is a mathematical representation that relates the object distance ( d_o ), the image distance ( d_i ), and the focal length ( f ) of a mirror. It provides a way to calculate unknown quantities when certain variables are known.
The equation is given by: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
For convex mirrors, remember that the focal length ( f ) is negative. This equation is crucial in determining where the image will be formed, which is further confirmed by the construction of a ray diagram.
By rearranging the formula, you can solve for the image distance: \[ \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \]
This step is essential in predicting the characteristics of the image created by the convex mirror.

Magnification

Magnification refers to how much larger or smaller the image is compared to the actual object. For mirrors, it is given by the ratio of the image height ( h_i ) to the object height ( h_o ) or by the negative ratio of the image distance to the object distance:
\[ m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \]
In the context of a convex mirror, the magnification will always be less than 1, since images are reduced in size.
A positive magnification indicates that the image is upright relative to the object. Since convex mirrors produce virtual images, the image keeps the same orientation as the object. In summary, magnification helps quantify the scale and orientation changes occurring when observing an image formed by a convex mirror.

• Formula: \[ m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \]
• Magnification < 1 for convex mirrors
• Image is smaller and upright

Virtual Image

A virtual image is an image that, unlike a real image, cannot be projected onto a screen. It appears to be located behind the mirror, where the reflected rays seem to emanate.
Convex mirrors always produce virtual images because the reflected light rays diverge, and the brain extrapolates them backward to form an image.
Characteristics of virtual images include:

• They are upright, maintaining the same orientation as the object.


• They are diminished, meaning they are smaller than the actual object.


• Despite being virtual, their position and size can be accurately calculated using the mirror equation and magnification formula.

Virtual images are invaluable in applications requiring a conscious understanding of spatial awareness without real projection ability, such as when using a passenger-side mirror on a car to expand the driver's field of view.