Problem 23 - Fundamentals of Physics

Question

Object \(O\) stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table \(34-4\) refers to (a) the type of mirror, (b) the focal distance \(f,\) (c) the radius of curvature \(r\), (d) the object distance \(p,\) (e) the image distance \(i\), and (f) the lateral magnification \(m\). (All distances are in centimeters.) It also refers to whether \((\mathrm{g})\) the image is real \((\mathrm{R})\) or virtual (V), (h) inverted (I) or noninverted (NI) from \(O\), and (i) on the same side of the mirror as object \(O\) or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.

Step-by-Step Solution

Answer

Apply the mirror formula \(\frac{1}{f} = \frac{1}{p} + \frac{1}{i}\) and relationships to find missing values. Use sign conventions for image characteristics.

Step 1: Understanding the Problem

We are given a situation with an object standing on the central axis of a mirror. We need to find the missing values for focal distance \(f\), radius of curvature \(r\), object distance \(p\), image distance \(i\), lateral magnification \(m\), and characteristics such as whether the image is real or virtual, inverted or noninverted, and the image's position relative to the object.

Step 1: Identify the Type of Mirror

Based on the problem, if the focal distance and the radius of curvature are positive, the mirror is concave. If they are negative, it is a convex mirror.

Step 2: Apply Mirror Formula

Use the mirror formula \(\frac{1}{f} = \frac{1}{p} + \frac{1}{i}\) to find the missing distances. Solve for the unknown value if \(p\), \(i\), or \(f\) is missing.

Step 3: Calculate the Radius of Curvature

Use the relationship \(r = 2f\) to find the radius of curvature from the focal length if needed.

Step 4: Find the Magnification

Calculate the magnification using the formula \(m = -\frac{i}{p}\). Determine the height and orientation of the image from the sign of \(m\).

Step 5: Determine Image Characteristics

Use the sign conventions. If \(i\) is positive, the image is real and on the same side as the object. Negative \(i\) means a virtual image on the opposite side. A positive \(m\) indicates a noninverted image, while a negative \(m\) indicates an inverted image.

Key Concepts

Concave Mirror

A concave mirror is a mirror with a surface that curves inward, resembling the interior of a sphere. These mirrors are used in a variety of applications such as telescopes, headlights, and shaving mirrors due to their ability to focus light to a point.
Concave mirrors have several distinct properties:

• Focal Point (F): The point where parallel rays of light either converge or appear to converge after reflecting off the mirror.
• Center of Curvature (C): The center of the sphere from which the mirror segment is taken.
• Principal Axis: The line passing through the center of curvature and the midpoint of the mirror.

When light rays parallel to the principal axis reflect off a concave mirror, they converge at the focal point. This property allows concave mirrors to form real images when the object is placed beyond the focal point. Real images are formed on the same side as the reflecting surface and can be inverted and magnified depending on the object's distance from the mirror.

Convex Mirror

A convex mirror, unlike a concave mirror, has a surface that curves outward. These mirrors are commonly used in vehicle side mirrors and security mirrors because they allow for a wider field of view.
Key features of convex mirrors include:

• Diverging Reflection: Light rays reflect off the surface and spread out, making all reflections appear to originate from a virtual point behind the mirror.
• Virtual Images: Since the reflected rays spread out, the images formed are always virtual, smaller, and upright compared to the actual object.
• Wider View: Because they diverge light, convex mirrors offer a wider field of view, making them useful for parking assistance and security.

In concave mirrors, the focal point and center of curvature lie behind the mirror, making the focal length and radius of curvature negative. The images formed by convex mirrors cannot be projected onto a screen as they are virtual.

Image Characteristics

The characteristics of an image formed by mirrors depend on several factors, including the type of mirror and the position of the object. These characteristics include:

• Real or Virtual: Real images are formed by concave mirrors when the object is beyond the focal point; they can be projected onto a screen. Virtual images, formed by convex mirrors or concave mirrors with objects placed within the focal point, cannot be projected.
• Inverted or Upright: Real images are typically inverted, whereas virtual images are upright.
• Same Side or Opposite Side: Real images by concave mirrors form on the same side as the object, while virtual images appear to be on the opposite side in the case of both concave and convex mirrors.

These characteristics help determine the suitability of mirrors in real-world applications, where specific image properties, like size or orientation, are required.

Magnification Formula

The magnification formula is a crucial aspect of understanding image formation with mirrors. This formula helps calculate how much larger or smaller, as well as orientational differences, an image is compared to the actual object. The formula is expressed as: \[ m = -\frac{i}{p} \]Where:

• m is the magnification factor.
• i is the image distance from the mirror.
• p is the object distance from the mirror.

A negative magnification indicates the image is inverted. Real images formed by concave mirrors often have negative magnification values.
Conversely, positive magnification represents an upright, usually virtual, image. Magnification also highlights whether the image is smaller (\( |m| < 1 \)) or larger (\( |m| > 1 \)) than the object. Understanding magnification is fundamental for predicting how an image will appear through mirrors.