Magnifying power of compound microscope, when image is formed at least distance of distinct vision

In compound microscope, the magnification is 95 to least distance of distinct vision and the distance of object from objective lens is (1/3.8) cm and focal length of objective is (1/4) cm. What is the magnification of eyepiece when final image is formed at least distance of distinct vision? A:5 B:10 C:100 D:none

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Hint: In this question we have been asked to draw a labelled ray diagram of image formation of a compound microscope. We have also been asked to calculate the magnifying power when the final image is formed at least distance of distinct vision. Therefore, we shall first draw a labelled ray diagram. We shall use the magnification formula and the lens formula to calculate the magnifying power.
Formula Used: \[\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}\]

Complete answer:

The compound microscope consists of two lenses as shown in the figure below. The first lens forms an inverted image as shown. The final image is observed through the ocular lens also known as eye piece. The following ray diagram shows the formation of an image by a compound microscope.

The magnifying power of a microscope is the ratio of angle subtended by image to the angle subtended by image placed at least distance of distinct vision. Therefore, from above diagram,\[m=\dfrac{\beta }{\alpha }\] …………….. (1)\[\therefore \alpha =\dfrac{AB}{D}\] ……………… (2)Where, D is the least distance of distinct vision i.e. distance between point O and Point BNow, angle made by image PQ can be given by,\[\beta =\dfrac{PQ}{EQ}=\dfrac{A'B'}{EB'}\]Let, \[{{u}_{e}}\] be the distance between image and eyes pieceTherefore,\[\beta =\dfrac{A'B'}{{{u}_{e}}}\] ……………. (3)Now, from (1), (2) and (3)We get,\[m=\dfrac{A'B'}{AB}\times \dfrac{D}{{{u}_{e}}}\]From the diagram above we can say that,\[\dfrac{A'B'}{AB}=\dfrac{OB'}{OB}\]Therefore,\[m=\dfrac{OB'}{OB}\times \dfrac{D}{{{u}_{e}}}\]OB and OB’ is the distance of object and image from point OTherefore,\[m=\dfrac{{{v}_{o}}}{-{{u}_{o}}}\times \left( \dfrac{-D}{-{{u}_{e}}} \right)\] …………………. (4)On solving,\[m=\dfrac{{{v}_{o}}D}{{{u}_{o}}{{u}_{e}}}\]Now, from lens formula we know,\[\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}\]The lens formula for given microscope can be written by,\[\dfrac{1}{{{f}_{e}}}=\dfrac{1}{-{{v}_{e}}}-\dfrac{1}{-{{u}_{e}}}\]We know that \[{{v}_{e}}\] is the distance DTherefore,\[\dfrac{1}{{{f}_{e}}}=\dfrac{1}{-D}+\dfrac{1}{{{u}_{e}}}\]Rearranging the termsWe get,\[\dfrac{1}{{{u}_{e}}}=\dfrac{1}{{{f}_{e}}}+\dfrac{1}{D}\]On solving,\[\dfrac{D}{{{u}_{e}}}=\dfrac{D}{{{f}_{e}}}+1\] …………….. (5)Now, from (4) and (5)We get,\[m=\dfrac{-{{v}_{o}}}{{{u}_{o}}}\left( 1+\dfrac{D}{{{f}_{e}}} \right)\]Therefore, the magnification power of compound microscope is given by, \[m=\dfrac{-{{v}_{o}}}{{{u}_{o}}}\left( 1+\dfrac{D}{{{f}_{e}}} \right)\].

Note:

The optical instrument that uses two lenses to obtain higher magnification of an image is known as compound microscope. A compound microscope provides two dimensional images. Magnification is the property by which the image size is enlarged. The compound microscopes are therefore used to view specimens as small as blood cells. The name compound microscope comes from the lens system of the microscope, that consists of a primary objective lens compounded with eyepiece or ocular lens.

With the help of a ray diagram obtain the expression for the magnifying power of a simple microscope when the image is formed at the least distance of distinct vision. 

Image is formed at the near point.

Magnifying Power (Angular Magnification). Magnifying power (M) is the ratio of the angle subtended at the eye by the image at the near-point, to the angle subtended at an unaided eye by the object, at the least distance of distinct vision.

In figure the image is formed at the near point. Let the angle subtended by the image at the near point at the eye be β. If the object is kept at the near point [marked as IE in figure], then the angle subtended by the object at the eye is,

Hence when the simple microscope is adjusted such that the image is formed at the near point, the angular magnification is equal to the linear magnification.Applying new cartesian sign convention, u is -ve, v is -ve and f is +ve.

The lens equation is, 

This is the required expression for magnification.