Problem 30 - College Physics

Question

A dentist uses a spherical mirror that produces an upright image of a tooth that is magnified four times. (a) The mirror is (1) converging, (2) diverging, (3) flat. Explain. (b) What is the mirror's focal length in terms of the object distance?

Step-by-Step Solution

Answer

(a) Mirror is diverging. (b) Focal length is \( \frac{4}{5}p \).

Step 1: Analyze Image Orientation and Type

The problem states that the image is upright and magnified. In optics, an upright and magnified image is typically formed by a diverging mirror when the object is located in front of the mirror. This is because a converging mirror only produces inverted images or real images unless the object is inside the focal point.

Step 2: Determine the Mirror Type

Given the information that the image is upright and magnified, we can conclude that the mirror used is a diverging mirror. Diverging mirrors can produce virtual, upright, and magnified images if the object is placed between the mirror and the focal point.

Step 3: Use the Magnification Formula

The magnification (\( M \)) is given by \( M = \frac{h'}{h} = \frac{q}{p} \) where \( h' \) and \( h \) are the image and object heights, and \( q \) and \( p \) are the image and object distances. Here, \( M = 4 \). This means \( \frac{q}{p} = 4 \).

Step 4: Apply the Mirror Equation

The mirror equation is \( \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \). Since \( q = 4p \), substitute into the equation to get \( \frac{1}{f} = \frac{1}{p} + \frac{1}{4p} \). This simplifies to \( \frac{1}{f} = \frac{5}{4p} \).

Step 5: Solve for the Focal Length

Rearrange the equation from Step 4 to solve for \( f \): \( f = \frac{4p}{5} \). This demonstrates that the focal length of the mirror can be expressed in terms of the object distance as \( \frac{4}{5}p \).

Key Concepts

Focal Length

The focal length of a mirror is a crucial element in determining how images are formed by that mirror. For spherical mirrors, the focal length (\( f \)) is defined as the distance between the mirror's surface and its focal point, where parallel rays of light either converge (in the case of a converging mirror) or appear to diverge from (in the case of a diverging mirror).
In the given problem, we're dealing with a diverging mirror since the image formed is upright and magnified, which means the mirror's focal point is virtual.

• The signer of a diverging mirror's focal length is negative, indicating that focal points are virtual, meaning they cannot be projected on a screen.

Knowing the focal length allows us to calculate the positions and characteristics of images formed by mirrors, essential for both photographers and dentists alike.

Magnification

Magnification (\( M \)) indicates the ratio of the height of the image (\( h' \)) to the height of the object (\( h \)). It is also the ratio of the image distance (\( q \)) to the object distance (\( p \)). Hence, \( M = \frac{h'}{h} = \frac{q}{p} \).

• In optics, a magnification value greater than 1 suggests a larger image, less than 1 suggests a reduced image.
• If the magnification is positive, the image is upright.

In our context, the dentist’s mirror gives a magnification of 4, implying the image of the tooth is four times larger than the actual tooth.
The mirror must be able to bend light in such a way that creates a virtual image in which details are clearer, which is incredibly useful for examination purposes.

Mirror Equation

The mirror equation is essential for calculating the relationship between object distance (\( p \)), image distance (\( q \)), and focal length (\( f \)) of a mirror:\( \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \).This equation allows us to find unknown variables about a mirror, such as focal length if the object and image positions are known.
In the exercise, the known magnification and image positioning lead to solving:

• Substituting \( q = 4p \) due to magnification:\( M = \frac{q}{p} \).
• After substitution into the mirror equation, simplifying results in \( f = \frac{4p}{5} \).

In simpler terms, by using the mirror equation coupled with magnification knowledge, it's proven that the focal length is four-fifths of the object distance. Understanding these concepts deepens one's grasp on the behavior of spherical mirrors and their application in practical scenarios like dental examinations.