Question

Use the thin-lens equation to show that the transverse magnification due to the objective of a microscope is \(m_{o}=-U f_{o} .\) [Hints: The object is near the focal point of the objective: do not assume it is at the focal point. Eliminate \(p_{0}\) to find the magnification in terms of \(q_{0}\) and $\left.f_{0} . \text { How is } L \text { related to } q_{0} \text { and } f_{0} ?\right].$

Short answer

Answer: The transverse magnification due to the objective of a microscope is given by \(m_o = -Uf_o\), where \(U = \frac{L-f_e}{f_o}\), \(L\) is the tube length, and \(f_o\) and \(f_e\) are the focal lengths of the objective and eyepiece, respectively.

Step-by-step solution

Define the thin-lens equation

The thin-lens equation is given by \(\frac{1}{f} = \frac{1}{p} + \frac{1}{q}\), where \(f\) is the focal length, \(p\) is the object distance, and \(q\) is the image distance.

Apply the lens equation to the microscope objective

For the microscope objective, we can write the lens equation as \(\frac{1}{f_o} = \frac{1}{p_o} + \frac{1}{q_o}\), where \(f_o\) is the objective focal length, \(p_o\) is the object distance, and \(q_o\) is the image distance.

Rearrange the equation to eliminate \(p_o\)

To eliminate \(p_o\), we can rearrange the equation as \(p_o = \frac{f_o q_o}{q_o - f_o}\).

Calculate the transverse magnification of the objective

The transverse magnification formula is \(m_o = \frac{-q_o}{p_o}\). Using the rearranged equation from Step 3, we get \(m_o = \frac{-q_o}{\frac{f_o q_o}{q_o - f_o}} = - \frac{q_o (q_o - f_o)}{f_o q_o}= -\frac{q_o - f_o}{f_o}\).

Find the relationship between \(L\), \(q_o\), and \(f_o\)

To find the relationship between \(L\), \(q_o\), and \(f_o\), we need to consider the geometry of the microscope. The tube length \(L\) is the distance between the objective and eyepiece focal points. As the object is near the focal point of the objective, we have \(L \approx q_o + f_e - f_o\). We can rearrange this expression for \(q_o\) as \(q_o = L + f_o - f_e\), where \(f_e\) is the focal length of the eyepiece.

Substitute the relationship into the magnification formula

Substituting the relationship from Step 5 into the magnification formula in Step 4, we have \(m_o = -\frac{(L + f_o - f_e) - f_o}{f_o} = -\frac{L-f_e}{f_o}\).

Show the final expression for the transverse magnification

To express the transverse magnification due to the objective of a microscope, we can now write \(m_o = -Uf_o\), where \(U = \frac{L-f_e}{f_o}\).