Microscopes are ubiquitous in science and engineering. You may have used a compound microscope, like the one in FIGURE 35.13, in a laboratory course. The essential idea is that a microscope can achieve a magnification up to $1000\times$ by a two-step magnification process.

FIGURE 35.13 A compound microscope.

FIGURE 35.14 is a two-lens model of a microscope. The object is located just slightly outside the focal point of the objective, a converging lens with a short focal length. The objective lens creates a magnified real image inside the microscope tube. That real image is then observed and further magnified by the eyepiece, which functions as a magnifier to produce a distant virtual image that is viewed by a relaxed eye. The overall magnification of a microscope, an angular magnification $M$, is the product of the lateral magnification of the objective and the angular magnification of the eyepiece:

FIGURE 35.14 The optics of a microscope.

The angular magnification of the eyepiece is given by Equation 35.9: $M_{\text{eye}}=(25\text{cm})\text{/}{f}_{\text{eye}}$. The most common microscope eyepiece is a $10\times$ magnifier, but $5\times$ and $20\times$ eyepieces are sometimes used.

We can find an approximate expression for the magnification of the objective by considering the two shaded triangles in the figure. The triangle on the left has height $h$, the object height, and width $f_{\text{obj}}$. The triangle on the right has height $h\prime$, the image height, and width $l$, the distance between the focal point and the image. These are similar triangles because they have the same angles, so $h\prime \text{/}h=l\text{/}{f}_{\text{obj}}$. But $h\prime \text{/}h$, the ratio of image height to object height, is $\left|m\right|$, the absolute value of the lateral magnification. We need to include a negative sign to show that the image is inverted, so the lateral magnification of the objective lens is

Unfortunately, there’s no easy way to determine $l$, and it varies from one objective lens to another.

One thing that doesn’t change, though, is the physical length of the microscope tube measured between the flanges at the ends where the objective and eyepiece are screwed in. This is called the tube length L. Most biological microscopes are standardized with tube length $L=160\text{mm}$. It turns out that length $l$ in Equation 35.11 differs from the tube length $L$ by only a few percent; thus $m_{\text{obj}}\approx -L\text{/}{f}_{\text{obj}}$ is a good approximation for the magnification of the objective. We can use this to see that a $20\times$ objective has a focal length $f_{\text{obj}}\approx (160\text{mm})\text{/}{m}_{\text{obj}}=8\text{mm}$.

In practice, the true magnifications of the objective (without the minus sign) and the eyepiece are stamped on the barrels, and the microscope’s overall magnification is the product of the two. A set of objectives on a rotating turret might include $10\times$, $20\times$, $40\times$, and $100\times$. When combined with a $10\times$ eyepiece, the microscope’s angular magnification ranges from $100\times$ to $1000\times$.

EXAMPLE 35.6 Viewing blood cells

A pathologist inspects a sample of $7\text{-}\mu \text{m}\text{-diameter}$ human blood cells under a microscope. She selects a $40\times$ objective and a $10\times$ eyepiece. What size object, viewed from 25 cm, has the same apparent size as a blood cell seen through the microscope?

MODEL Angular magnification compares the magnified angular size to the angular size seen at the near-point distance of 25 cm.

SOLVE The microscope’s angular magnification is $M=-\left(40\right)\times$ $\left(10\right)=-400$. The magnified cells will have the same apparent size as an object $400\times 7\mu \text{m}\approx 3\text{mm}$ in diameter seen from a distance of 25 cm.

REVIEW 3 mm is about the size of a capital O in a printed book, so a blood cell seen through the microscope will have about the same apparent size as an O seen from a comfortable reading distance.