# 2.2 Spherical mirrors  (Page 5/20)

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${d}_{\text{o}}=\text{−}{d}_{\text{i}}$

which is the same as [link] obtained earlier.

Notice that we have been very careful with the signs in deriving the mirror equation. For a plane mirror, the image distance has the opposite sign of the object distance. Also, the real image formed by the concave mirror in [link] is on the opposite side of the optical axis with respect to the object. In this case, the image height should have the opposite sign of the object height. To keep track of the signs of the various quantities in the mirror equation, we now introduce a sign convention.

## Sign convention for spherical mirrors

Using a consistent sign convention is very important in geometric optics. It assigns positive or negative values for the quantities that characterize an optical system. Understanding the sign convention allows you to describe an image without constructing a ray diagram. This text uses the following sign convention:

1. The focal length f is positive for concave mirrors and negative for convex mirrors.
2. The image distance ${d}_{\text{i}}$ is positive for real images and negative for virtual images.

Notice that rule 1 means that the radius of curvature of a spherical mirror can be positive or negative. What does it mean to have a negative radius of curvature? This means simply that the radius of curvature for a convex mirror is defined to be negative.

## Image magnification

Let’s use the sign convention to further interpret the derivation of the mirror equation. In deriving this equation, we found that the object and image heights are related by

$-\frac{{h}_{\text{o}}}{{h}_{\text{i}}}=\frac{{d}_{\text{o}}}{{d}_{\text{i}}}.$

See [link] . Both the object and the image formed by the mirror in [link] are real, so the object and image distances are both positive. The highest point of the object is above the optical axis, so the object height is positive. The image, however, is below the optical axis, so the image height is negative. Thus, this sign convention is consistent with our derivation of the mirror equation.

[link] in fact describes the linear magnification    (often simply called “magnification”) of the image in terms of the object and image distances. We thus define the dimensionless magnification m as follows:

$m=\frac{{h}_{\text{i}}}{{h}_{\text{o}}}.$

If m is positive, the image is upright, and if m is negative, the image is inverted. If $|m|>1$ , the image is larger than the object, and if $|m|<1$ , the image is smaller than the object. With this definition of magnification, we get the following relation between the vertical and horizontal object and image distances:

$m=\frac{{h}_{\text{i}}}{{h}_{\text{o}}}=\text{−}\frac{{d}_{\text{o}}}{{d}_{\text{i}}}.$

This is a very useful relation because it lets you obtain the magnification of the image from the object and image distances, which you can obtain from the mirror equation.

## Solar electric generating system

One of the solar technologies used today for generating electricity involves a device (called a parabolic trough or concentrating collector) that concentrates sunlight onto a blackened pipe that contains a fluid. This heated fluid is pumped to a heat exchanger, where the thermal energy is transferred to another system that is used to generate steam and eventually generates electricity through a conventional steam cycle. [link] shows such a working system in southern California. The real mirror is a parabolic cylinder with its focus located at the pipe; however, we can approximate the mirror as exactly one-quarter of a circular cylinder.

#### Questions & Answers

what is bohrs model for hydrogen atom
Swagatika Reply
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
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Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
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Sivalakshmi Reply
what is energy
Isiguzo Reply
কাজের একক কী
Jasim
কাজের একক কী
Jasim
friction ka direction Kaise pata karte hai
Rahul Reply
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
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b) symmetric of time only
Swagatika
fundamental note of a vibrating string
fasoyin Reply
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
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he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
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it is key of quantum
Manikant
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Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
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de broglie wave equation
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when you consider systems consisting of fixed charges
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