A point object is placed in air at a distance of 40 cm from a concave refracting surface of refractive index 1.5. If the radius of curvature of the surface is 20 cm, then the position of the image is –

Options

A. in air and at 30 cm from pole
B. in refracting medium and at 30 cm from pole
C. in air and at infinity
D. in refracting medium and at infinity

Answer

To solve the problem of finding the position of the image formed by a concave refracting surface, we will use the formula for refraction at a spherical surface. Here’s a step-by-step solution:

### Step 1: Identify the Given Values
- Object distance (u) = -40 cm (the negative sign indicates that the object is on the same side as the incoming light)
- Refractive index of the medium (n1) = 1 (for air)
- Refractive index of the concave surface (n2) = 1.5
- Radius of curvature (R) = -20 cm (the negative sign indicates that it is a concave surface)

### Step 2: Write the Refraction Formula
The formula for refraction at a spherical surface is given by:
\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \]
Where:
- \( v \) = image distance
- \( u \) = object distance
- \( n_1 \) = refractive index of the first medium (air)
- \( n_2 \) = refractive index of the second medium (concave surface)
- \( R \) = radius of curvature

### Step 3: Substitute the Known Values into the Formula
Substituting the known values into the formula:
\[ \frac{1.5}{v} - \frac{1}{-40} = \frac{1.5 - 1}{-20} \]
This simplifies to:
\[ \frac{1.5}{v} + \frac{1}{40} = \frac{0.5}{-20} \]

### Step 4: Simplify the Equation
Calculating the right side:
\[ \frac{0.5}{-20} = -\frac{1}{40} \]
So the equation becomes:
\[ \frac{1.5}{v} + \frac{1}{40} = -\frac{1}{40} \]

### Step 5: Combine the Terms
Now, we can combine the terms:
\[ \frac{1.5}{v} = -\frac{1}{40} - \frac{1}{40} = -\frac{2}{40} = -\frac{1}{20} \]

### Step 6: Solve for v
Now we can solve for \( v \):
\[ \frac{1.5}{v} = -\frac{1}{20} \]
Cross-multiplying gives:
\[ 1.5 \cdot 20 = -v \implies v = -30 \text{ cm} \]

### Step 7: Interpret the Result
The negative sign indicates that the image is formed on the same side as the object, which is consistent with the properties of a concave surface. Thus, the image is located 30 cm from the pole of the surface.

### Final Answer
The position of the image is at a distance of 30 cm from the pole of the concave surface.

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