Problem 13 - College Physics

Question

An object is placed \(50 \mathrm{~cm}\) in front of a convex mirror and its image is found to be \(20 \mathrm{~cm}\) behind the mirror. What is the focal length of the mirror? What is the lateral magnification?

Step-by-Step Solution

Answer

Focal length is approximately -33.33 cm; magnification is 0.4.

Step 1: Identify the Known Values

We are given that the object distance \(d_o = 50\ cm\) and the image distance \(d_i = -20\ cm\). Note that the image distance is negative because the image is formed behind the mirror, which is a convention for convex mirrors.

Step 2: Mirror Equation Setup

The mirror equation is \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]. We will use this equation to find the focal length \(f\).

Step 3: Substitute Known Values

Substitute the known values into the mirror equation: \[ \frac{1}{f} = \frac{1}{50} + \frac{1}{-20} \].

Step 4: Calculate the Focal Length

Solve for \(\frac{1}{f}\): \[ \frac{1}{f} = \frac{1}{50} - \frac{1}{20} = \frac{2 - 5}{100} = \frac{-3}{100} \]. Thus, \(f = -\frac{100}{3}\) cm or \(f \approx -33.33\ cm\).

Step 5: Magnification Formula Setup

The magnification \(m\) is given by \(m = -\frac{d_i}{d_o}\).

Step 6: Calculate the Magnification

Substitute the known values into the magnification formula: \(m = -\frac{-20}{50} = \frac{20}{50} = 0.4\).

Key Concepts

Focal Length

Understanding the focal length of a mirror, particularly a concave mirror, is crucial for studying optics. The focal length (\(f\)) of a concave mirror is the distance between the mirror's surface and its focal point, where parallel rays of light meet after reflection. In concave mirrors, the focal length is positive when dealing with real focal points and negative for virtual ones.
For convex mirrors, like in our example, the focal length is always negative, as these mirrors diverge light, creating a virtual focus behind the mirror.

• The mirror equation connects object distance (\(d_o\)), image distance (\(d_i\)), and focal length (\(f\)): \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
• Convex mirrors have negative image distances because the image forms on the side opposite the object.
• Finding the negative focal length confirms the divergent nature of convex mirrors.

By substituting the object and image distances into the formula, we discover our convex mirror's focal length as approximately \(-33.33 \text{ cm}\).

Lateral Magnification

Lateral magnification is a measure of how much larger or smaller the image size is compared to the object. It is an essential concept when dealing with mirrors, as it helps determine the nature and orientation of an image.
The lateral magnification (\(m\)) formula is:

• \[ m = -\frac{d_i}{d_o} \]
• "\(-\)" signifies image orientation. A positive magnification means an upright image, whereas a negative value indicates an inverted image.

For the convex mirror scenario, substituting our values showed a magnification of \(0.4\). This means the image is 40% the size of the object and upright (as indicated by the positive value). Convex mirrors typically create images smaller than the object, suitable for wide-field viewing, like in car side mirrors.

Mirror Equation

The mirror equation is pivotal in connecting object distance, image distance, and focal length. Understanding this relationship enables one to predict image characteristics in concave and convex mirrors. Applicable for both types, the equation is:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]It expresses the inverse relationship, wherein adjusting either the object or image distances alters the focal length accordingly.
Key aspects to remember:

• The object distance (\(d_o\)) is positive if the object is in front of the mirror and negative if behind.
• The image distance (\(d_i\)) follows the same sign rule: positive for real images and negative for virtual images.

In our convex mirror example, \(d_i = -20\ \text{cm} \), emphasizing a virtual image forming behind the mirror. This negative image distance helps us solve the equation to find the negative focal length, solidifying the nature and behavior of convex mirrors as diverging systems.