An optical instrument uses a 25 D objective and 20 D eyepeice with a tube length of 25 cm when eye is least strained

Options

A. (a)The instrument is a telescope with angular magnification 20.
B. (b)The instrument is a microscope with angular magnification 20.
C. (c)The instrument is a telescope with angular magnification 24.
D. (d)The instrument is a microscope with angular magnification 24.

Answer

To solve the problem, we need to determine whether the optical instrument described is a microscope or a telescope based on the given parameters. We will also calculate the angular magnification of the instrument.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Objective power (P₀) = 25 D
- Eyepiece power (Pₑ) = 20 D
- Tube length (L) = 25 cm

2. **Calculate the Focal Lengths:**
- The focal length (f) in meters can be calculated from the power (P) using the formula:
\[ f = \frac{1}{P} \]
- For the objective:
\[ f₀ = \frac{1}{25} = 0.04 \text{ m} = 4 \text{ cm} \]
- For the eyepiece:
\[ fₑ = \frac{1}{20} = 0.05 \text{ m} = 5 \text{ cm} \]

3. **Determine the Image Distance for the Objective:**
- The image formed by the objective lens (V₀) is at the focal length of the eyepiece (focal length of eyepiece is 5 cm). Since the tube length is 25 cm, the distance from the objective to the image formed (V₀) is:
\[ V₀ = L - fₑ = 25 \text{ cm} - 5 \text{ cm} = 20 \text{ cm} \]

4. **Calculate the Object Distance for the Objective:**
- Using the lens formula:
\[ \frac{1}{f₀} = \frac{1}{V₀} + \frac{1}{U₀} \]
- Rearranging gives:
\[ \frac{1}{U₀} = \frac{1}{f₀} - \frac{1}{V₀} \]
- Substituting the values:
\[ \frac{1}{U₀} = \frac{1}{4} - \frac{1}{20} \]
- Finding a common denominator (20):
\[ \frac{1}{U₀} = \frac{5}{20} - \frac{1}{20} = \frac{4}{20} = \frac{1}{5} \]
- Therefore:
\[ U₀ = 5 \text{ cm} \]

5. **Calculate the Angular Magnification (M):**
- The angular magnification (M) of the instrument can be calculated using the formula:
\[ M = -\frac{V₀}{U₀} \cdot \frac{L}{fₑ} \]
- Substituting the values:
\[ M = -\left(\frac{20}{5}\right) \cdot \frac{25}{5} = -4 \cdot 5 = -20 \]
- The negative sign indicates that the image is inverted.

6. **Determine the Type of Optical Instrument:**
- Since the focal length of the objective (f₀ = 4 cm) is less than the focal length of the eyepiece (fₑ = 5 cm), this indicates that the instrument is a microscope.

### Conclusion:
The angular magnification of the optical instrument is 20, and it is classified as a microscope.