2 One end of a long glass rod ( µ=1.5) is formed into the shape of a convex surface of radius 6.0 cm. An object is located in air along the axis of the rod, at distance of 10 cm from the end of the rod. Find

(a) How far apart the object and image formed by the glass rod?

(b) For what range of distance from the end of the rod must the object be located in order to produce a virtual image?

To tackle this problem, we need to apply the principles of optics, specifically the lens maker's formula and the thin lens equation. The glass rod behaves like a lens due to its curved surface, and we can analyze the situation step by step.

Understanding the Setup

We have a glass rod with a refractive index (μ) of 1.5, and one end is shaped into a convex surface with a radius of curvature (R) of 6.0 cm. An object is placed in air, 10 cm away from the convex end of the rod. The goal is to find the distance between the object and the image formed by the rod, as well as the range of distances for which a virtual image can be produced.

Step 1: Determine the Focal Length

For a lens, the focal length (f) can be calculated using the formula:

• f = R / (μ - 1)

Substituting the values:

• R = 6.0 cm
• μ = 1.5

Now, we can calculate:

• f = 6.0 cm / (1.5 - 1) = 6.0 cm / 0.5 = 12.0 cm

Step 2: Use the Thin Lens Equation

The thin lens equation relates the object distance (u), image distance (v), and focal length (f):

• 1/f = 1/v - 1/u

Here, the object distance (u) is negative because we follow the sign convention (object on the same side as incoming light is negative). Thus, u = -10 cm. Now we can substitute the values into the equation:

• 1/12 = 1/v - 1/(-10)

Rearranging gives:

• 1/v = 1/12 + 1/10

Finding a common denominator (60):

• 1/v = 5/60 + 6/60 = 11/60

Now, taking the reciprocal:

• v = 60/11 ≈ 5.45 cm

Step 3: Calculate the Distance Between Object and Image

The distance between the object and the image can be found by adding the absolute values of u and v:

• Distance = |u| + |v| = 10 cm + 5.45 cm = 15.45 cm

Finding the Range for Virtual Images

For a virtual image to be formed, the object must be within the focal length of the lens. Since we have calculated the focal length to be 12.0 cm, the object must be placed closer than this distance from the lens. Therefore:

• Object distance (u) must be greater than -12 cm (in terms of the sign convention).

This means the object must be located between 0 cm (at the lens) and -12 cm (inside the focal length). Thus, the range for producing a virtual image is:

• -12 cm < u < 0 cm

Summary of Results

In summary:

• (a) The distance between the object and the image is approximately 15.45 cm.
• (b) The object must be located between 0 cm and -12 cm from the end of the rod to produce a virtual image.

This analysis demonstrates how the principles of optics can be applied to understand the behavior of light through a curved medium. If you have any further questions or need clarification on any point, feel free to ask!