An astronomical telescope is to be designed to have a magnifying power of 50 in normal adjustment ,If the length of tube is 102cm,Find the power of objective.Find the power of the objective and the eyepiece

To determine the power of the objective and the eyepiece for an astronomical telescope designed to have a magnifying power of 50 with a tube length of 102 cm, we can use some fundamental formulas related to telescopes. The magnifying power (M) of a telescope in normal adjustment is given by the ratio of the focal lengths of the objective (f₀) and the eyepiece (fₑ):

Understanding the Relationships

The magnifying power can be expressed as:

M = f₀ / fₑ

We also know that the length of the telescope tube (L) is approximately equal to the sum of the focal lengths of the objective and the eyepiece:

L = f₀ + fₑ

Setting Up the Equations

Given that the magnifying power (M) is 50 and the length of the tube (L) is 102 cm, we can set up the following equations:

• Equation 1: f₀ / fₑ = 50
• Equation 2: f₀ + fₑ = 102

Solving the Equations

From Equation 1, we can express f₀ in terms of fₑ:

f₀ = 50 * fₑ

Now, substitute this expression into Equation 2:

50 * fₑ + fₑ = 102

This simplifies to:

51 * fₑ = 102

Now, divide both sides by 51:

fₑ = 102 / 51 = 2 cm

Finding the Focal Length of the Objective

Now that we have the focal length of the eyepiece, we can find the focal length of the objective using our earlier expression:

f₀ = 50 * fₑ = 50 * 2 = 100 cm

Calculating the Powers

The power of a lens is defined as the reciprocal of its focal length in meters:

Power (P) = 1 / f (in meters)

Now, let's calculate the powers of both the objective and the eyepiece:

• Power of the Objective: P₀ = 1 / (100 cm) = 1 / (1 m) = 1 D (diopter)
• Power of the Eyepiece: Pₑ = 1 / (2 cm) = 1 / (0.02 m) = 50 D (diopters)

Final Results

To summarize:

• The power of the objective lens is 1 diopter.
• The power of the eyepiece lens is 50 diopters.

This design allows the telescope to achieve the desired magnifying power of 50 while maintaining a tube length of 102 cm. If you have any further questions or need clarification on any of the steps, feel free to ask!