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In optics, the term focal length refers to the distance between the center of a lens and its focal point. A lens's ability to bend light is determined by its focal length. A short focal length indicates a more powerful lens that can bend light to a greater degree, converging rays closer to it. On the other hand, a longer focal length is linked to a lens that bends light less, converging rays at a greater distance. In our microscope example, the focal length of the objective lens is 12.0 mm. This relatively short focal length suggests the lens is quite strong, designed to produce significant magnification of the object being viewed, which is crucial for detailed examination of small samples.

The lens formula is a fundamental equation used in optics to relate the focal length (\( f \)), object distance (\( d_o \)), and image distance (\( d_i \)) in a lens system. The lens formula is expressed as:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

This equation helps us determine the position of the image formed by the lens, given the lens's focal length and the object's position. In the context of our microscope problem, we use this formula to solve for the image distance (\( d_i \)) when the object is placed at a known distance from the lens. By rearranging the formula:

• \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \)

we can substitute the given values to find that the image forms at approximately 192 mm from the objective lens.

Magnification determines how much larger (or smaller) an image appears compared to the object's actual size. For a lens, magnification (\( M \)) can be calculated with the formula:

• \( M = -\frac{d_i}{d_o} \)

where the negative sign indicates that the image formed is inverted. For the objective in the microscope, using the calculated image distance (\( d_i = 192 \) mm) and object distance (\( d_o = 12.8 \) mm), the magnification is:

• \( M_o = -\frac{192}{12.8} \approx -15 \)

This high magnification means the image appears 15 times larger than the actual size of the object, which is typical for a microscope used to study tiny details.

Angular magnification describes how much larger an object appears when viewed through an optical instrument compared to viewing it with the naked eye at the near point (usually around 25 cm for a human eye). For the eyepiece of a microscope, the angular magnification (\( M_e \)) is determined by:

• \( M_e = \frac{25}{f_e} \)

where (\( f_e \)) is the focal length of the eyepiece. Given that the eyepiece's focal length is 2.5 cm, the angular magnification is:

• \( M_e = \frac{25}{2.5} = 10 \)

To find the total angular magnification of the microscope, you multiply the magnifications of the objective and the eyepiece:

• \( M_{total} = M_o \times M_e = -15 \times 10 = -150 \)

This significant magnification allows the observer to examine tiny details of objects that would otherwise be impossible to see with the naked eye.