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33. Geometric Optics

Mirror Equation

Physics

33. Geometric Optics

Mirror Equation

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6:41 minutes

Problem 52

Textbook Question

(a) In order to measure distances with parallax at 100 ly, what minimum angular resolution (in degrees) is needed?
(b) What diameter mirror or lens would be needed?

Verified step by step guidance

To solve part (a), recall that parallax is the apparent shift in the position of a nearby star against the background of distant stars due to Earth's orbit around the Sun. The parallax angle \( \theta \) (in radians) is related to the distance \( d \) (in parsecs) by the formula \( \theta = \frac{1}{d} \). Convert the distance from light-years to parsecs using the conversion factor: \( 1 \, \text{parsec} = 3.26 \, \text{light-years} \).

Next, calculate the parallax angle \( \theta \) in radians for a distance of 100 light-years. Use the formula \( \theta = \frac{1}{d} \), where \( d \) is the distance in parsecs. Then, convert the angle from radians to degrees using the conversion factor \( 1 \, \text{radian} = 57.3 \, \text{degrees} \). This gives the minimum angular resolution needed in degrees.

For part (b), use the Rayleigh criterion to determine the diameter of the mirror or lens. The angular resolution \( \theta \) (in radians) is related to the wavelength \( \lambda \) of light and the diameter \( D \) of the aperture by the formula \( \theta = 1.22 \frac{\lambda}{D} \). Rearrange this formula to solve for \( D \): \( D = 1.22 \frac{\lambda}{\theta} \).

Assume a typical wavelength of visible light, such as \( \lambda = 550 \, \text{nm} \) (nanometers). Convert this wavelength to meters: \( 1 \, \text{nm} = 10^{-9} \, \text{m} \). Substitute the value of \( \lambda \) and the angular resolution \( \theta \) (in radians, calculated in part (a)) into the formula for \( D \).

Finally, calculate the required diameter \( D \) of the mirror or lens in meters. Ensure all units are consistent throughout the calculation, and express the result in meters or another appropriate unit of length.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parallax

Parallax is the apparent shift in position of an object when viewed from different angles. In astronomy, it is used to measure distances to nearby stars by observing their position against distant background stars at different times of the year. The parallax angle is the angle subtended at the star by the baseline of the Earth's orbit, and it is inversely proportional to the distance to the star.

Angular Resolution

Angular resolution refers to the smallest angle over which details can be distinguished in an image. It is crucial in astronomy for determining how closely two objects can be positioned before they appear as a single point of light. The minimum angular resolution required for a given distance can be calculated using the formula θ = d / D, where θ is the angular resolution in radians, d is the distance to the object, and D is the diameter of the lens or mirror.

Diameter of Lens or Mirror

The diameter of a lens or mirror directly affects its ability to resolve fine details and gather light. A larger diameter allows for better angular resolution and greater light-gathering power, which is essential for observing faint objects in the universe. The relationship between diameter and resolution can be quantified using the Rayleigh criterion, which states that the minimum resolvable angle is inversely proportional to the diameter of the aperture.

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Related Practice

Textbook Question

A 2.0-cm-tall object is placed in front of a mirror. A 1.0-cm-tall upright image is formed behind the mirror, 150 cm from the object. What is the focal length of the mirror?

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Textbook Question

The place you get your hair cut has two nearly parallel mirrors 5.0 m apart. As you sit in the chair, your head is 2.0 m from the nearer mirror. Looking toward this mirror, you first see your face and then, farther away, the back of your head. (The mirrors need to be slightly nonparallel for you to be able to see the back of your head, but you can treat them as parallel in this problem.) How far away does the back of your head appear to be? Neglect the thickness of your head.

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Textbook Question

BIO A keratometer is an optical device used to measure the radius of curvature of the eye's cornea—its entrance surface. This measurement is especially important when fitting contact lenses, which must match the cornea's curvature. Most light incident on the eye is transmitted into the eye, but some light reflects from the cornea, which, due to its curvature, acts like a convex mirror. The keratometer places a small, illuminated ring of known diameter 7.5 cm in front of the eye. The optometrist, using an eyepiece, looks through the center of this ring and sees a small virtual image of the ring that appears to be behind the cornea. The optometrist uses a scale inside the eyepiece to measure the diameter of the image and calculate its magnification. Suppose the optometrist finds that the magnification for one patient is 0.049. What is the absolute value of the radius of curvature of her cornea?

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Textbook Question

(II) You are standing 2.7 m from a convex security mirror in a store. You estimate the height of your image to be half what it would be in a plane mirror at the same place. Estimate the radius of curvature of the mirror.

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Textbook Question

Suppose the mirrors in a Michelson interferometer are perfectly aligned and the path lengths to mirrors M₁ and M₂ are identical. With these initial conditions, an observer sees a bright maximum at the center of the viewing area. Now one of the mirrors is moved a distance x. Determine a formula for the intensity at the center of the viewing area as a function of x, the distance the movable mirror is moved from the initial position.

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Multiple Choice

A 4 cm tall object is placed in 15 cm front of a concave mirror with a focal length of 5 cm. Where is the image produced? Is this image real or virtual? Is it upright or inverted? What is the height of the image?

You want to produce a mirror that can produce an upright image that would be twice as tall as the object when placed 5 cm in front of it. What shape should this mirror be? What radius of curvature should the mirror have?