Problem 12 - Applied Physics

Question

An object \(12.6 \mathrm{~cm}\) in front of a convex mirror forms an image \(6.00 \mathrm{~cm}\) behind the mirror. What is the focal length of the mirror?

Step-by-Step Solution

Answer

The focal length of the mirror is approximately \(-11.45\) cm.

Step 1: Understand the Problem

We need to find the focal length of a convex mirror given the object distance and the image distance. The object distance \( (d_o) \) is 12.6 cm in front of the mirror, and the image distance \( (d_i) \) is -6.00 cm behind the mirror. In a convex mirror, the image is virtual and formed behind the mirror, hence the negative sign for \(d_i\).

Step 2: Mirror Equation

The mirror equation is given by:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance.

Step 3: Substitute the Values

Plug the given values into the mirror equation.\[\frac{1}{f} = \frac{1}{12.6} + \frac{1}{-6.00}\]

Step 4: Calculate Each Fraction

Calculate each individual fraction:\[\frac{1}{12.6} \approx 0.07937\]\[\frac{1}{-6.00} = -0.16667\]

Step 5: Add the Fractions

Add the calculated fractions:\[\frac{1}{f} = 0.07937 - 0.16667 = -0.0873\]

Step 6: Calculate the Focal Length

To find \(f\), take the reciprocal of \(-0.0873\):\[f = \frac{1}{-0.0873} \approx -11.452\]Thus, the focal length is approximately \(-11.45\) cm.

Key Concepts

Mirror Equation

The mirror equation is a fundamental formula used in optics to establish the relationship between the object distance, image distance, and focal length for mirrors. It is represented as:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Where:

• \( f \) is the focal length of the mirror.
• \( d_o \) refers to the object distance, the distance from the object to the mirror.
• \( d_i \) is the image distance, the distance from the image to the mirror.

The mirror equation is crucial because it allows us to determine various unknown parameters if we have the values for two out of the three variables. In the case of convex mirrors, it is important to note that the equations and formulas use the sign convention. For convex mirrors, images are formed behind the mirror, making the image distance \(d_i\) negative, as seen in the provided example exercise.
To solve an optical problem using this equation, one must substitute the known values into the formula and resolve for the unknown. This systematic approach aids in understanding the position and nature of the image being reflected by the mirror.

Focal Length Calculation

Calculating the focal length of a mirror, particularly a convex one, involves using the mirror equation and substituting the known object distance and image distance values. Given our example problem, with an object distance \(d_o = 12.6 \text{ cm}\) and an image distance \(d_i = -6.0 \text{ cm}\), we apply these to the mirror equation.
By plugging these values into \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \], we start by calculating each term individually.

• The reciprocal of the object distance is \( \frac{1}{12.6} \approx 0.07937\).
• The reciprocal of the image distance is \( \frac{1}{-6.0} = -0.16667\).
• Adding these, we get: \( \frac{1}{f} = 0.07937 - 0.16667 = -0.0873 \).
• Finally, take the reciprocal of \(-0.0873\) to find the focal length \(f\), resulting in \(f \approx -11.45 \text{ cm}\).

The negative sign indicates that the focal point of a convex mirror is virtual, situated on the mirror's imaginary or virtual side. Understanding this calculation process is vital, as it provides clarity on how curved mirrors actually operate both in theoretical and practical applications.

Virtual Image Formation

Convex mirrors, unlike concave mirrors, always form virtual images. These mirrors bulge outward, reflecting light in a way that causes the extensions of the reflected rays to meet behind the mirror. This phenomenon gives rise to virtual images, which cannot be projected on a screen, because they do not really exist in space.
In practical terms, when light from an object strikes a convex mirror, the reflected light appears to come from a point behind the mirror. Since human eyes interpret light as traveling in straight lines, the brain is tricked into perceiving a virtual image.

• The image formed is always upright (not inverted).
• It is also smaller than the actual object, due to the way the light rays diverge and reflect.
• This consistency in virtual image formation makes convex mirrors useful for various applications like vehicle rearview mirrors, where a wide field of view is necessary.

Understanding the characteristics of virtual images can help in anticipating the way images will behave in different optical setups, and why certain mirror types are chosen for specific tasks.