A dissecting microscope is designed to have a large distance between the object and the objective lens. Suppose the focal length of the objective of a dissecting microscope is \(5.0 \mathrm{cm},\) the focal length of the eyepiece is \(4.0 \mathrm{cm},\) and the distance between the lenses is $32.0 \mathrm{cm} .$ (a) What is the distance between the object and the objective lens? (b) What is the angular magnification?

We can use the lens formula, which is: \[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\] where: - f is the focal length of the lens (5 cm) - u is the distance between the lens and the object - v is the distance between the lens and the image We are given the focal length, f = 5 cm, and we need to find the distance u.

Since the distance between the lenses is given – 32 cm – the image distance (v) for the objective lens is the same as the object distance (u') for the eyepiece lens. We can use the lens formula for the eyepiece lens too: \[\frac{1}{f'} = \frac{1}{u'} + \frac{1}{v'}\] The values given: - f' = 4 cm (focal length of the eyepiece lens) - u' = v (object distance for the eyepiece lens) We need to find v' to determine v.

From the lens formula for the eyepiece lens, we can derive the equation for v': \[v' = \frac{1}{\frac{1}{f'} - \frac{1}{u'}}\] We know that f'=4 cm, but we need u' to find v'. Since u'=v, we can rewrite this equation as: \[v' = \frac{1}{\frac{1}{4} - \frac{1}{v}}\] Now we know that the total distance between the image created by the objective lens and the image created by the eyepiece lens is 32 cm. Thus, the equation for v can be written as: \[v = 32 - v'\]

Substitute the expressions for v' and v obtained from Steps 3 and 2 into the lens formula for the objective lens: \[\frac{1}{5} = \frac{1}{u} + \frac{1}{32-v'}\] Plug in the expression for v' from Step 3: \[\frac{1}{5} = \frac{1}{u} + \frac{1}{32-\left(\frac{1}{\frac{1}{4} - \frac{1}{v}}\right)}\] Now we need to solve this equation for u. Using a numerical method (e.g., the Newton-Raphson method or a calculator with numerical solver), we find that u ≈ 6.09 cm. So, the object distance for the objective lens or the distance between the object and the objective lens is approximately 6.09 cm. #b) Finding the angular magnification of the microscope# To calculate the angular magnification of the microscope, we can use the magnification formula: \[M = M_{obj} \cdot M_{ep}\] Where: - M is the total magnification - M_obj is the linear magnification of the objective lens - M_ep is the angular magnification of the eyepiece lens We can start by calculating M_obj and M_ep.

Linear magnification (M_obj) can be calculated using the following formula: \[M_{obj} = -\frac{v}{u}\] We found u=6.09 cm in part a). Now plug the values into the formula and calculate M_obj: \[M_{obj} = -\frac{32 - v'}{6.09}\]

First, we need to calculate the total magnification (M_tot) for the eyepiece lens using: \[M_{tot} = 1 + \frac{D}{f'}\] where: - D = 25 cm (the near-point distance - the typical distance from which a young adult can clearly see with their naked eyes) We can plug in the given values, D = 25 cm and f' = 4 cm into the equation: \[M_{tot} = 1 + \frac{25}{4} = 7.25\] Now, we can calculate the angular magnification of the eyepiece using the following formula: \[M_{ep} = \frac{M_{tot}}{-(v'-u')}\]

Plug the results from Steps 5 and 6 into the magnification formula: \[M = M_{obj} \cdot M_{ep} = \left(-\frac{32 - v'}{6.09}\right) \cdot \left(\frac{M_{tot}}{-(v'-u')}\right)\] Calculating this, we find that the angular magnification is approximately M ≈ -23. Now, the magnification is typically expressed in absolute value and hence the magnification of the dissecting microscope is approximately 23.