Explaining the Lens Equation: Why h' & h Positive, s' Negative & s Positive?

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In summary, the equation for magnification is h'/h = -s'/s, where h and h' are the horizontal and vertical distances from the lens to where the image is formed, and s is the magnification. The - sign is necessary in order that both versions of the equation yield the same result.
  • #1
negation
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The lens equation is given as M = h'/h = -s'/s

It's not explained in the book as to why h' and h are positive and s' is negative while s is positive.

Could someone explain?
 
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  • #2
negation said:
The lens equation is given as M = h'/h = -s'/s

It's not explained in the book as to why h' and h are positive and s' is negative while s is positive.

Could someone explain?

With some pictures it gets clearer.

leneq3.gif



http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html
 
  • #3
Malverin said:

I'm taking issue with the book. As the intersection of the light ray where the image formed is inverted, reduced and real, the vertical distance to the horizontal axis is labelled -h'.

But the given equation is h'/h. For some reason, the -ve sign has vanished.
 
  • #4
The equation is correct. h'/h is the definition of the Magnification M. If you plug in a positive value for h and a negative value for h' (Inverted image) you get a negative value For M. A negative magnification means you get an upside down image (inverted image).
 
  • #5
negation said:
The lens equation is given as M = h'/h = -s'/s

It's not explained in the book as to why h' and h are positive and s' is negative while s is positive

First: you need to beware that there are two different sign conventions (rules for assigning + and - signs) that are used in different optics textbooks:

  • Gaussian: Real objects and images have positive distances from the lens. Virtual objects and images have negative distances from the lens.
  • Cartesian: Objects and images on the right side of the lens have positive distances. On the left side of the lens, they have negative distances.

(In both cases we assume the light travels through the lens from left to right.)

Some equations have + and - signs placed differently, depending on which convention you're using. The magnification equation you gave above is for the Gaussian convention. However, the diagram you included in a later post shows the Cartesian convention! (as evidenced by the "Note: object distance normally negative.")

In your diagram, h is a positive number because the object arrow points upward, and h' is a negative number because the image arrow points downward. Therefore M = h'/h is a negative number, which reflects the fact that the image is inverted with respect to the object.

Using the Gaussian convention, s is a positive number because the object is real, and s' is a positive number because the image is real. Therefore M = -s'/s is a negative number, in agreement with h'/h. The - sign in the formula is necessary in order that both versions of the formula give the same result.

Using the Cartesian convention, s would be negative, s' would be positive, and the magnification equation would be M = s'/s (without the added - sign).
 
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  • #6
jtbell said:
First: you need to beware that there are two different sign conventions (rules for assigning + and - signs) that are used in different optics textbooks:

  • Gaussian: Real objects and images have positive distances from the lens. Virtual objects and images have negative distances from the lens.
  • Cartesian: Objects and images on the right side of the lens have positive distances. On the left side of the lens, they have negative distances.

(In both cases we assume the light travels through the lens from left to right.)

Some equations have + and - signs placed differently, depending on which convention you're using. The magnification equation you gave above is for the Gaussian convention. However, the diagram you included in a later post shows the Cartesian convention! (as evidenced by the "Note: object distance normally negative.")

In your diagram, h is a positive number because the object arrow points upward, and h' is a negative number because the image arrow points downward. Therefore M = h'/h is a negative number, which reflects the fact that the image is inverted with respect to the object.

Using the Gaussian convention, s is a positive number because the object is real, and s' is a positive number because the image is real. Therefore M = -s'/s is a negative number, in agreement with h'/h. The - sign in the formula is necessary in order that both versions of the formula give the same result.

Using the Cartesian convention, s would be negative, s' would be positive, and the magnification equation would be M = s'/s (without the added - sign).


I have managed to derive it.

Thanks
 

Related to Explaining the Lens Equation: Why h' & h Positive, s' Negative & s Positive?

1. Why is h' positive in the lens equation?

In the lens equation, h' represents the height of the image formed by the lens. It is positive because the image formed by the lens is always upright, meaning the height of the image will have the same direction as the object.

2. Why is h positive in the lens equation?

In the lens equation, h represents the height of the object being viewed through the lens. It is positive because the object is always assumed to be placed above the principal axis of the lens, resulting in a positive height value.

3. Why is s' negative in the lens equation?

In the lens equation, s' represents the distance of the image from the lens. It is negative because the image formed by the lens is always located on the opposite side of the lens from the object, which is considered the positive direction in the equation.

4. Why is s positive in the lens equation?

In the lens equation, s represents the distance of the object from the lens. It is positive because the object is assumed to be placed on the same side of the lens as the observer, which is considered the positive direction in the equation.

5. How does the lens equation explain the formation of images?

The lens equation, s' = (h'/h)s, relates the object distance, image distance, and the magnification of the lens. It helps to explain how light rays passing through a lens are refracted and focused to form an image of an object. The equation shows that the image distance and size are determined by the object distance and size, as well as the properties of the lens being used.

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