An equiconcave lens has power P. Find power of plano concave lens when given lens is cut in such a way that two plano concave lens are formed.

Options

A. `P`
B. `P/2`
C. `2P`
D. `P/4`

Answer

To solve the problem, we need to find the power of a plano-concave lens formed by cutting an equiconcave lens with power P into two equal plano-concave lenses. Let's go through the solution step by step.

### Step 1: Understand the Given Lens
We start with an equiconcave lens that has a power \( P \). The power of a lens is given by the formula:
\[ P = \frac{1}{f} \]
where \( f \) is the focal length of the lens.

### Step 2: Determine the Focal Length of the Equiconcave Lens
Since the power of the equiconcave lens is \( P \), we can express the focal length \( f \) as:
\[ f = \frac{1}{P} \]

### Step 3: Cutting the Lens
When the equiconcave lens is cut into two equal plano-concave lenses, each lens will have one flat surface (the plano side) and one concave surface.

### Step 4: Focal Length of the Plano-Concave Lens
For a plano-concave lens, the focal length can be derived using the lens maker's formula. The formula for the focal length \( f' \) of a plano-concave lens 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 concave side (which is negative),
- \( R_2 \) is the radius of curvature of the flat side (which is infinite).

Assuming the radius of curvature of the concave surface is \( r \), we have:
- \( R_1 = -r \)
- \( R_2 = \infty \)

Substituting these values into the lens maker's formula gives:
\[ \frac{1}{f'} = (n - 1) \left( \frac{1}{-r} - 0 \right) = -\frac{(n - 1)}{r} \]
Thus, the focal length of the plano-concave lens is:
\[ f' = -\frac{r}{(n - 1)} \]

### Step 5: Relating the Focal Lengths
Since the original equiconcave lens has a focal length \( f = -\frac{r}{2(n - 1)} \) (because it is a double concave lens), we can relate the focal lengths:
\[ f = 2f' \]
This means:
\[ -\frac{r}{2(n - 1)} = 2 \left(-\frac{r}{(n - 1)}\right) \]

### Step 6: Finding the Power of the Plano-Concave Lens
Now, using the relationship between power and focal length:
\[ P' = \frac{1}{f'} \]
From the previous steps, we know:
\[ f' = \frac{f}{2} \]
Thus, the power of the plano-concave lens \( P' \) is:
\[ P' = \frac{1}{f'} = \frac{2}{f} = \frac{2}{\frac{1}{P}} = \frac{2P}{1} = \frac{P}{2} \]

### Final Answer
The power of each plano-concave lens formed is:
\[ P' = \frac{P}{2} \]