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Compound Microscope

Compound Microscope :- A microscope made by combining two or more lenses to magnify extremely small objects is called a compound microscope. In this, the effect (magnification) of one lens is further increased by the other lens.

Structure of Compound Microscope

In this, a convex lens (O) of short focal length and small aperture is fitted at one end of a cylindrical metal tube, which is called objective lens, which is towards the object. At the other end of this tube, another larger tube is attached, on the outer end of which another convex lens (E) is attached, whose focal length and aperture are greater than those of the objective lens. This lens is close to the eye, hence it is called eye piece. The distance between these two lenses can be increased or decreased by sliding the smaller tube inside the larger tube through the rack and pinion arrangement.

Ray Diagram of Compound Microscope

Working of Compound Microscope

The object AB is placed in front of the objective lens a little beyond its focus. The rays emanating from the object are refracted by the objective lens to form real, inverted and magnified images A’B’. This image acts as an object for the eye lens.

The distance between the two lenses is adjusted using the rack and pinion arrangement in such a way that the image A’B’ is formed between the focus (F_E) and the optical center (O₂) of the eye lens. Now the eye lens acts like a simple microscope and forms a virtual, highly magnified image A”B” of the intermediate image A’B’. This final image A”B” is formed inverted with respect to the original object AB.

Magnifying Power of Compound Microscope

The magnifying power of a compound microscope is defined by the ratio of the angle subtended by the final image A”B” at eye (β) to the angle subtended by the object AB at eye (α) when placed at the least distance of distinct vision (D). Therefore, the magnifying power of a compound microscope

m = angle subtended by the final image A”B” at eye (β)/angle subtended by the object AB at the eye (α) when placed at a distance D

$\displaystyle m=\frac{\beta }{\alpha }$

The angles β and α are very small, hence $\displaystyle \beta \approx \text{tan}\beta ,\text{ }\alpha \approx \text{tan}\alpha$

$\displaystyle \Rightarrow m=\frac{\text{tan}\beta }{\text{tan}\alpha }$ …..(1)

In the triangle A’B’O₂ in the figure,

$\displaystyle \text{tan}\beta =\frac{A'B'}{{{O}_{2}}B'}$ …..(2)

Similarly in triangle ABE,

$\displaystyle \text{tan}\alpha =\frac{AB}{EB}$ …..(3)

Putting the values of equations (2) and (3) in equation (1),

$\displaystyle m=\frac{{}^{A'B'}\!\!\diagup\!\!{}_{{{O}_{2}}B'}\;}{{}^{AB}\!\!\diagup\!\!{}_{EB}\;}$

$\displaystyle \Rightarrow m=\frac{A'B'}{{{O}_{2}}B'}\times \frac{EB}{AB}$

$\displaystyle \Rightarrow m=\frac{A'B'}{AB}\times \frac{EB}{{{O}_{2}}B'}$ …..(4)

Triangles ABO₁ and triangle A’B’O₁ are similar, hence

$\displaystyle \frac{A'B'}{AB}=\frac{{{O}_{1}}B'}{{{O}_{1}}B}$ …..(5)

Substituting the value of equation (5) into equation (4), we get

$\displaystyle m=\frac{{{O}_{1}}B'}{{{O}_{1}}B}\times \frac{EB}{{{O}_{2}}{B}'}$ …..(6)

Using sign convention,

O₁B’ = +v_o

O₁B = -u_o

EB = -D

O₂B’ = -u_E

Putting all these values in equation (6),

$\displaystyle m=\frac{\left( +{{v}_{o}} \right)}{\left( -{{u}_{o}} \right)}\times \frac{\left( -D \right)}{\left( -{{u}_{E}} \right)}$

$\displaystyle \Rightarrow m=-\frac{{{v}_{o}}}{{{u}_{o}}}\times \frac{D}{{{u}_{E}}}$ …..(7)

Equation (7) is the general formula for the magnifying power of a compound microscope. For different values of u_E, different values of magnifying power will be obtained. Let us now find the value of magnifying power in some special cases.

Special Cases

(1) When the final image is formed at the least distance of distinct vision (D)

For the eye lens, when the final image is formed at D,

u = -u_E

v = v_E = -D

f = +f_E

Using lens formula on the eye lens,

$\displaystyle \frac{\text{1}}{f}=\frac{1}{v}-\frac{1}{u}$

$\displaystyle \Rightarrow \frac{\text{1}}{\left( +{{f}_{E}} \right)}=\frac{1}{\left( -D \right)}-\frac{1}{\left( -{{u}_{E}} \right)}$

$\displaystyle \Rightarrow \frac{1}{{{u}_{E}}}=\frac{1}{{{f}_{E}}}+\frac{1}{D}$

$\displaystyle \Rightarrow \frac{D}{{{u}_{E}}}=\frac{D}{{{f}_{E}}}+\frac{D}{D}$

$\displaystyle \Rightarrow \frac{D}{{{u}_{E}}}=1+\frac{D}{{{f}_{E}}}$

Putting this value in equation (7),

$\displaystyle m=-\frac{{{v}_{o}}}{{{u}_{o}}}\left[ 1+\frac{D}{{{f}_{E}}} \right]$ …..(8)

Here the negative sign indicates that the final image (A”B”) is inverted with respect to the object (AB). In this situation the magnifying power of the compound microscope reaches its maximum.

Length of the tube (The distance between the optical center O₁ of the objective lens and the optical center O₂ of the eye lens is called the length of the tube) :-

$\displaystyle L={{v}_{o}}+\left| {{u}_{E}} \right|$ …..(9)

While determining the length of the tube in equation (9), the value of u_E is to be taken positive.

(2) When the final image is formed at infinity (∞)

For the final image to be formed at infinity, the position of the image A’B’ in front of the eye lens (E) should be at its focus point (f_E). To do this, the eyepiece is moved back a little using the rack and pinion arrangement, so that the image A’B’ comes at the focus (f_E) of the eyepiece. Putting u_E = f_E in equation (7),

$\displaystyle m=-\frac{{{v}_{o}}}{{{u}_{o}}}\times \frac{D}{{{f}_{E}}}$ …..(10)

Here the value of u_E is not taken negative because while deriving equation (7) we have already kept the negative sign for u_E.

Length of the tube:-

$\displaystyle L={{v}_{o}}+{{f}_{E}}$ …..(11)

Note :-

(1). In equations (7), (8) and (10) the negative sign indicates that the final image A”B” is inverted with respect to the object AB.

(2). The aperture of the objective lens should be small and the aperture of the eye lens should be large.

(3). For higher magnification, from equations (8) and (10) : –

The value of u_o should be less and for this to happen it is necessary that the focal length f_o of the objective lens should be less.
The focal length (f_E) of the eye lens should be short.
The distance of image A’B’ from the objective lens (O) should be greater. For this, the object AB should be kept near the objective lens i.e. the value of u_o should be less.

Compound Microscope

Structure of Compound Microscope

Ray Diagram of Compound Microscope

Working of Compound Microscope

Magnifying Power of Compound Microscope

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