Problem 13 - University Physics with Modern Physics

Question

Dental Mirror. A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she wants an erect image with a magnification of 2.00 when the mirror is 1.25 \(\mathrm{cm}\) from a tooth. (Treat this problem as though the object and image lie along a straight line.) (a) What kind of mirror (concave or convex) is needed? Use a ray diagram to decide, without performing any calculations. (b) What must be the focal length and radius of curvature of this mirror? (c) Draw a principal-ray diagram to check your answer in part (b).

Step-by-Step Solution

Answer

A concave mirror with a focal length of 2.5 cm and a radius of 5.0 cm is needed.

Step 1: Identify Type of Mirror

Since we are given that the image is erect and has a magnification of +2, the image must be virtual. In optics, concave mirrors can produce virtual images that are magnified and erect when the object is located within the focal length of the mirror. Therefore, the dentist needs a concave mirror.

Step 2: Use Mirror Formula for Focal Length

We start with the magnification formula, where magnification \( M = \frac{-d_i}{d_o} \), and is given as 2.00. Here, \( d_i \) is the image distance and \( d_o = 1.25 \mathrm{cm} \). Rearrange for \( d_i \): \( d_i = -2 \times d_o = -2.50 \mathrm{cm} \). The negative sign indicates a virtual image. Now use the mirror formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Substitute \( d_o = 1.25 \mathrm{cm} \) and \( d_i = -2.50 \mathrm{cm} \).

Step 3: Calculate Focal Length

Substituting in the mirror equation gives: \( \frac{1}{f} = \frac{1}{1.25} + \frac{1}{-2.50} = \frac{2}{2.5} - \frac{1}{2.5} = \frac{1}{2.5} = 0.4 \). So, \( f = 2.5 \mathrm{cm} \).

Step 4: Determine Radius of Curvature

The radius of curvature \( R \) is related to the focal length \( f \) by the formula \( R = 2f \). Since \( f = 2.5 \mathrm{cm} \), \( R = 2 \times 2.5 = 5.0 \mathrm{cm} \).

Step 5: Draw Principal-Ray Diagram

Draw the concave mirror with its principal axis. Place the object (tooth) 1.25 cm from the mirror, and use the focal point at 2.5 cm from the mirror. The first ray goes parallel to the axis and reflects through the focal point behind the mirror, and the second ray goes through the origin and reflects parallel to the axis. The intersection of reflected rays behind the mirror locates the virtual image, confirming our calculations.

Key Concepts

Concave Mirror

In the world of optics, mirrors play a pivotal role. When examining concave mirrors, one can understand how they form images. Concave mirrors, also known as converging mirrors, have reflective surfaces that curve inward. They resemble part of the interior of a sphere. This curvature allows them to redirect light inward towards a focal point. Concave mirrors are useful in various applications, including dentistry. When a dentist uses a concave mirror, it helps magnify the teeth, making them easier to observe closely. This is important for accurate diagnoses and treatments.
The properties of concave mirrors that make them indispensable in dental work include:

• Ability to produce magnified images
• Capability to form virtual images when objects are positioned within the focal length
• Providing clear and erect images of objects close to the mirror

Image Magnification

Image magnification is the process of enlarging the appearance of an object using optical instruments such as mirrors or lenses. With concave mirrors in dental practices, magnification ensures that tiny details are visible to the dentist. This is vital for spotting cavities or monitoring the progress of dental treatment.The magnification (M) is given by the ratio of the image distance (-d_i) over the object distance (d_o). This relation is expressed as:\[M = \frac{-d_i}{d_o}\]When the magnification is positive and greater than 1, as in our problem where it’s +2, the image is:

• Erect, appearing right-side up
• Virtual, since the calculations indicate a negative image distance
• Twice the size of the object, enhancing visibility

This magnified representation helps dentists thoroughly evaluate oral regions that are otherwise hard to access.

Mirror Formula

The mirror formula is central to understanding how concave mirrors create images. This formula relates the focal length (f), the object distance (d_o), and the image distance (d_i):\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]Using the mirror formula helps determine the focal length necessary for creating an image with the desired characteristics. For a desired magnified and erect image in dental inspection, understanding and using this formula allows precise calculations for mirror usage. During calculations, remember:

• A positive focal length implies a concave mirror
• Image distance being negative shows a virtual image
• Accurate substitution reveals the mirror’s focal length and its utility

This formula's balancing act ensures dentists can visualize images that aid in patient care effectively.

Virtual Image

A virtual image is a classic outcome when using concave mirrors in close quarters, such as dental inspections. In contrast to real images, which are formed on the opposite side of the mirror, a virtual image appears to be located "inside" or "behind" the mirror. This is significant as it affects how we perceive image orientation and magnification. In the context of a concave mirror, a virtual image offers these characteristics:

• It is erect, making interpretation straightforward
• Forms when the object is within the focal length of the mirror
• Perceived on the same side as the object, enhancing convenience

Virtual images, being non-tangible, cannot be projected on a screen. Yet, they are invaluable for detail-oriented examinations, particularly in medical fields like dentistry.

Optics

Optics is the branch of physics that deals with light. It encompasses the behavior and properties of light, including its interactions with matter through lenses, mirrors, and other media. Understanding optics is crucial for designing and using dental mirrors effectively. In the case of concave mirrors within optical systems:

• Light rays get focused towards a single point – the focal point
• Mirrors can manipulate light to enhance visibility and image clarity
• The mirror’s curvature affects how light reflects and thus image characteristics

These principles are pivotal in practical applications, allowing dentists and other medical professionals to navigate small, challenging spaces with ease. By grasping the basics of optics, they can better utilize tools that make complex tasks simpler and more accurate.