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Chapter 32: Problem 12

(II) A small candle is 35 \(\mathrm{cm}\) from a concave mirror having a radius of curvature of 24 \(\mathrm{cm} .(a)\) What is the focal length of the mirror? (b) Where will the image of the candle be located? (c) Will the image be upright or inverted?

Short Answer

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(a) Focal length is 12 cm. (b) Image is approximately 18.28 cm from the mirror. (c) Image is inverted.

Step by step solution

01

Calculate the Focal Length

To find the focal length of the mirror, use the formula: \( f = \frac{R}{2} \) where \( R \) is the radius of curvature. Given \( R = 24 \ \text{cm} \), substitute it into the formula: \( f = \frac{24}{2} = 12 \ \text{cm} \). Therefore, the focal length of the mirror is \( 12 \ \text{cm} \).
02

Locate the Image Position

Use the mirror equation to locate the image: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) where \( d_o \) is the object distance (35 cm) and \( f \) is the focal length (12 cm). Substitute the values: \( \frac{1}{12} = \frac{1}{35} + \frac{1}{d_i} \). Solve for \( \frac{1}{d_i} \) by simplifying: \( \frac{1}{d_i} = \frac{1}{12} - \frac{1}{35} \), which gives \( \frac{1}{d_i} = 0.0833 - 0.0286 = 0.0547 \). Therefore, \( d_i = \frac{1}{0.0547} \approx 18.28 \ \text{cm} \). The image is located at approximately 18.28 cm from the mirror.
03

Determine Image Orientation

For concave mirrors, when the object is placed beyond the focal length, the image formed is real and inverted. Since the object distance is 35 cm, which is greater than the focal length of 12 cm, the image will be inverted.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Focal Length
The focal length of a mirror is a key characteristic that defines its optical power. In the context of concave mirrors, it's particularly important. The focal length, denoted by \( f \), is the distance between the mirror's surface and the focal point, where parallel rays of light converge after reflection.

For a concave mirror, you can find the focal length using the formula: \( f = \frac{R}{2} \). Here, \( R \) represents the radius of curvature, which is the radius of the sphere of which the mirror is a part. In simpler terms, it's twice the focal length. So, if you know the radius of curvature, you can easily calculate the focal length.

Understanding the concept of focal length helps in predicting how the mirror will form images and the type of images it will create.
Image Formation
Image formation in concave mirrors involves understanding how light converges to create an image. When you place an object at different distances from a concave mirror, it reflects light and forms images at various positions and with different characteristics.

There are a few key scenarios to consider:
  • **Beyond Focal Length:** If an object is placed beyond the focal length, the mirror forms a real and inverted image. This happens because light rays actually meet at a point.
  • **Inside the Focal Length:** If the object is positioned inside the focal length, the image is virtual, upright, and larger.
  • **At the Focal Point:** No distinct image is formed when an object is placed exactly at the focal point, as reflected rays travel parallel to each other.
Knowing these scenarios helps in predicting the nature and orientation of the image produced by concave mirrors under various setups.
Mirror Equation
The mirror equation is a fundamental tool for determining the relationships between object and image distances, and the focal length of a mirror. The equation is expressed as follows:

\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]

In this equation:
  • \( f \) is the focal length of the mirror.
  • \( d_o \) represents the object distance from the mirror.
  • \( d_i \) denotes the image distance from the mirror.
You can manipulate this equation to find any of these parameters as long as you know the other two.

Using the mirror equation allows you to determine exactly where the image will form relative to the mirror. It's crucial for calculations in optics, especially when dealing with image height and orientation.

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Most popular questions from this chapter

(I) How long does it take light to reach us from the Sun, \(1.50 \times 10^{8} \mathrm{km}\) away?

(II) A laser beam of diameter \(d_{1}=3.0 \mathrm{mm}\) in air has an incident angle \(\theta_{1}=25^{\circ}\) at a flat air-glass surface. If the index of refraction of the glass is \(n=1.5,\) determine the diameter \(d_{2}\) of the beam after it enters the glass.

A slab of thickness \(D,\) whose two faces are parallel, has index of refraction \(n .\) A ray of light incident from air onto one face of the slab at incident angle \(\theta_{1}\) splits into two rays A and B. Ray A reflects directly back into the air, while \(B\) travels a total distance \(\ell\) within the slab before reemerging from the slab's face a distance \(d\) from its point of entry (Fig. \(60 ) .(a)\) Derive expressions for \(\ell\) and \(d\) in terms of \(D, n,\) and \(\theta_{1},\) (b) For normal incidence (i.e., \(\theta_{1}=0^{\circ} )\) show that your expressions yield the expected values for \(\ell\) and \(d\)

Two identical concave mirrors are set facing each other \(1.0 \mathrm{~m}\) apart. A small lightbulb is placed halfway between the mirrors. A small piece of paper placed just to the left of the bulb prevents light from the bulb from directly shining on the left mirror, but light reflected from the right mirror still reaches the left mirror. A good image of the bulb appears on the left side of the piece of paper. What is the focal length of the mirrors?

(I) Rays of the Sun are seen to make a \(33.0^{\circ}\) angle to the vertical beneath the water. At what angle above the horizon is the Sun?
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