A section of a spherical shell of outer radius `R_(0)` and inner radius `R_(1)` is cut, and used as a lens. The light first strikes what was originally the outer surface of the glass shell. Which of the following statements about the resulting lens is true ?

Options

A. The lens is converging
B. The lens is diverging
C. The lens has zero focal length
D. The lens has infinite focal length

Answer

To solve the problem regarding the lens formed by a section of a spherical shell with outer radius \( R_0 \) and inner radius \( R_1 \), we can follow these steps:

### Step 1: Understand the Geometry of the Lens
The lens is formed by cutting a section of a spherical shell. The outer surface of the shell has a radius \( R_0 \) and the inner surface has a radius \( R_1 \). When light strikes the outer surface first, it will refract according to the lens maker's formula.

**Hint:** Visualize the lens by sketching the spherical shell and marking the outer and inner radii.

### Step 2: Apply the Lens Maker's Formula
The lens maker's formula is given by:
\[ \frac{1}{F} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where \( n \) is the refractive index of the lens material, \( R_1 \) is the radius of curvature of the first surface (outer surface), and \( R_2 \) is the radius of curvature of the second surface (inner surface).

**Hint:** Identify which radius corresponds to which surface based on the direction of light travel.

### Step 3: Determine the Sign of the Radii
For the outer surface (where light first strikes), the radius \( R_1 \) is positive because it is convex. For the inner surface, \( R_2 \) is negative because it is concave when viewed from the direction of incoming light.

**Hint:** Remember the sign convention: radii are positive for convex surfaces and negative for concave surfaces when following the direction of light.

### Step 4: Substitute the Values into the Formula
Using the sign convention:
- \( R_1 = R_0 \) (positive)
- \( R_2 = -R_1 \) (negative)

Substituting these values into the lens maker's formula gives:
\[ \frac{1}{F} = (n - 1) \left( \frac{1}{R_0} - \left(-\frac{1}{R_1}\right) \right) = (n - 1) \left( \frac{1}{R_0} + \frac{1}{R_1} \right) \]

**Hint:** Ensure you keep track of the signs while substituting values into the formula.

### Step 5: Analyze the Resulting Lens Type
Since both terms \( \frac{1}{R_0} \) and \( \frac{1}{R_1} \) are positive, the overall result of \( \frac{1}{F} \) will also be positive. This indicates that the lens is converging.

**Hint:** A positive focal length indicates a converging lens, while a negative focal length indicates a diverging lens.

### Conclusion
The resulting lens formed by the section of the spherical shell is a converging lens.

**Final Statement:** The correct statement about the resulting lens is that it acts as a converging lens.