Question

Convex mirrors are often used in side view mirrors on cars. Many such mirrors display the warning "Objects in mirror are closer than they appear." Assume a convex mirror has a radius of curvature of \(14.0 \mathrm{~m}\) and that there is a car that is \(11.0 \mathrm{~m}\) behind the mirror. For a flat mirror, the image distance would be \(11.0 \mathrm{~m}\) and the magnification would be 1\. Find the image distance and magnification for this mirror.

Short answer

Answer: The image distance is approximately 4.29 m, and the magnification is approximately -0.39.

Step-by-step solution

Identify the given information

We are given: 1. Radius of curvature (\(R\)) for the convex mirror: \(14.0 \mathrm{m}\) 2. Object distance (\(o\)): \(11.0 \mathrm{m}\)

Find the focal length of the convex mirror

The focal length (\(f\)) of a mirror can be obtained from its radius of curvature (\(R\)) using the following formula: $$ f = | \frac{R}{2} | $$ Since it is a convex mirror, the focal length is positive, so no need for absolute value: $$ f=\frac{14.0}{2}=7 \mathrm{m} $$

Apply the mirror formula

The mirror formula is given by: $$ \frac{1}{f} = \frac{1}{o} + \frac{1}{i} $$ Where \(f\) is the focal length, \(o\) is the object distance, and \(i\) is the image distance. Rearrange the formula to find the image distance (\(i\)): $$ \frac{1}{i} = \frac{1}{f} - \frac{1}{o} $$ Substitute the values of \(f\) and \(o\): $$ \frac{1}{i} = \frac{1}{7} - \frac{1}{11} $$

Calculate the image distance

Now, solve for \(i\): $$ i = \frac{1}{\frac{1}{7} - \frac{1}{11}} = 4.2941 \mathrm{m} $$ So, the image distance is approximately \(4.29 \mathrm{m}\).

Find the magnification of the convex mirror

The magnification (\(M\)) of a mirror can be obtained using the formula: $$ M = - \frac{i}{o} $$ Substitute the values of \(i\) and \(o\): $$ M = - \frac{4.2941}{11} = -0.39037 $$ So the magnification is approximately \(-0.39\).

Interpret the magnification

Since the magnification is negative, it indicates that the image is inverted. The magnitude of the magnification being less than 1 indicates that the image is smaller compared to the object which means "Objects in mirror are closer than they appear." In conclusion, the image distance is approximately \(4.29 \mathrm{m}\), and the magnification is approximately \(-0.39\).

Key concepts

Convex Mirror

A convex mirror is a curved glass or metallic surface that bulges outward. This unique shape is known as a convex surface. Convex mirrors have several important applications, primarily due to the way they reflect light. They can form images of objects that make them appear smaller and more upright than they actually are. This makes them ideal for use in safety devices like side view mirrors on cars.
Convex mirrors allow a wider field of view than flat mirrors, enabling drivers to see more of their surroundings, albeit at the cost of distortion. This is why the warning "Objects in mirror are closer than they appear" is often inscribed on these mirrors to caution drivers that objects seem further away than their actual position.

Focal Length

The focal length of a mirror is an important characteristic that indicates how strongly the mirror converges or diverges light. In the case of a convex mirror, the focal length is positive because it spreads out light beams, effectively behaving as if the light rays are coming from a point behind the mirror.
The relationship between a mirror's focal length (\(f\)) and its radius of curvature (\(R\)) is given by the formula:

• \(f = \frac{R}{2}\)

For a convex mirror with a radius of curvature of \(14.0 \, \text{m}\), the focal length would be \(7.0 \, \text{m}\). This means that if parallel light rays hit the mirror, they will appear to spread out as if they originated from a focal point located \(7.0 \, \text{m}\) behind the mirror.

Mirror Formula

The mirror formula is a fundamental equation in optics that connects the focal length (\(f\)), object distance (\(o\)), and image distance (\(i\)) for a mirror. It is expressed as:

• \(\frac{1}{f} = \frac{1}{o} + \frac{1}{i}\)

By using this equation, one can calculate the location of an image formed by the mirror. For instance, a convex mirror with \(f = 7.0 \, \text{m}\) and an object positioned \(11.0 \, \text{m}\) from it, allows us to find the image distance (\(i\)) with the rearranged formula:

• \(\frac{1}{i} = \frac{1}{f} - \frac{1}{o}\)

Substituting the values, we find \(i \approx \frac{1}{\frac{1}{7} - \frac{1}{11}} \approx 4.29 \, \text{m}\). This calculation indicates where the image appears behind the mirror.

Magnification

Magnification describes how much larger or smaller an image is compared to the actual object. It is crucial for understanding how the image formed by a mirror appears to us. The magnification (\(M\)) formula for mirrors is given by:

• \(M = -\frac{i}{o}\)

In this formula, the negative sign indicates that the image formed by a mirror is inverted relative to the object's orientation. For a convex mirror with image distance \(i = 4.29 \, \text{m}\) and object distance \(o = 11.0 \, \text{m}\), the magnification is \(-\frac{4.29}{11} \approx -0.39\).
The magnitude of magnification being less than 1 shows that the image is smaller than the object itself. In this case, a magnification of approximately \(-0.39\) means that the image appears smaller and upright. This is consistent with the warning on convex mirrors that objects may appear closer than they actually are.