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Chapter 2: Problem 9

A meter stick lies along the optical axis of a convex mirror of focal length \(40 \mathrm{cm},\) with its nearer end \(60 \mathrm{cm}\) from the mirror surface. How long is the image of the meter stick?

Short Answer

Expert verified
The image length is approximately 66.67 cm.

Step by step solution

01

Identify the Problem

We have a meter stick placed along the optical axis of a convex mirror. The convex mirror has a focal length of \( f = -40 \) cm (negative because it's a convex mirror) and the nearer end of the stick is \( 60 \) cm from the surface of the mirror. We need to find the length of the image of the meter stick.
02

Calculate Image Position of Nearer End

Using the mirror formula \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \( f = -40 \) cm and \( u = -60 \) cm (the negative sign indicates the object distance is against the direction of the light), solve for \( v \), the image distance for the nearer end:\[\frac{1}{-40} = \frac{1}{v} + \frac{1}{-60}\]Simplifying, we find:\[\frac{1}{v} = \frac{1}{-40} - \frac{1}{-60} = \frac{-3 + 2}{120} = -\frac{1}{120}\]Thus, \( v = -120 \) cm.
03

Calculate Image Position of Farther End

The farther end of the meter stick is \( 100 \) cm away from the nearer end, making it \( 60 + 100 = 160 \) cm from the mirror. Using the mirror formula again for this distance:\[\frac{1}{-40} = \frac{1}{v_2} + \frac{1}{-160}\]Solving for \( v_2 \), we get:\[\frac{1}{v_2} = \frac{1}{-40} - \frac{1}{-160} = \frac{-4 + 1}{160} = -\frac{3}{160}\]Thus, \( v_2 = -\frac{160}{3} \approx -53.33 \) cm.
04

Calculate the Length of the Image

The image of the meter stick will be the difference between the image position of the nearer end and the farther end:\[|v_2 - v_1| = |-53.33 - (-120)| = |_53.33 - 120| = |66.67| = 66.67 \, \text{cm}\]
05

Conclusion

The length of the image of the meter stick is approximately \( 66.67 \) cm, which means the image is reduced in size because of the nature of the convex mirror.

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Key Concepts

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

Convex Mirror
Convex mirrors are commonly found in everyday applications such as rearview mirrors on cars and in hallways for security purposes. Their defining feature is their outward-curved reflective surface. This curvature spreads light rays out, causing them to diverge.
  • This divergent behavior means that convex mirrors create images that are always virtual, upright, and smaller than the object.
  • These images appear to be located behind the mirror.

The nature of these mirrors helps in viewing a larger field of view. Hence, they are ideal for situations where a broad perspective is needed.
Mirror Formula
The mirror formula is a fundamental equation in geometrical optics, used to relate object distance, image distance, and focal length for mirrors. It is expressed as: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] where:
  • \( f \) is the focal length of the mirror.
  • \( v \) is the image distance from the mirror.
  • \( u \) is the object distance from the mirror.

In the context of convex mirrors, both the focal length \( f \) and the image distance \( v \) are negative. This is because convex mirrors always form virtual, erect images behind the mirror. The negative sign in the mirror formula plays a crucial role in understanding the nature of the image.
However, when applying this formula, you need to pay close attention to sign conventions, which indicate the direction of the distances involved.
Image Formation
Image formation in convex mirrors is a fascinating process due to the mirror's unique characteristics. Since these mirrors cause divergence of light rays, they always form virtual images.
When you draw ray diagrams:
  • You'll notice that light rays reflecting off a convex mirror appear to come from a point behind the mirror.
  • The image is always diminished, making the object appear smaller.


    • In practical terms, this means that the height and dimensions of the image are less than those of the actual object. Hence, in our example, even though the meter stick is 100 cm long, the image created is only about 66.67 cm.
    This reduction in size helps drivers judge the distance and speed of cars behind them, crucial for safety.
Focal Length
The focal length of a mirror determines how it converges or diverges incoming light rays. In the case of a convex mirror, the focal length is always considered negative due to the diverging nature of the mirror.
Given that the focal point of a convex mirror is not real, licht cannot actually converge at this point. Instead, the focal length (measured from the mirror's surface to the focal point) serves as an important parameter for calculations.
  • Focal length helps derive key properties like image size and distance using the mirror formula.
  • In the provided exercise, the focal length of \(-40\) cm is essential in determining the image size and position.
The negative sign simply reaffirms that the mirror's curvature redirects light rays outward, confirming its diverging nature.

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Most popular questions from this chapter

Light rays emanating in air from a point object on axis strike a plano- cylindrical lens with its convex surface facing the object. Describe the line image by length and location if the lens has a radius of curvature of \(5 \mathrm{cm},\) a refractive index of \(1.60,\) and an axial length of \(7 \mathrm{cm} .\) The point object is \(15 \mathrm{cm}\) from the lens

A ray of light makes an angle of incidence of \(45^{\circ}\) at the center of the top surface of a transparent cube of index 1.414 Trace the ray through the cube.

A small source of light at the bottom face of a rectangular glass slab \(2.25 \mathrm{cm}\) thick is viewed from above. Rays of light totally internally reflected at the top surface outline a circle of \(7.60 \mathrm{cm}\) in diameter on the bottom surface. Determine the refractive index of the glass.

A parallel beam of light is incident on a plano-convex lens that is \(4 \mathrm{cm}\) thick. The radius of curvature of the spherical side is also \(4 \mathrm{cm} .\) The lens has a refractive index of 1.50 and is used in air. Determine where the light is focused for light incident on each side.

To determine the refractive index of a transparent plate of glass, a microscope is first focused on a tiny scratch in the upper surface, and the barrel position is recorded. Upon further lowering the microscope barrel by \(1.87 \mathrm{mm},\) a focused image of the scratch is seen again. The plate thickness is \(1.50 \mathrm{mm} .\) What is the reason for the second image, and what is the refractive index of the glass?
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