Problem 87 - Physics

Question

In a compound microscope, the focal length of the objective is \(3.50 \mathrm{~cm}\) and that of the eyepiece is \(6.50 \mathrm{~cm} .\) The distance between the lenses is \(26.0 \mathrm{~cm}\). (a) What is the angular magnification of the microscope if the person using it has a near point of \(35.0 \mathrm{~cm} ?\) (b) If, as usual, the first image lies just inside the focal point of the eyepiece (see Figure \(26-33\) ), how far is the object from the objective? (c) What is the magnification (not the angular magnification) of the objective?

Step-by-Step Solution

Answer

(a) 39.94, (b) 4.25 cm, (c) 4.59.

Step 1: Formula for Angular Magnification of the Microscope

The angular magnification \(M\) of a compound microscope is given by the formula:\[ M = M_o \times M_e \]where \(M_o\) is the magnification of the objective, and \(M_e\) is the magnification of the eyepiece. Remember, for a final virtual image at the near point, \(M_e = \frac{N}{f_e} + 1\), where \(f_e\) is the focal length of the eyepiece and \(N\) is the near point of the person using the microscope.

Step 2: Calculate Eyepiece Magnification

Given: \(N = 35 \mathrm{~cm}\) and \(f_e = 6.5 \mathrm{~cm}\).Substitute these values into the eyepiece magnification formula:\[ M_e = \frac{35 \mathrm{~cm}}{6.5 \mathrm{~cm}} + 1 = 5.38 \]

Step 3: Calculate Magnification of the Objective

The magnification for the objective \(M_o\) is given by:\[ M_o = \frac{L}{f_o} \]where \(L\) is the tube length and \(f_o = 3.5 \mathrm{~cm}\) is the focal length of the objective. Substitute these, \(L = 26.0 \mathrm{~cm}\):\[ M_o = \frac{26.0}{3.5} \approx 7.43 \]

Step 4: Calculate Angular Magnification

Now calculate the angular magnification:\[ M = M_o \times M_e = 7.43 \times 5.38 \approx 39.94 \]

Step 5: Distance from Object to Objective

For this step, use the lens formula:\[ \frac{1}{f_o} = \frac{1}{v_o} + \frac{1}{u} \]where \(v_o\) is the image distance (nearly equal to \(L - f_e\), because image forms just inside the focal point of the eyepiece) and \(u\) is the object distance. Substitute the values:\[ \frac{1}{3.5} = \frac{1}{26 - 6.5} + \frac{1}{u} \]\[ \frac{1}{3.5} = \frac{1}{19.5} + \frac{1}{u} \]\[ \frac{1}{u} = \frac{1}{3.5} - \frac{1}{19.5} \]Solving for \(u\), we find \(u \approx 4.25 \mathrm{~cm} \).

Step 6: Calculate Objective Magnification

The magnification of the objective \(m_o\) is given by:\[ m_o = \frac{v_o}{u} \]Substitute the values \(v_o \approx 19.5 \mathrm{~cm}\) and \(u = 4.25 \mathrm{~cm}\):\[ m_o = \frac{19.5}{4.25} \approx 4.59 \]

Key Concepts

Angular Magnification

When we talk about the angular magnification of a compound microscope, we're referring to how much the microscope enlarges the angle under which we view the image of the object compared to the naked eye. This magnification allows one to navigate the microscopic world with greater clarity.
The total angular magnification of the microscope is the product of the magnifications provided by both the objective lens and the eyepiece. Mathematically, it’s expressed as:

• Formula: \( M = M_o \times M_e \)
• Where, \( M_o \) is the magnification of the objective lens
• \( M_e \) is the magnification of the eyepiece

Angular magnification is crucial because it forms the bridge between simply making an image larger and making it possible to study small details that are otherwise not visible.

Focal Length

The focal length of a lens is the distance between the lens and its focus point. In a compound microscope, there are two important focal lengths:

• Objective focal length (f_o)
• Eyepiece focal length (f_e)

The focal length of the objective lens determines how much the lens can magnify and what its resolving power is. Shorter focal lengths mean higher magnification.
For a clearer understanding in this context:

• The objective lens in our example has a focal length of \(3.50\; \mathrm{~cm}\)
• The eyepiece lens has a focal length of \(6.50\; \mathrm{~cm}\)

These values show how the microscope is set up to start the magnification process strongly with the objective, while also playing a role in the overall resolving power of the device.

Objective Lens

The objective lens is the first and perhaps the most influential lens in a compound microscope. It's responsible for the initial magnification of the specimen. The objective gathers light from the specimen and creates a real image inside the microscope.
Crucial points about the objective lens include:

• It typically has a short focal length, leading to more significant magnification.
• In the given example, the focal length is \(3.50\, \mathrm{~cm}\).
• The magnification provided by the objective is calculated by: \( \frac{L}{f_o} \), where \(L\) is the distance between the lenses.
• This lens must precisely adjust to observe detailed features and ensure accurate focus.

The construction and quality of the objective lens significantly influence the overall performance of a microscope.

Eyepiece

The eyepiece, also known as the ocular lens, is the second key component in a compound microscope, through which the observer views the magnified image. It further magnifies the image produced by the objective lens.
Key aspects of the eyepiece include:

• It has a longer focal length compared to the objective lens to moderately magnify the already enlarged image.
• The eyepiece examined here has a focal length of \(6.50\, \mathrm{~cm}\).
• The formula for calculating eyepiece magnification considers the user's near point \(N\) and is expressed as: \(M_e = \frac{N}{f_e} + 1\).

In essence, the eyepiece works together with the objective lens to provide the overall magnification and is essential for delivering a clear and focused large-scale view of the image.

Magnification

Magnification in the realm of a compound microscope is about making small objects appear large enough to be studied in detail. The job of the microscope's lenses, both objective and eyepiece, is to enhance this tiny image into a viewable size.
Key points about magnification:

• The overall magnification is the combination of the objective and eyepiece magnifications.
• Objective magnification is determined by the formula: \( \frac{L}{f_o} \).
• Eyepiece magnification is computed with: \( \frac{N}{f_e} + 1\).
• The total magnification is a multiplication of the two values.

Understanding magnification helps users know what level of detail they can expect from their microscope and how closely they can view the specimens they're studying.