Question

Convex mirrors are often used as sideview mirrors on cars. Many such mirrors display the warning "Objects in mirror are closer than they appear.” Assume that a convex sideview mirror has a radius of curvature of \(14.0 \mathrm{~m}\) and that a car 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 the magnification for this convex mirror.

Short answer

Answer: The image distance is 19.25 m, and the magnification is -1.75.

Step-by-step solution

Find the focal length of the convex mirror

We are given the radius of curvature (R) of the convex mirror as \(14.0 \mathrm{~m}\). For a convex mirror, the focal length (f) is half of the radius of curvature. Therefore: \(f = \frac{R}{2}\) \(f = \frac{14.0}{2}\) \(f = 7.0 \mathrm{~m}\)

Use the mirror equation to find the image distance

The mirror equation is given by: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) We know the focal length (f) and the object distance (\(d_o\)). We can plug these values into the equation and solve for the image distance (\(d_i\)): \(\frac{1}{7.0} = \frac{1}{11.0} + \frac{1}{d_i}\) Now, we need to solve for \(d_i\).

Solve for the image distance

In order to find the image distance, we can follow these steps: 1. Subtract \(\frac{1}{11.0}\) from both sides of the equation: \(\frac{1}{7.0} - \frac{1}{11.0} = \frac{1}{d_i}\) 2. Find a common denominator for the fractions: \(\frac{11-7}{7 \cdot 11} = \frac{1}{d_i}\) 3. Simplify the fraction: \(\frac{4}{77} = \frac{1}{d_i}\) 4. Take the reciprocal of both sides to find \(d_i\): \(d_i = \frac{77}{4}\) \(d_i = 19.25 \mathrm{~m}\)

Find the magnification using the magnification equation

Now we can use the magnification equation to find the magnification (M) of the convex mirror: \(M = -\frac{d_i}{d_o}\) We already found the image distance in step 3 (\(d_i=19.25 \mathrm{~m}\)). Plug in the values and solve: \(M = -\frac{19.25}{11.0}\) \(M = -1.75\) The image distance is \(19.25 \mathrm{~m}\), and the magnification is \(-1.75\). Since the magnification is negative, the image is virtual, and objects appear closer than they actually are in the convex mirror.

Key concepts

Radius of Curvature

Understanding the radius of curvature is essential when dealing with mirrors in optics. The radius of curvature (\r) refers to the radius of the spherical surface from which the mirror is a section. For a convex mirror, this surface bulges outwards, like the exterior of a sphere.

\r A valuable property of the radius of curvature is that it is directly related to the focal length (\f) of a mirror. The relationship is simple: the focal length is half the radius of curvature. This relation is expressed as:
\r\[ f = \frac{R}{2} \]
\rIn our example involving a convex sideview mirror on a car with a given radius of curvature of 14.0 meters, the focal length is 7.0 meters. This information is critical for solving problems related to image formation by such mirrors.

Mirror Equation

The mirror equation is a fundamental tool in optics, describing the relationship between the object distance (\r\(d_o\)), the image distance (\r\(d_i\)), and the focal length (\f) of a mirror. It's given by:
\r\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
\rFor convex mirrors, this equation allows us to compute where the image formed by the mirror will appear with respect to the mirror's surface. Although a convex mirror always forms virtual images, the location where these images appear to be is vital for applications like vehicle mirrors, which is why understanding and applying this equation can help predict how these images will look.

Using the given object distance and known focal length, we can rearrange the equation to solve for the image distance. By following the algebraic steps outlined in the solution, we find that the image distance for the car in the mirror's example would be 19.25 meters. This distance is where the image would appear to be located behind the convex mirror.

Magnification of Mirror

Magnification in the context of mirrors refers to how much larger or smaller the image is compared to the actual object. For mirrors, the magnification (\r\(M\)) can be calculated using the equation:
\r\[ M = -\frac{d_i}{d_o} \]
\r The negative sign indicates that if the image formed is virtual, it will appear upright as opposed to the inverted images typical in real images. However, the magnitude of the magnification tells us about the relative size.

In the case of the convex mirror on a car, where the object (another car) is 11.0 meters away, and the image distance is calculated to be 19.25 meters, the magnification comes out to be -1.75. This means the image is smaller than the object (since the absolute value of magnification is less than 1), and the negative sign confirms that the image is virtual. In practical terms, this makes objects in the convex mirror appear closer than they are, leading to the common safety warning on vehicle mirrors.