Draw a neat, labeled ray diagram of a compound microscope showing the formation of an image at the least distance of distinct vision. Derive an expression for its magnifying power.

Q: Draw a neat, labeled ray diagram of a compound microscope showing the formation of an image at the least distance of distinct vision. Derive an expression for its magnifying power.

Introduction\nA compound microscope is an optical instrument used to observe highly magnified images of tiny objects. It consists of two convex lenses: an objective lens of short focal length and aperture, and an eyepiece of relatively larger focal length and aperture. Ray Diagram Description & Working The objective lens $O$ forms a real, inverted, and magnified image $A'B'$ of the object $AB$. This image lies close to the focus of the eyepiece. The eyepiece $E$ acts as a simple magnifying glass, taking $A'B'$ as its object and forming a virtual, magnified, and inverted final image $A''B''$ at the least distance of distinct vision ($D$). Derivation of Magnifying Power\nMagnifying power ($m$) is defined as the ratio of the angle $\beta$ subtended by the final image at the eye to the angle $\alpha$ subtended by the object when placed at the least distance of distinct vision: $$m = \frac{\beta}{\alpha} \approx \frac{\tan \beta}{\tan \alpha}$$ From the eyepiece triangle: $$\tan \beta = \frac{h'}{ue}$$ Where $h'$ is the height of image $A'B'$ and $ue$ is the object distance for the eyepiece. From the standard object position at distance $D$: $$\tan \alpha = \frac{h}{D}$\text{where } h \text{ is the object height.}$$ Substitute into magnifying power formula: $$m = \frac{h' / ue}{h / D} = \left(\frac{h'}{h}\right) \cdot \frac{D}{ue}$$ Linear magnification of objective ($mo$): $$\frac{h'}{h} = \frac{vo}{uo}$$ Therefore: $$m = \frac{vo}{uo} \cdot \frac{D}{ue}$$ Special Case (Image at Least Distance of Distinct Vision, $D$): Using lens formula for eyepiece: $\frac{1}{ve} - \frac{1}{ue} = \frac{1}{fe}$, with $ve = -D$: $$-\frac{1}{D} - \frac{1}{ue} = \frac{1}{fe} \implies \frac{D}{ue} = 1 + \frac{D}{fe}$$ Substituting this back: $$m = \frac{vo}{uo}\left(1 + \frac{D}{fe}\right)$$

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Introduction
A compound microscope is an optical instrument used to observe highly magnified images of tiny objects. It consists of two convex lenses: an objective lens of short focal length and aperture, and an eyepiece of relatively larger focal length and aperture. Ray Diagram Description & Working The objective lens $O$ forms a real, inverted, and magnified image $A'B'$ of the object $AB$. This image lies close to the focus of the eyepiece. The eyepiece $E$ acts as a simple magnifying glass, taking $A'B'$ as its object and forming a virtual, magnified, and inverted final image $A''B''$ at the least distance of distinct vision ($D$). Derivation of Magnifying Power
Magnifying power ($m$) is defined as the ratio of the angle $\beta$ subtended by the final image at the eye to the angle $\alpha$ subtended by the object when placed at the least distance of distinct vision: $$m = \frac{\beta}{\alpha} \approx \frac{	an \beta}{	an \alpha}$$ From the eyepiece triangle: $$	an \beta = \frac{h'}{ue}$$ Where $h'$ is the height of image $A'B'$ and $ue$ is the object distance for the eyepiece. From the standard object position at distance $D$: $$	an \alpha = \frac{h}{D}$	ext{where } h 	ext{ is the object height.}$$ Substitute into magnifying power formula: $$m = \frac{h' / ue}{h / D} = \left(\frac{h'}{h}ight) \cdot \frac{D}{ue}$$ Linear magnification of objective ($mo$): $$\frac{h'}{h} = \frac{vo}{uo}$$ Therefore: $$m = \frac{vo}{uo} \cdot \frac{D}{ue}$$ Special Case (Image at Least Distance of Distinct Vision, $D$): Using lens formula for eyepiece: $\frac{1}{ve} - \frac{1}{ue} = \frac{1}{fe}$, with $ve = -D$: $$-\frac{1}{D} - \frac{1}{ue} = \frac{1}{fe} \implies \frac{D}{ue} = 1 + \frac{D}{fe}$$ Substituting this back: $$m = \frac{vo}{uo}\left(1 + \frac{D}{fe}ight)$$

NCERT · Class 12 · Physics · Ray Optics and Optical InstrumentsDraw a neat, labeled ray diagram of a compound microscope showing the formation of an image at the least distance of distinct vision. Derive an expression for its magnifying power.

Introduction\nA compound microscope is an optical instrument used to observe highly magnified images of tiny objects. It consists of two convex lenses: an objective lens of short focal length and aperture, and an eyepiece of relatively larger focal length and aperture.

Ray Diagram Description & Working

Derivation of Magnifying Power\nMagnifying power ($m$) is defined as the ratio of the angle $\beta$ subtended by the final image at the eye to the angle $\alpha$ subtended by the object when placed at the least distance of distinct vision: