Problem 71 - University Physics with Modern Physics

Question

A mirror on the passenger side of your car is convex and has a radius of curvature with magnitude 18.0 cm. (a) Another car is behind your car, 9.00 m from the mirror, and this car is viewed in the mirror by your passenger. If this car is 1.5 m tall, what is the height of the image? (b) The mirror has a warning attached that objects viewed in it are closer than they appear. Why is this so?

Step-by-Step Solution

Answer

The image height is 1.485 cm, and objects look smaller in convex mirrors, appearing farther.

Step 1: Understanding the Problem

A convex mirror has a radius of curvature with absolute value \( R = 18.0 \) cm. The object (another car) is at a distance \( d_o = 900 \) cm (converting from 9.00 m to cm) from the mirror, and the height of this car \( h_o = 150 \) cm (converting from 1.5 m to cm). We need to find the height of the image formed by the mirror.

Step 2: Find the Focal Length

The focal length \( f \) of a mirror is given by \( f = \frac{R}{2} \). For a convex mirror, the focal length is negative, so \( f = -\frac{18.0}{2} = -9.0 \) cm.

Step 3: Use the Mirror Equation

The mirror equation is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( d_i \) is the image distance. Substituting the known values we have \( \frac{1}{-9.0} = \frac{1}{900} + \frac{1}{d_i} \). Solve for \( d_i \).

Step 4: Solve for Image Distance

Rearrange the equation to find \( \frac{1}{d_i} = \frac{1}{-9.0} - \frac{1}{900} \). Calculating these gives \( \frac{1}{d_i} = -0.1111 - 0.0011 = -0.1122 \). So, \( d_i = -8.91 \) cm.

Step 5: Apply the Magnification Formula

The magnification \( m \) of a mirror is given by \( m = -\frac{d_i}{d_o} \) and also by \( m = \frac{h_i}{h_o} \). Using \( m = -\frac{-8.91}{900} \approx 0.0099 \). Thus, the height of the image \( h_i = m \cdot h_o = 0.0099 \times 150 = 1.485 \) cm.

Step 6: Explanation of Warning

Convex mirrors cause images to appear smaller, making them seem farther away. This is due to the diverging nature of the mirror, which compresses the field of view into a smaller area.

Key Concepts

Convex Mirror

A convex mirror is a type of spherical mirror that curves outward, like the exterior of a sphere. It is often used in vehicles, such as on the passenger side of cars, due to its ability to provide a wider field of view. This is because light rays diverge after reflecting off the convex surface, making it easier for drivers to see more area behind them.

One of the peculiar features of convex mirrors is that they always produce virtual, upright, and diminished images of objects, regardless of their position. These qualities make convex mirrors particularly useful for safety and surveillance purposes in situations where a broad field of vision is necessary.

Mirror Equation

The mirror equation relates the focal length of the mirror, the object distance, and the image distance. It is an essential tool in optics for analyzing mirrors' behavior. The equation is written as:

\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]

Here, \( f \) is the focal length, \( d_o \) is the object distance from the mirror, and \( d_i \) is the image distance. The focal length for convex mirrors is negative because they cause parallel rays to diverge.

By substituting the known values into this equation, one can solve for the unknowns, like the image distance. This equation shows that when an object is positioned very far from a convex mirror, the image becomes increasingly smaller and appears closer to the focal point.

Image Magnification

Image magnification in optics tells us how much larger or smaller an image appears compared to the original object. It can be determined using the formula:

\[ m = -\frac{d_i}{d_o} \]

It is also defined as the ratio of the image height \( h_i \) to the object height \( h_o \):
\[ m = \frac{h_i}{h_o} \]

In the context of a convex mirror, since the image height is smaller than the object height, the magnification factor will be less than one. This is the reason why objects reflect smaller than their actual size in a convex mirror.

Focal Length

Focal length is a critical property of mirrors and lenses, representing the distance from the mirror to the focal point. For a spherical mirror, the focal length \( f \) can be calculated from the radius of curvature \( R \) using the formula:

\[ f = \frac{R}{2} \]

With convex mirrors, the focal length is considered negative, which reflects the diverging nature of these mirrors. Diverging means that parallel rays of light that hit the mirror spread out after reflection.

Understanding the focal length is crucial for predicting how images will form, especially in determining whether they will appear larger or smaller, closer or farther than they are. This explains why objects viewed in convex mirrors seem farther away than they actually are, as stated in warnings often printed on vehicle mirrors.